7 eng.pdfcontents introduction 5 preliminaries 9 § 1. kegular operators and their products 11 § 2....

I N S T Y T U T MATEMATYCZNY P O L S K I E J A K A D E M I I NAUK

DISSERTATIONES MATHEMATICAE

ROZPRAWY MATEMATYCZNE

K O M I T E T R E D A K C Y J N Y

KAROL BORSUK redaktor BOGDAN BOJARSKI, ANDRZEJ MOSTOWSKI,

MARCELI STARK, STANISŁAW TURSKI

LVIII

A. PEŁCZYtfSKI

Linear extensions, linear averagings, and their applications to linear topological classification

of spaces of continuous functions

W A R S Z A W A 1968 P A Ń S T W O W E W Y D A W N I C T W O N A U K O W E

COPYRIGHT 1968 by

P A Ń S T W O W E W Y D A W N I C T W O N A U K O W E WARSZAWA ( P O L A N D ) , ul. M i o d o w a 10

All Eights Keserved

No part of this book may be translated or reproduced in any form, by mimeograph or any other means, without permission in writing from the publishers.

P R I N T E D I N P O L A N D

W R O C Ł A W S K A D R U K A R N I A N A U K O W A

CONTENTS

Introduction 5

Preliminaries 9

§ 1. Kegular operators and their products 11

§ 2. Exaves. Extension and averaging operators 15

§ 3. Linear multiplicative exaves and retractions. Localization principle . . . 21

§ 4. Integral representations and compositions of linear exaves 22

§ 5. Milutin spaces 27

§ 6. Dugundji spaces 34

§ 7. Exaves and topological groups 37

§ 8. Application to linear topological classification of spaces of continuous func

tions 40

§ 9. Linear averaging operators and projections onto spaces of continuous

functions 47

Notes and Remarks 59

Appendix: Category-theoretical approach 75

Bibliography 80

INTRODUCTION

Initially this paper (*) was to be an exposition of the following result due to Milutin (cf. Milutin [1], [2]).

If Sx and $2 are uncountable compact metric spaces, then the spaces C(Si) and C(S2) are linearly homeomorphic (*).

This result settles a question raised by Banach ([1], p. 185): Are the spaces of all continuous scalar-valued functions on the unit interval and on the unit square linearly homeomorphic f

Milutin's method is based on a very clever construction (cf. Lemma 5.5 in the present paper) of a map W from the Cantor set ^ onto the unit interval I such that there exists a projection n of norm one from C(^) onto its subspace consisting of all composed functions/ = goWioY geC{I).

Then the problem can be reduced via the Borsuk-Dugundji theorem on linear extensions to the standard decomposition method.

The projection n is an example of the Birkhoff's averaging operator (cf. Birkhoff [1]).

Subsequently I discovered that averaging operators and extension operators admit a common generalization to the operators which will be called in the sequel "linear exaves". This paper is devoted to a development of the theory of linear exaves acting between spaces of continuous functions on compact Hausdorff spaces, together with some applications including Milutin's result.

We define linear exaves as follows. If y: S -> T is a continuous map (8, T — compact), then y°: C(T) -> C(S) denotes the induced operator defined by y°{g) = #oc> for geC{T). A linear operator u: C{S) ->C(T) is called a linear exave if (p°U(p° = (p0. That condition is always satisfied whenever u is either left or right inverse for <p°. In the first case u is in fact Birkhoff's averaging operator, in the second u is a linear extension operator.

Indeed if uq>° = idc ( r ) (where idx denotes the identity on X), then G(T) may be identified with <p°[C(T)]f the subalgebra of G(S) consisting of all functions which are constant on each set q>~1 (t) for t eT. Hence A = utp° is a projection from G(S) onto (p°[C(T)]. If this projection is positive

(*) This research was supported in part by National Science Foundation Grant (GP-3579) and the Division of Sponsored Research, Florida State University.

(*) For the terminology and notation see "Preliminaries".

6 Linear extensions, linear averagings

(i.e. takes non-negative functions into non-negative functions), then u is of the form

(uf)(t) = j (p{s)jut(ds) for teT and for fcC(S), <p-\t)

where fit is a probability measure concentrated on y~ 1(t) for teT. Now it is easily seen that A satisfies Birkhoff's averaging condition, A[f1-A(f2)) = Afl-Af, ioTf19ft in C(8).

If <p°u = ido(^), then c>° maps C(T) onto C{S) and therefore cp is one-to-one. The compactness of S and T implies then that <p is a homeo-morphic embedding of S into T. Identifying S with its homeomorphic image <pS in T it is natural to regard 99 ° as the operator of restriction of function on T to the function on <pS. Then the condition q>°u =* idc^s) means nothing else but the fact that for each / in C(S), uf is an extension of / .

In the present paper we are mainly concerned with the following three types of exaves (a) linear exaves, (b) regular exaves = linear exaves which are regular operators (cf. § 1 for definition), (c) linear-multiplicative exaves = linear exaves which are linear-multiplicative operators.

Roughly speaking, the existence of an extension operator (resp. averaging operator) from C(8) into C(T) of type either (a) or (b) is closely related to the existence of a bounded linear projection from C(T) onto its.subspace isometric to C(S) (resp. from C(S) onto its subspace isometric to C{T)), while the existence of linear-multiplicative extension (resp. averaging) is equivalent to the existence of certain retraction from T onto S (resp. from S onto T).

The advantage of regular operators is that they admit some infinite operations (cf. § 1). This enables us to reduce some general theorems on existence of certain exaves to the assertion of the existence of a very particular regular exaves (like Milutin Lemma). On the other hand the non-existence theorems seem to be properly stated for general linear exaves.

Summary of the contents. § 1 is devoted to study regular operators* The formal part of the theory is developed in § 2-§ 4. In § 5 and § 6 we study two classes of compact spaces: Milutin spaces and Dugundji spaces. The first are continuous images of a generalized Cantor set ( = product of two point spaces) by an epimorphism admitting linear averaging operator. The second are those which admit an embedding into Tichonov cube such that there exists a linear extension operator. Both classes include compact metric spaces, their products and are, in a certain sense, "closer" to metric spaces than other compact spaces. The proof that each metric space is a Milutin space is rather complicated. A key tool is Lemma 5.5 (the Milutin Lemma).

Introduction 7

In § 7 we study linear exaves commuting with certain group of mappings. We conclude this paragraph with a proof that every compact topological group is a Milutin space (an improvement of earlier results of Ivanovskii [1] and Kuzminov [1]).

§ 8 and § 9 are devoted to applications. In § 8 we prove Milutin's result (Theorem 8.3). This, together with some results of C. Bessaga and the author (cf. Bessaga and Pełczyński [1]), enables us to get the complete linear topological classification of spaces C{S) for compact metric S. In the non-metric case we extend Milutin's result to the case of product of compact metric spaces and of arbitrary compact groups as well. It is shown that if G is a compact topological group, then the linear topological type of C{G) is entirely determined by the topological weight of G (Theorem 8.9).

§ 9 is mainly devoted to the study of a class of epimorphisms of order two (i.e. such that the inverse image of each point has at most two points) which have no linear averaging operator. This enables us to obtain some results of Amir [1], [2].

In the Notes and Bemarks we discuss some counterexamples and state some open questions. We also discuss the relationships of some of our results and methods to those in literature as well as make some general comments about the references. In the Bibliography we have attempted to include a complete list of references on the subject of extension of continuous functions and related topics.

In this paper we restrict ourselves to spaces C(8) with S compact though extension and averaging operators also appear in different contexts, e.g. in differential equations (cf. Adams, Aronszajn and Smith [1]) and in the group representation theory (if J? is a subgroup of a locally compact group G, then jf{xy)dy yields an averaging operator from C{G) onto

H C(G/H) (cf. § 7 for details)). In the Appendix we try to describe a "general concept of an exave" in terms of the theory of categories and we enlist some important models. Also in the Notes and Bemarks the reader can find some information concerning various types of exaves.

Acknowledgments. This work would have been almost impossible without the help and general support of my friends. I am very much indebted to M. I. Kadec from whom I learned Milutin's Lemma, and a discussion with him in 1962 gave me the main idea of Proposition 9.11. I wish to express my gratitude to H. H. Corson and E. Michael for several valuable remarks and for permission to include some of their unpublished results in this paper. I am also indebted to C. Bessaga, B. Engelking, B. B. Phelps and Z. Semadeni for several conversations. I would like to thank V. L. Klee and O. G. Harrold for arranging my stay at the University of Washington and at the Florida State University where I found


extremely good working condition, and to S. Mazur for arranging my leave of absence from the University of Warsaw. Finally I am indebted to Mrs. Eachel Jones and especially to Mrs. Elizabeth Kurrle for their excellent typing jobs,

UNIVERSITY OF WARSAW U N I V E R S I T Y OF WASHINGTON T H E F L O R I D A STATE U N I V E R S I T Y

PRELIMINARIES

Topological spaces. Throughout this paper by "topological space" we mean "topological Hausdorff space". A map q>: 8 ~> T means a continuous transformation from 8 into T. If q>: 8 -> Q and ip:Q -> T are maps, then the composition w — yxp: 8 -> T (sometimes we shall write ipoy) is the map defined by co(s) = y>(<p(s)) for seS. A map <p: 8 -> T is called an epimorphism (resp. a homeomorphic embedding) if y8 = T (if <p regarded as a map from S onto cpS is a homeomorphism). The identity map on 8 will be denoted by ids.

If & is a topological space, then the topological weight, in shorter form, the weight, of S is the smallest cardinal number m such that there exists a base for the topology of 8 (see Kelley [2], p. 46) of power m.

If (8a)aeA is a family of spaces, then P 8a, in short PSa, denotes aeA

the Cartesian product of the opac^s #a (i.e. the set of all functions s = (sa)aeA

on the coordinate set A such that sae8a for aeA) endowed with the weakest topology in which the natural projections

pb:P8a->Sb defined by pb((sa)) = $b

are continuous functions for all be A. The cardinality of A will be denoted by A. The Cartesian product of two spaces S and T will be denoted by 8 x T. If m is a cardinal number, then 8m will denote the Cartesian product of m copies of 8.

If (8a)aeA and (Ta)aeA are two families of topological spaces and {<pa> ®a -> Ta)aeA is a family of maps, then the product map

P<pa:P8a^PTa

is the unique map with the property that the diagram

P<fa PSa +~PTn

Pb Qb

Sh

<pb -*»Tb

commutes for each be A, where pb and qb denote the natural projections onto 8b and onto Tb respectively.

In the sequel the capital letters S, T,Q will be reserved for denoting compact spaces. The following symbols will be used for denoting special compact spaces:

I = [0 ,1] — the closed unit interval. J) = {0} w {1} — the two-point discrete space. [a] — the space of all ordinals < a endowed with the usual order

topology (cf. Kelley [2], pp. 57, 266-271). Im — the Cartesian product of m copies (m is a cardinal number)

of I, the Tichonov cube of the topological weight m; Dm — the Cartesian product of m copies of two-point spaces, the

generalized Cantor set of power m. A general point of Dm will be denoted by | = (Ca)aeA) where | a = 0 or 1, aeA, and A is an arbitrary set of indices of the power A = m.

<% = x>so _ the ordinary Cantor set. In this case we choose as A the set of all positive integers. A general point of ^ is a sequence £ = (£*)?! i where & = 0 or 1.

Banach spaces. In the sequel X, Y,Z and E will denote Banach spaces unless otherwise specified. If X is a Banach space, then X* denotes the dual (conjugate) space to X. The elements of X* are linear functionals on X and will be denoted by x*, y*, z*, ... The weak-star topology on X* is the weakest topology on X* such that for each xeX the function x (x*) = x* x is continuous on X*. The symbols u, v, w are reserved for denoting linear operators. Linear operators are assumed to be bounded, therefore continuous. If u: X -> Y is a linear operator, then u* is the adjoint operator of u, i.e., u* is the linear operator from Y* into X* defined by (u*y*){x) = y*ux for xeX and y*e Y*.

A linear operator u: X ^ Y is called an epimorphism if uX = Y, a linear homeomorphism if it is an epimorphism and is one-to-one and has a bounded inverse (isomorphism in the terminology of Banach [1]), a linear homeomorphie embedding if it is a linear homeomorphism onto a subspace of Y, isometric embedding if it is one-to-one and \\ux\\ = \\x\\ for xeX, an isometry if it is an isometric embedding and an epimorphism.

A Banach space X is said to be linearly homeomorphie (isometric) to a Banach space Y if there is a linear homeomorphism (an isometry) w.X-^ Y.

The Cartesian product I x Yof Banach spaces X and Y is the space of all pairs (x,y), xeX, y*Y, with the usual operations of addition and multiplication by scalars; we admit \$x, y)\\ = max(||a?||, \\y\$.

If E is a subspace of X, then XjE denotes the quotient space X by E', the elements of XjE are cosets modulo E, i.e. the sets [x] = {ztX: z = x-\- e

Preliminaries 11

for some eeE}. The norm in X/E is defined by ||[a?]|| = inf ||a? + c||. The eeE

epimorphism x -> [x] is called the quotient map. A subspace E of a Banach space X is said to be complemented in X if there exists a projection u ( = a bounded linear idempotent, i.e. u2 — u) from X onto E. Let us recall that E is complemented in X if and only if there exists a linear homeomorphism v from X onto the Cartesian product X/Ex E such that ve = (0, e) for eei£.

Any unexplained terminology and notation will be that of Dunford-Schwartz [1].

Spaces of continuous functions and measure spaces. If $ is a compact space, then C(8) (respectively CR(8)) denotes the Banach space of all continuous complex (respectively real) valued functions on 8 with the norm ||/|| = sup|/(s)|, and M(S) denotes the space of all complex finite

SeS

regular Borel measures on 8 with the norm HH = the total variation of /u on 8. According to the Eiesz representation theorem (Dunford-Schwartz [1], p. 265) we identify M(S) with the space dual to C(S). We shall employ the notation /*(/) = ff(s)[t{ds) for [i in M{8) and feC{S). A ^ in M{8)

s is said to be non-negative provided fz{f) > 0 whenever / is non-negative, i.e. f(s) ^ 0 for 8eS. The set of all non-negative measures in M(8) is called the positive cone of M(S). If \\/x\\ = 1, then JU, is called normalized. A measure pi in M{8) is said to be concentrated on a (closed) subset F of 8 provided ff(s)/j,(ds) = ff(s)fj,(ds) for every / in C{8). Por s in 8 we

F s denote by <5S the unit point mass at the point s, i.e. ds(f) =f(s) for feC{8). By ls we denote the function on 8 which is identically one. If / and g are in C(8), then / > g means that f(s) > g(s) for all SeS.

If <p:8-^*T is a map, then cp°: C(T) -> C(8) denotes the induced operator defined by q>°(f) = fo<p for feC(T). The induced operator <p° is linear and multiplicative; it is an isometric embedding (an epimorphism) if and only if C(T) is said to be regular provided ||w|| = 1 and uls = ly.

1.2. PROPOSITION. For every linear operator u: C{S) -> C(T) the following conditions are equivalent:

(1.2.1) u is regular.

uls — IT and uf > 0 whenever / > 0.

The adjoint operator u* maps isometrically the positive cone of M(T) into the positive cone of M(S).

For t in T and for f in C(8)

s

where t -> fit — u*dt is a continuous function from T into M(S) endowed with the weak-star topology, and for each t in T, jut is a positive normalized measure.

Proof. (1.2.1) => (1.2.2). Let 0^feC{S) and let / > 0. Then for every e with ||/||_1 ^ e > 0 we have ps—e/| | < 1. The regularity of u implies that \{uls—uf){t)\ =* \l-e{uf){t)\ < 1 for t in T and for \\f\\~x

^ e > 0. Clearly the last inequality implies that (uf)(t)^0 for teT. Hence uf > 0 whenever / ^ 0.

(1.2.2) => (1.2.3). Let veM(T) and let v > 0. Then since the condition / > 0 implies uf ^ 0, we have

(«%)(/) = v(uf) > 0 for 0 </eC(fl) .

Hence w% > 0. Finally, for the non-negative measures v and w*V we have

||i*Ml = (u*v)(ls) = v(uls) = *(l r) = |M|.

This shows that w maps isometrically the positive cone of M(T) into the positive cone of M(S).

(1.2.3) => (1.2.4). Let us put pt = u*dt for teT. Since dt > 0 and II H = 1, condition (1.2.3) implies that [xt is a positive normalized measure in M(8). Clearly for feC(S) and teT we have (uf)(t) = dt(uf) — (u*dt)(f) = /it(f) = f f(s)ftt(ds). Finally, since t -> dt is a continuous

s function from T into M(T) equipped with the weak-star topology and since the adjoint operator u*: M(T) -» M{8) is continuous if both spaces M{T) and M(S) carry their weak-star topologies, the composed function t -> nt is also weak-star continuous.

(1.2.4) => (1.2.1). Since fit is a continuous function on t in the weak-star topology of M(S), for every feC(S) the function (it(f) is continuous on T. Hence the formula (uf)(t) = ff(s)/it(ds) for feC(S) and teT defines

an operator from C(S) into G{T). Clearly u is linear and ||«|| ||ttl«|| = 1. Hence ||w|| = 1. That completes the proof.

Regular operators and their products 13

1.3. PROPOSITION. Let (Sa)aeA and (Ta)atA be families of compact spaces. Let ua: C(Sa)-> C(Ta) be regular linear operators (aeA). Then there is the unique regular linear operator

u= ®ua:C(PSa)-+C(PTa)

such that for arbitrary finite subset B a A

(1.3.1) U ([Jplfa) =[]q0a(Uafa) (fa*C(Sa); aeB) aeB acB

where pa and qa denote natural projections from PSa onto Sa and from PTa

onto Ta respectively (ffga denotes the ordinary product of functions ga (aeB)). aeB

Proof. By (1.2.4) for every ta in Ta (aeA) there is a positive normalized measure fit in M(Sa) such that

(UafWa) = ff(Sa)f*la(dsa) for fcC(Sa).

Let us set ju{ta) = <g> pt for (ta)ePTa. Clearly the product measure fi{ta) aeA

s a positive normalized measure in M(PSa). i Let us set

(1.3.2) uf((ta))= ffW/HtJM) tor feC(PSa) and for (ta)ePTa. PSa

Clearly u (defined by (1.3.2)) is a linear operator from C(PSa) into the space B(PTa) of all bounded complex valued functions on PTa with the usual sup-norm. Moreover, \\u\\ = sup \\u(t J| = 1. It follows from

Fubini's Theorem and from the well-known properties of product measures that the operator u satisfies (1.3.1). Hence in particular ufeC(PTa) for each function feC(PSa) of the form

(1.3.3) f = [Jplfa {fa*C(Sa); aeB; B is a finite subset of A). aeB

Let C0 (PSa) denote the smallest linear manifold spanned by the functions of the form (1.3.3). By the linearity of u, ufis a continuous function on PTa

for every feC0(PSa). Since (by the Stone-Weierstrass Theorem) C0(PSa) is dense in C(PSa) and since C(PTa) can be regarded as a closed linear subspace of B(PTa), the continuity of u implies that u[C(PSa)] c C(PTa). Thus u can be treated as a linear operator from C(PSa) into C(PTa). Clearly u is regular and, as it has been observed, u satisfies (1.3.1).

Finally observe that (1.3.1) determines uniquely the linear operator from C(PSa) into C(PTa), because the functions of the form (1.3.3) span a dense linear manifold in C(PSa).

Proposition 1.3 can be generalized to the case of inverse systems of compact spaces (cf. Eilenberg-Steenrod [1], pp. 213-220). We employ in the next proposition the terminology and notation of that treatise.

R e m a r k . Let us observe that the Cartesian product of a family (8a)aeA of compact spaces can be regarded as the inverse limit of the system {8, n) over the set M of all finite subsets of A directed by inclusion. If 5 c M, then 8B = P Sa, and if B' => B, then TIB is the natural

aeB

projection of SB> onto 8B. Therefore the proof of Proposition 1.3 can be reduced verifying the hypothesis of Proposition 1.4 tha t the diagrams (1.4.1) are commutative, which verification amounts to proving Proposition 1.3 for finite products.

1.4. PROPOSITION. Let {8, n] and {T, a} be an inverse systems of compact spaces over a directed set M with inverse limits 8^ and T^ respectively. Let us suppose that the limit maps, na: 8^ -> 8a and aa: T^ -> Ta, are epimorphisms (aeM). Let further ua: C{8a) -> C(Ta) be regular operators such that the diagram

C(Sa) - • C ( Ą )

(1.4.1)

K<ra) C(Ta) ^C(TP)

is commutative for a < (3 (a, jSeJIf).

Then there is the unique regular linear operator

u =limua:C(800)->C(T00)

such that the diagram

C(S0) •WoJ

(1.4.2)

C(Ta) - ^ C ( T 0 0 )

flvO / D \

Up

I \C

commutes for each aeM.

Regular operators and their products 15

Proof. Let us set

oteM

Since the subalgebra C0($oo) separates the points of 8^ and with each function / contains the adjoint function /, the Stone-Weierstrass Theorem implies that C0(£oo) is dense in C (#«,). Since na are epimorphisms, the relation na = n^Tip implies that 7tf are epimorphisms for fl > a (ae31). Therefore (7ra)° and {np

af are invertible. Moreover, if a < /?, then {na)°[C{8a)] <={np)°tC(Sfi)l Let fl€C(Sa) and let / = W ° / , = W°(«! )7 i . Then (1.4.1) implies

= ( )°%( J)7i = (<^)Vf)X/i

This shows that the formula

(1.4.3) « / = ( e r « ) X [ ( * « ) T 7 *>r /«(*a)O[C(0«)]; aeilf

well defines a linear operator from C0($oo) into (/(T^,). Since C0($oo) is dense in 0(8^) and since ||w/|| < ||/|| for / in C0(#«,), the operator w can be uniquely extended to a linear operator from 0(8^) into CiT^). Clearly-such defined u is regular. Let f^C{8a). Then (1.4.3) implies

u(7ta)°fi = K ) X [ ( r c a ) 0 r ł K ) 7 i = M°uafi (aeM). Hence u satisfies (1.4.2).

Conversely, if u satisfies (1.4.2), then for each fe(aa)°[C(Sa)] we have (1.4.3). Therefore u is uniquely defined on the dense subset O0(#oo) of CiSgo). That completes the proof.

§ 2. EXAVES. EXTENSION AND AVERAGING OPERATORS

2.1. DEFINITION. Let cp\ 8 -> T be a map (S, T are compact spaces). A linear operator u: C(8) -> C(T) is said to be a linear exave for cp provided cp°u is the identity on (p°[C{T)] or equivalently <p°tHp0 = 990. A regular exave is a linear exave which is a regular operator. If 9? is a homeomorphic embedding, then a linear exave (regular exave) for 99 is called linear extension operator {regular extension operator). If 99 is an epimorphism, then a (regular) linear exave for <p is called linear averaging operator (regular averaging operator).

A linear exave u for 99 is said to be normal if u<p°u = u. Let us observe that if ux is an arbitrary (regular) linear exave for 99, then u = u^u^ is a normal (regular) linear exave for 99. Linear extension and averaging operators are always normal.

We leave to the reader the simple proofs of the next proposition and corollaries.

2.2. PROPOSITION. Let <p: 8 -> T be a map. Then a linear operator u: C{8) ->C(T) is a linear extension operator (averaging operator) if and only if cp°u = idC(S) (respectively u<p° = idC(T))-

2.3. COROLLARY. An epimorphism <p:8-+T lias a linear (regular) averaging operator of norm < A if and only if there is a projection of norm < 1 (of norm one) from C(S) onto its subspace q>°[C(T)] isometric to C(T).

2.4. COROLLARY. If u: C(8) -> (7(T) is a linear (regular) extension operator for q>: 8 -> T, then u is a linear homeomorphism (linear isometry) from C(S) onto a complemented subspace of C(T).

2.5. PROPOSITION. Let u be a linear exave for a map <p: 8 ->- T. Let Tx be a closed subset of T such that q>8 c: Tx. Then i°u is a linear exave for the map <px: 8 ~> Tx, where (pxs = q>s for se8 and ix denotes the natural (identical) embedding of Tx into T.

Proof. Since ixq>x = q>, we have <pl(h.u)(p° = (i^ifuyl = (p°U(p°. Now, let us choose gx in C(TX). By the Tietze-Urysohn extension theorem there is g in C(T) such that i\g = gv Hence cp[gx = (pli^g = (h<Pi)°g = <p°g-Therefore, since (p°u<p° — <p°, we get

<Pi(£u)<plgi = <p0wp°i9i = <P°wp09 = <P°9 — <P°i°\9 = <Pi9i-

Hence <pl(i°u)q>l = 9?°. That completes the proof. If Q is a subset of T, then T/Q denotes the compact space obtained

from T by identification of all points of Q. The natural epimorphism from T onto T/Q will be denoted i/Q. Clearly (i/Q)° is an isomorphism from the algebra C(T/Q) onto the subalgebra of C(T) consisting of all continuous functions on T which are constant of Q. In the sequel we shall identify this subalgebra with C(TjQ).

2.6. PROPOSITION. A map q>\ 8 -> T admits a linear extension operator if and only if ij(p8: T -> Tjq>8 admits a linear averaging operator.

Proof. Let u be a linear extension operator for (p. Let us fix tf0 in cp8 and set

Pf=f-u<P°f+f(to)u>ls for feC(T).

Clearly P is a projection from C(T) onto C(Tjcp8). Thus, by Corollary 2.4, i\<p8 admits a linear averaging operator.

Conversely, if ij<p8 admits a linear averaging operator, then again by Corollary 2.4 there is a projection P from C(T) onto C(Tjcp8). Let u be an operator (not necessarily linear or continuous) from C(8) into C(T)

such that ||M/|| = ||/|| and <p°uf = / for feC{8). The existence of u follows from the Tietze-Urysohn extension theorem. Let us fix tQ in <p8 and let us set

Uf = uf-Puf+(Puf)(t0)lT for feC(S).

Since PufeC(T/(pS), (Puf)((ps) = (Puf)(t0) for each s in 8. Hence uf is an extension of / , because cp°uf ==/ for feC{8). Since ||«/|| = ||/||, the definition of u implies that

W K (2 ||P|| +1)11/11 for / i n C(8).

Therefore to complete the proof that u is a linear extension operator for 9? it is enough to establish the linearity of u. Since u is bounded, it is enough to show that if fx and / 2 are in G(S), then u(f1-\-f2) = ufx-\-uf2. To do this, first we note that for arbitrary gx in C(T) if

(2.6.1) <p°g1 = <p°g2

then

(2.6.2) gi-Pgi+(Pgi)(to)iT = g^-Pg^+iPg^)^)^-

Indeed, (2.6.1) implies that (g1—g2)(t) = 0 for tc<pS. Thus gx — g%eC{T\<pS), and consequently

(2.6.3) 9l-gt = P(g1-g2) = Pgl-Pg2-

In particular

(2.6.4) o = (gi-g2)(t0) =Pgi{to)-PgM-

Clearly (2.6.3) and (2.6.4) imply (2.6.2). Now if gx = ufx-\-uf2 and g2 = wf/j+Za) for /j and /2 in C(S), then such defined gx and g2 satisfy (2.6.1) and therefore (2.6.2). But for this particular choice of gx and g2, (2.6.2) together with the definition of u imply u(f1+f2) = uf1-\-uf2. That completes the proof.

E e m a r k . If Q is a two-point subset of T, then C{TjQ) is subspace of C{T) of codimension one. Hence for every e > 0 there exists a projection of norm < 2 + e (cf. Bohnenblust [1], Griinbaum [1], Isbell and Semadeni [1]). Thus if Q is a two-point subset of a compact space T and if e > 0, then ijQ admits a linear averaging operator of norm < 2 + £•

2.7. DEFINITION. By C(S, E) (S compact, E — a Banach space) we denote the space of all continuous functions from 8 into E with the norm ||/|| = sup||/(s)||.

SeS

A linear operator u: C(S, E) ->C(T, E) is said to be an E-valued linear exave for a map 9?: 8 -> T provided

<p°EU(p°E = (p°E, where c&(/) = /oc> for feC{T, E).

Dissertationes Mathematicae LVIII 2

The next proposition shows that for a given cp an E-valued linear exave exists if and only if there exists a (scalar valued) linear exave.

2.8. PROPOSITION. For arbitrary map cp: S -> T the following conditions are equivalent: (2.8.1) For some Banach space E0, cp has an E0-valued linear exave. (2.8.2) cp has a (scalar valued) linear exave. (2.8.3) For every Banach space E, cp has an E-valued linear exave.

Proof. (2.8.1) =>• (2.8.2) Let u be an i£0-valiied linear exave for cp. Let us choose e in E and e* in E* such that \\e\\ = ||e*|| = e*e = 1. Let us put

uf = e*u(f-e) for /«C(S),

where (f-e)(s) = f(s)e for seS. Clearly u is a linear exave for cp. (2.8.2) => (2.8.3). Let u: C(8) -> C(T) be a linear exave for cp and

let E be an arbitrary Banach space. Let G0(8, E) denote the linear manifold in C(S, E) consisting of all functions of the form

n

(2.8.4) g = gft-et, faCiSy, e^E (i = l , 2 , . . . , n ; n = l , 2 , . . . ) . i = i

We set for g defined by (2.8.4), n

(2.8.5) ug = ^ufi-ei. i= l

I t is easily seen that ug does not depend on the representation (2.8.4) n m n m

of g, i.e., if g = 2fi-et = 2 7 r 4 t h e n 2ufi'ei = Eufi'e'i-i=\ j' = l i = l 7 = 1

Further we have n n

\\ug\\ = supII Y(ufi)(t)eĄ = sup sup I Y(ufa)(t)e*et\

= sup |U( J '0*^)11 < INI sup suyle* I Yfi(s)ei)\ n

< INI||^/ i-«c|| = INIIIsrl|.

Hence (2.8.5) defines a bounded linear operator from C0(S, E) into 0(T, JE?) (actually into C0(T,E)). Since C0(8,E) is dense in C(S,E), u has the unique extension to a bounded linear operator from C(S, E) into C(T, E). I t follows from the assumption that u is a linear exave for cp and from (2.8.4) that this extension is the E-valued linear exave for cp.

(2.8.3) => (2.8.1). This implication is trivial.

Be m a r k . In terms of tensor product the construction used in the proof of the implication (2.8.2) => (2.8.3) can be described as follows. If E is a Banach space and 8 is compact, then C(S, E) can be identified with

ft the weak tensor product C(8)®E (cf. Grothendieck [1], Chap. I, p. 90).

ft Thus, if w is a linear exave for cp: 8 -> T, then u = u®idE, where id# denotes the identity on E, is an E-valued linear exave for cp (for the defini-

ft tion of Vi® v2 — the weak tensor product of linear operators «x and v2, cf. Grothendieck [1], Chap. I, p. 93).

The next proposition shows that there is no essential difference between the "real" and "complex theory" of linear exaves. By CR(8) we denote the real Banach space of all real valued continuous functions on 8.

2.9. PROPOSITION. Let c > 1. Then for arbitrary map cp: 8 -> T the following conditions are equivalent: (2.9.1) There exists a linear exave u: C(8) -+C{T) for cp with \\u\\ < e. (2.9.2) There exists a real linear exave uR: CR(8) -> CR(T) for (p with

\\uR\\ < e. Moreover, u is regular if and only if uR is regular. Proof. (2.9.1) => (2.9.2). We put

( W ) (*) = re [(«/) (*)] for feCR(8).

Clearly uR is a linear exave for cp and \\uR\\ < \\u\\. (2.9.2) => (2.9.1). We put

uf = uB{ref) - iuR(re (if)) for / e C(8).

Using literally the same arguments as in the Bohnenblust-Sobczyk proof of the "complex" Hahn-Banach extension theorem (cf. Dunford and Schwartz [1], p. 63) we verify that u is a complex linear operator from C(8) into C(T) with ||u|| = \\uR\\. Moreover, if feCR(8), then uf = Unf. Finally since uR is a real exave for q>, i.e. <p°uR(p° = <p°, we get <p°ucp0 = q>°. Hence u is a complex linear exave for cp.

For regular exaves we have a statement slightly stronger than Proposition 2.9.

2.10. DEFINITION. Let C+(8) denote the cone of all non-negative functions in C(8). An operator v: C+(8) ->C+(T) is said to be an affine exave for a map cp: 8 -> T if v is continuous, and v(a1f1 + a2f2) = a1vf1Ą-#2^/2 for alf a2 ^ 0 and for / u / 2 in G+(8), and (<p°V(p°)(f) = 9?0/ for feC+(S).

2.11. PROPOSITION. A map cp: 8 -> T has a regular exave if and only if there is an affine exave for cp.

Proof. Clearly the restriction of a regular exave to the cone of non-negative functions is an affine exave (cf. Proposition 1.2).

Conversely, let v: C+{8) ->C+(T) be an affine exave for <p. Let us fix sQ in 8 and let U = {teT: (vls)(t) > 2 - 1}. Then U is open and U => <p8. Choose A in C+(T) with 0 < A < 1 such that A_1(l) => c>£ and A-ł(0) = T\Z7. Clearly Aoc> = l s and (lT-A)oc> = 0. Thus t^C+Off) -* 0+ (T) defined by

vj = lf+ ( I T - A)/(«0) for /e<7+ (£)

is an affine exave for 9?.We have (-y1l/S)(<) ^ 4 _ 1 for teT. Indeed, if X(t) > 2_ 1 , then *eZ7 and («ls)(«) > 2~\ Therefore (MsKO > >l(t)(f>ls)(t) > 4 - 1 . If A(t) < 2_ 1 , then (Ms)(*) > (lT-A)(*)-l /g(s0) > 2 - 1 . Let us set

vj^-^j- for /«C+(S).

Since vx ls is a positive function on T and since vx is an affine exave for 9?, it is easily seen that v2: C+(S)-> C+ (T) is a well defined affine exave for <p. Clearly v2ls = IT- We extend v2 on CR(8) in the standard way. We set

ug = v2g+-v2g~ for g€CR(8)

where g+ — max(<jr, 0) and g~ = { — g)+. Clearly uf = v2f for feC+(8) and uag = aug for every real a and for

geCR(S). To prove the additivity of u first observe that if g = fl—f2 for some fx and f2 in 0+(#) , then w<7 = ufl — uf2 = ,o2fx — v2f2. Indeed, since g = g+ — g~ = / i — / 2 , we get g++f2 = flT+/v Thus, since v2 is an affine operator,

^2(fl,+ +/2) = v2g+ + vj2 = v2g~ + vjl = v2{g~+fx).

Hence w# = v2g+ — v2g~ = v2f1 — v2f2. Now, let </x and #2 b© m 0R{8). We have

1+ 2 = gt-gT+gt-gT = (sf+s^-ter+fc"). Since w is an extension of v2 and since v2 is additive on the cone of

non-negative functions, the preceding remark implies

* (0i+g%) = u (gt+gt)-u (gi+fc") = «0i" + ^2+ - ugT - ug* = ugt — ugi +w#2

+ — ug2 = ugx + ug2.

This shows that u is additive. Finally, since u takes the positive cone of C+{8) into the positive cone of C+{T), for each geCR{S) with ||</|| < 1 we have ls±g^C+(S). Thus tt(ls±flf) = uls±ug = v2ls±ug = 1T± ±ugeC+(T). Hence ||w#|| < 1. This shows that ||u|| < 1. Clearly, by the respective property of v2, we have qfucp0 = 99°. Thus u is a regular real exave for 9? from CJJ(#) into CR{T). To complete the proof we apply Proposition 2.9.

§ 3. Linear multiplicative exaves and retractions 21

§ 3 . LINEAR MULTIPLICATIVE EXAVES AND RETRACTIONS. LOCALIZATION PRINCIPLE

3.1. DEFINITION. A map cp: 8 -> T is a coretraction corresponding to a retraction r: T -> 8 provided rap = id#. Clearly coretractions are homeomorphic embeddings and retractions are epimorphisms.

A neighbourhood coretraction is a map (p: 8 -» T such that for each s in 8 there is a closed neighbourhood of s, say V, such that (pv: V -> T is a coretraction, where <pv denotes the restriction of cp to V.

We recall that a compact space 8 is said to be an absolute retract (a neighbourhood absolute retract) if (p: 8 -> T is a coretraction (a neighbourhood corectraction) for every homeomorphic embedding cp of 8 into an arbitrary compact space T.

3.2. DEFINITION. A regular linear-multiplicative exave for a map cp: S -> T is a regular linear exave for <p, say u: C{8) -> C(T), such that u(fif2) = ufi'uf2 for fuf2m C(8).

3.3. PROPOSITION. Tfte following conditions are equivalent: (3.3.1) cp: 8 -> T is a coretraction. (3.3.2) There is a regular linear-multiplicative extension operator for c>.

Proof. (3.3.1) => (3.3.2). Let r: T -> 8 be a retraction corresponding to <p. Then u = r° is a regular linear-multiplicative extension operator for <p.

(3.3.2) => (3.3.1). Let u be a regular linear-multiplicative extension operator for <p: 8 -> T. Then u is an isomorphism of the algebra C(8) into C{T) such that uls = 1T. Hence (cf. Gillman and Jerison [1], p. 141) there is a map r: T -> 8 such that u = r°. Since u is a linear extension operator, <p°u = idC(lS). Therefore y0r° = (r<p)° = (ids)°. Thus rep = id#. That completes the proof.

3.4. PROPOSITION. The following conditions are equivalent:

(3.4.1) r is a retraction.

(3.4.2) There is a linear-multiplicative averaging operator for r.

The proof is analogous to that of Proposition 3.3.

3.5. PROPOSITION. If <p is a neighbourhood coretraction, then there is a regular extension operator for (p.

This proposition is an immediate consequence of Proposition 3.3 and the following lemma.

3.6. LEMMA (Localization Principle). Let <p:8->T. Let {Ta}a€A

be a family of closed subsets of T the interiors of which cover <pS. Let 8a

= (p~1(Ta r\ cpS) and let q>a: 8a -> Ta denote the restriction of cp to 8a(aeA).

Let us assume that for each aeA there exists a (regular) linear exave uafor(pa. Then there is a (regular) linear exave for <p.

Proof. Since T is compact, there is a finite subfamily {Ta.}1<i<N

of {Ta}aeA such that [J Ui •=> <p8 where Ui denotes the interior of TH = Ti.

Let us set 8t = 8a., (pi — cv and % = ua. (i = 1, 2,.... JV). Let U0 be an % i % N

open subset of T such that T\<pS => U0 => T\{J Ui. Then the family

{UiJo^i^N is an open cover for T. Let {Xi}0<i<N be a partition of unity such that Xi vanishes outside Vi for i = 1,2, ..., N. Let s0 be a fixed point in 8. Let us set

N

(3.6.1) « /=/ (*„) J o + ^ t t J f o r fe(JW

where (i*</)(<) = 0 for teT\Ti and (utf)(t) = Wi/i(«) for k ^ (/< denotes the restriction o f / t o Si). Clearly XtUifeC(T) for every / in C(8) and for i = 1,2, ..., N. Furthermore <p% = 0, because {teT: X0(t) # 0} <=z U0

c T \ ^ . Thus N N

(3.6.2) c ^ ^ ^ V ^ l s .

Let /6^0[(7(T)]. Then /,erf [CfT,)]. Hence (?•%/<)(*) = /,(*) = / (* ) for every s in #i? because ut is a linear exave for q>t (i = 1, 2, . . . , JV). Since ( U T : ^ ) ^ 0} c ^ c T<, we get <p°(XiUif)(s) = Xi((ps)-f(s) for every s in # and for * = 1, 2, . . . , JV. Hence, (3.6.1) and (3.6.2) imply, for each s in #,

(<p°uf)(s)=[(pO{f(S0)X0)}(s)+[(p

O{^XiUif)](s) = 2Xi(<f8)'f(8) = / ( « ) .

Therefore w is a linear exave for q>. If all Ui are regular, then |(AiM-i/)(<)| < h(t)\\Uifi\\ < (011/11 (f° r

teT and for feC(S)) and A ^ l ^ = ^ (i = 0 , 1 , . . . , N). Thus |(i*/)(t)| TV N

< 2X«)11/11 = 11/11 for /eT and for / e C ^ ) , and uls = J£ A< = 1T. Hence u i=Q i=o

is regular.

§ 4. INTEGRAL REPRESENTATIONS AND COMPOSITIONS OF LINEAR EXAVES

4.1 . PROPOSITION. Let u: C(8) -> C(T) be a regular exave for a map cp : 8 -> T. Let u* : M(T) -> 31(8) be the adjoint operator to u and let <5f

denote the unit point-mass at t (teT). Then

(4.1.1) (uf)(t) = ff(s)fit(ds) for teT and for feC(S),

where

(4.1.2) in = u*bt for teT,

and the following conditions are satisfied: (i) t -> iii is a continuous function from T into M(S) endowed with

the weak-star topology and [it is a non-negative normalized measure in M(8) for each t in T.

(ii) if te(pS, then [xt is concentrated on the set q>~l{t)1 i.e.

(4.1.3) (uf)(t) = J f{s)Mds) for U(PS and /*£(#)•

Conversely any function t -> /xt satisfying (i) and (ii) uniquely determines by (4.1.1) a regular exavefor y such that (4.1.2) holds.

Proof. I t u: C(S) ->• C(T) is a regular operator, then, by Proposition 1.2, it has the representation (4.1.1) with [it defined by (4.1.2) and satisfying (i). To verify (ii) let us fix t in q>S. Let F be a closed subset of S disjoint with c>-1(tf). Then t4<p{F) = F1 and, by Urysohn's Lemma, there exists a non-negative function g in G(T) such that <jf *(1) => <pF and g{t) = 0. Let / = q>°g. Since <p°u<p° = <p°, for seep ~1(t) we obtain

(**/)(<) = (uf)(<ps) = (VflOfo*) = {cp°u<p°g){s) = {<p°g)(8) = g(<ps) = g(t) = 0.

Clearly / > 0 and /($) = 1 for seF a c T 1 ^ ) . Thus since ^ is a non-negative measure, representation (4.1.1) implies

0 < pt(F) < jf(s)Mds) = (uf)(t) = 0. s

Hence ftt{F) = 0 for arbitrary closed subset F oi S disjoint with y~l{t). Therefore pt is concentrated on cp~l{t).

Conversely, if a function t -> [xt satisfies (i) and if u is defined by (4.1.1), then (by Proposition 1.2) u is a regular operator from C(8) into C(T) and pt = u* dt for teT.

Finally, let m satisfy (ii). Let us fix seS and let t = <ps. Then, by (i), (ii) and (4.1.3), for arbitrary g in C{T) we have:

((p°u<p°g)(s) = (u(p°g)((ps) = J g(cps)^(ds) = j g{t)[xt{ds)

= g{t) j ^(ds) = 9W = (<P°9)(s)-

Hence <p°W9?° = <p°. This completes the proof that u is a regular exave for q>. Be mark . One can easily show that every linear operator u : C(S)

-+C{T) has the representation (4.1.1) where [xt — u* 6t. If u is a linear

exave for a map <p: 8 -> T, then the carrier of fit contains (p~l{t), but in general these two sets do not coincide (cf. Example 2 in Notes and Eemarks). However, in the case of linear extension operators we have

4.2. PROPOSITION. Let y>: 8 -> T be a homeomorphic embedding. Let u:C(8) ->C{T) be a linear extension operator for <p. Let fit = u* dt for teT. Then u has the representation (4.1.1) and the function t ~> fit from T into M(8) endowed with the weak-star topology is continuous and

(iii) fięs = 68 for se8. Conversely, if t -> fit is a continuous function from T into 31(8) endowed

with the weak-star topology and satisfying (iii), then there is the unique linear extension operator u : C(S) -> C(T) for y such that the formulas (4.1.1) and (4.1.2) hold.

Proof. For arbitrary linear operator u:C(8)^C(T) the function t -> fit = u* dt from T into M(S) endowed with the weak-star topology is continuous (cf. the proof of Proposition 1.2, the implication (1.2.3) => (1.2.4)). Clearly u has the representation (4.1.1). Now, if u is a linear extension operator for cp, then by (4.1.1) for s in 8 and fo r / in C(8)

Pvs(f) = (^*<W(/) = M ^ / ) = ufi<P*) = / (*) •

This proves (iii). The second part of the proposition is obvious. We are now ready to prove the basic facts on compositions and prod

uct of linear exaves. The results are much more complete for regular exaves than for arbitrary exaves.

4.3. PROPOSITION. Let v and u be regular exaves for maps y^.Q -> 8 and <p: 8 -> T respectively. Let (ptQ — <p~1T0 for some subset TQ of T. Then w = uov is a regular exave for xp = cpoyx.

Proof. Let us set fit = u*dt for teT and vs — v*dsior se8. Let / = \p° g for some geC(T). Then for any fixed t in ipQ

/(?) =9(t) = c f o r e v e i T qey>~l{i)-

Clearly the assumption y>xQ = <p~1T0 implies that T0 = ipQ, and implies that if *ec)~1(<), then ^ f 1 ^ ) <= V~l{t)- Hence, by Proposition 4.1,

(vf)(s)= J f(q)v.(dq)=c for 8«p-l(t).

Again applying Proposition 4.1 we obtain

M)(t) = J vf(s)fit(ds) =c. <p~\t)

Since / = \p° g and since c — g(t), the last formula means that {wip°g)(t) = g(t) for texpQ and for geC{T). Thus if qeQ, then

(v>°g)(q) = gin) = (wy>°g)(n) = iv>°W)ig) for g*C{T).

Hence ip°w\p° = y>°. Thus w is a linear exave for y. The regularity of w is trivial.

Examples 3 and 4 in Notes and Eemarks show that the assumptions of Proposition 4.3 (the regularity of u and v and the condition (pxQ = <p~lT0) are in general essential. The next proposition shows however that in certain cases these assumptions can be dropped.

4.4. PROPOSITION. Let v and u be linear exaves for maps yx: Q -> S and <p: S -> T respectively. Let either (a) v be a linear averaging operator, or (b) u be a linear extension operator, then w = uov is a linear exave for v> = <p<pv

Proo f , (a) implies t ha t V(p°x = idcf(s). Thus, since xp° = <p\q>°, we get wy>° = uv<p\<p° = u<p°. Since <p°u<p° = q>°, we obtain:

o o o o o o o o

ip wip = (pxcp u<p = (px<p = ip .

(b) implies <p°u — i&wsy Hence ip°w — <p°<p°uv = <p°xv. Thus, since o o o

o o o o o o o o ip wip = <pxv<px(p = (px<p — ip .

Hence in both cases w is a linear exave for ip. Example 3 in Notes and Eemarks shows that the composition of

a linear exave with a linear extension operator, as well as with a linear averaging operator, taken in the opposite order as in Proposition 4.4, is not in general a linear exave. However, we have:

4.5. PROPOSITION. Let u be a {regular) linear exave for a map <p: S -> T and let (px: Sx -> S be a contraction corresponding to a retraction r: S -» Sv

If r°[C{Sx)] 3 9?°[0(T)], then ur° is a {regular) linear exave for q><px.

Proof. By the assumption for every g in C{T) there is an feC(Sx) such that cp°g = r°f. Since q>\r° = ido^) and since (p°u<p° = q>°, for arbitrary g in C{T) we have

/ \°....° I \ ° ~ O O O O O O O O O O / . O O Or

(Wi) ur (Wj) g = <Pi ur f

= <p0i<p°u<p0g = (p°x(p0g = {<p<px)°g.

Hence ur° is a linear exave for (pq>x. Clearly, if u is regular, then ur° has the same property.

4.6. PROPOSITION. Let

Vi

8 *~T

A A 'V2

be a commutative diagram, where tpx and ip2 are homeomorphic embeddings, cp is an epimorphism and y)x80 = q>~1(y)2T0). If there exist a regular extension operator ux for ipx and a regular averaging operator v for <p, then there exist a regular averaging operator v0 for yQ and a regular extension operator u2 for ip2.

Proof. Let us set

v0 = %p\vux, u2 = vuxq%.

Clearly vQ and u2 are regular operators. Since v and ux are regular operators and since y>xS0 = (p~1(y>2TQ), if follows from Proposition 4.3 that vux

is a regular exave for cpipx. Since y>2 is a homeomorphic embedding and since (yy>i)S0 = ip2TQ and <p0y>2 = cp\px, Proposition 2.5 implies that v0 = xp°2vux

is a regular exave for cp0. Since <p0S0 = TQ, vQ is a regular averaging operator for <pQ. Hence vQq>°0 = idc«(So). Finally we have

yS aVa = ipo2vux(p°0\pl = v0<ply)°2 = \p°2.

Hence u2 is a regular exave for ip2. Since ip2 is a homeomorphic embedding, u2 is a regular extension operator. That completes the proof.

E e m a r k . In general vux ^ u2v0, but we have the identity

u2v0 = vux(y)2rp0)°vux.

Thus, if the exave vux is normal (cf. Definition 2.1), then vux = u2v0. 4.7. PROPOSITION. Let {<pa: 8a ->- Ta)aeA be a family of maps. Let

wa: C(Sa) -> C{Ta) be a regular exave for cpa (aeA). Then u = ®ua is a regular exave for the map Pcpa : PSa -> PTa. Moreover, if all ua are regular extension operators resp. regular aver

aging operators, then u has the same property. Proof. Let 9? = Pq>a. By the definition of cp, if pa: PSa -> Sa and

qa : PTa -> Ta denote the natural projections, then

(4.7.1) <PaPa = qa°g = cp°g for g in a linearly dense subset G of C(T). Let G be the set of all functions g of the form

g = [1 qlga (ga€C(Ta)] aeB, B is a finite subset of A). aeB

Using the fact that <p° is a linear-multiplicative operator and applying formulas (1.3.1) of Proposition 1.3 and (4.7.1), we obtain

O O O 1 I" O O O I T O O O 1 I O O 1 T O O O

cp ucp g = (p uf l aeB aeA aeB aeB

This completes the proof that u is a regular exave for (p. The second part of the Proposition follows from the fact that if cpa

are homeomorphic embeddings (resp. epimorphisms) for all aeA, then q> has the same property.

§ 5. MILUHN SPACES

We recall that a compact space is called dyadic if it is a continuous image of a generalized Cantor set Bm.

This section is devoted to study some subclass of the class of all dyadic spaces.

5.1. DEFINITION. A Milutin space (resp. an almost Milutin space) is a compact space T such that there exists an epimorphism <p : Dm -+ T which has a regular averaging operator (a linear averaging operator).

The next proposition is a modification of a result of Sanin [1]. 5.2. PROPOSITION. If T is a Milutin space (almost Milutin space)

of weight n, then there exists a regular averaging operator (a linear averaging operator) for an epimorphism cp : Dn -> T.

Proof. The case where n is finite is trivial. Let us suppose that n is infinite. Let T be an epimorphism. Clearly m > n. Then (cf. Engelking and Pełczyński [1], Theorem 1) there is a coretraction <p1:Bn-^Bm such that r°[C(Dn)] =xp°[C(T)], where r° : Dm -> Dn is a retraction corresponded to <px. Hence, by Proposition 4.5, if u is a regular averaging operator (a linear averaging operator) for <p, then ur° is a regular averaging operator (a linear averaging operator) for y> = cpcpx.

For sake of completness we describe the construction of the core-traction <px: Dn -> Dm. Let Dm = PDa where A = m and Da is a two

aeA point space (aeA). If B is a subset of A, then pB : PDa ~> P^a denotes

aeA aeB ,

the natural projection. Clearly pB is a retraction which corresponds to a coretraction <pB : PDa -> PBa such that for rj = (r}a)ePBa, cpBr\ = (Ca)aeA

aeB aeA aeB

where fa = r\a for aeB and Ca = 0 for aeA\B. If feC(Bm), then Bf = O [^\f€V°B(G(PBa))\ is the smallest set of coordinates on which /

aeB

"essentially depends", i.e. if £0 = lafor aeBf, t hen / ( | ) = /(£') for £ = (Ja) and i' = (tj'a) in Dm. Since (by the Stone-Weierstrass theorem) the set of all functions depending on a finite number of coordinates is dense in C(Bm), Bf is at most countable for each / in C(Dm). Since the weight of T is equal to the density character of C(T) and since <p°[C(T)] is isometric to C(T) (because <p is an epimorphism), there is in <p°[C(T)] a dense subset, say W, such that W = n. Let A0 = [J Bf. Clearly A = n and Bf a A0 for each

nw f«p°[C(T)l Hence (*) <P°[C(T)] c p%[C(Dn)l where Dn = P Da.

We put 9?! = q>j0. Since c?J restricted to p\{C{Dn)] is an algebraic isomorphism, it follows from (*) that (qxp^f = <p\q>° is an algebraic isomorphism from C{T) into C(Dn), equivalently cpcpx(D

n) = T.

5.3. PROPOSITION. The Cartesian product of an arbitrary family of Milutin spaces is a Milutin space. More precisely, if {cpa : B

ma -> Ta}afA is a family of epimorphisms and if ua is a regular averaging operator for (pa(aeA), then ®ua is a regular averaging operator for the epimorphism P<pa:PDm»-+PTa.

Proof. This is an immediate consequence of Proposition 4.7 and of the fact that P DWa is homeomorphic to Dm where m = max (A, supma).

OeA OeA

We shall say that a compact space T is locally Milutin [locally almost Milutin) if each t in T has a closed neighbourhood which is a Milutin space (an almost Milutin space).

5.4. PROPOSITION. Every locally Milutin space [locally almost Milutin space) is a Milutin space {an almost Milutin space).

Proof. By the standard arguments the proof reduces to verify that if T is a compact space with the property that there is a finite open cover {UijiLi of T such that the closure Ti of each Ut is a Milutin space (an almost Milutin space) for i — 1, 2 , . . . , N, then T is a Milutin space (an almost Milutin space).

Let the epimorphisms cpi: Dmi -> Tt admit regular averaging operators (linear averaging operators). Let 8 denote the discrete sum of spaces Dmi and let cp: S -> T be the map induced by q>^ i.e.

q>(x)=<Pi(x) for xeDmi c 8 (i = 1,2, ...,N). N

Since (J Ti = T and since <Pi{Dmi) = Tt for i = 1, 2, . . . , N, <p is an i = l

epimorphism. Clearly {Dmi}£L1 is an open and closed covering of 8. The restriction of cp to Dmi being cpi admits a regular averaging operator (a linear averaging operator). Hence, by Lemma 3.6, cp has a regular averaging operator (a linear averaging operator). Let Z^ denote the discrete sum

of N copies of Bm, where m = max(mn m 2 , . . . , mN, K0). Since m > mt, each of the copies of Dm admits a retraction onto Bmi (i = 1, 2 , . . . , N). These retractions induce the retraction r from Zffl onto 8. Hence, by-Propositions 3.4 and 4.4, there exists a regular averaging operator (a linear averaging operator) for cpr. To complete the proof it is enough to observe that if m > g0, then Zffl is homeomorphic to Dm.

Let ur+>n denote the relation "is homeomorphic to". Then clearly Zffl /— [N] x Dm, where [N] denotes the discrete space consisting of N points. Since DK° x [N] is a zero-dimensional compact perfect metrizable space, it is homeomorphic to Ds° (cf. Kuratowski [2], p. 58). If m > X0, then m + K0 =s m. Hence

Dm ~ Dm+*o ~Dmx DK° ~ Bm x Ds° X [N] ~ Dm x [N].

Therefore if m > K0, then 35?> ~ Dm x [ # ] ~ Bm. That completes the proof.

We recall that # = DK° denotes the Cantor set, that is the countable product of two-point spaces A = D = {0} w {1} (t = 1, 2, . . . ) . A general point of # is denoted by I = (£*) where & = 0 or 1 (i = 1, 2 , . . . ) . If i and rj are in #, then £ < 97 means that either £ = ?y, or there is an index i0 such that & = rji for i < £0 and &0 < ^ 0 (that is the lexicographical order).

By the product measure on ^ we mean the product measure <g>mi where m = mt (a = 1,2, . . . ) is the measure Qn i> such that m({0}) = m({l}) = 2_1 . We shall write / / (£)d£ instead of / /(£)&«**(<*£).

5.5. LEMMA (Milutin [1], [2]). Tfte dosed! interval I = [0,1] is a Milutin space, i.e. there exists an epimorphism W from the Cantor discontinuum <% onto I which admits a regular averaging operator.

Proof. Since ^ x # is homeomorphic to #, we can use # x # instead of #. Let us set

h(C)=]?2-1Ci for * = (&)«*. *=i

I t is well known that the u Cantor map" h is continuous and maps # onto I. Let us put h^.l{s) — max £ for seJ. Then Tnjl1: 7 -> ^ i s a meaSUr-able map which has a countable set of points of discontinuity (the set of all dyadic points in (0,1)). Moreover,

1

(5.5.1) j 9{Mt = j g(K\*))** «* 0

for every complex valued function g on ^ integrable with respect to the product measure on #.

Let us define the epimorphism W: ^ x # -->- 7 and the regular averaging operator u : C{tfx V) -> C(I) for W by:

(5.5.2) !F(f,f) fe(l)-i+^(l)-i)2 + 4fe(|)/i(q for f # 0

2ft(£) U(C) for 1 = 0,

(5.5.3) («/)(*) - J*/(|, ^1[A2A(^) + A(1-7KI))]) df

for 0 < A < 1 and for feC(VxV).

The map IF can be described as follows. It is the composition of the map hxh'.&xtf-^lxl and the epimorphism Wx: Ix I -» 7 which assings to each point (s, t) of the square 7 x 1 the unique Ae7 such that the point {s, t) lies in the interval joining the points (0, A) and (1, A2). Hence !F(£, £) is the non-negative root of the quadratic equation h[£) = (A2 — A)ft(f) + A. Clearly W is continuous and maps ^ x *# onto 7.

We shall show that u is a regular averaging operator for W. First, let us consider the integrand in the right hand of (5.5.3). If y(C, A) = A2h{C) + l(l-h(C)), then 0 < A < 1 and 0 < /* (£ )< 1 implies that A > 2/(|, A) > A2 for | in «\ Clearly y{£, 0) = 0 and 2/(1,1) = 1 for f etf. If 0 < A < 1, then for every fixed A, y (£, A) as a function of £, is continuous, decreasing, and strictly decreasing except a countable set of £. Thus

^+x(2/(£> ^)) and/If , ^+1(y (I , A))l as functions of | have at most a countable set of points of discontinuity. For A = 0 ,1 , we have

/ ( * , Kl(ytt, 0))) = / ( £ , 0) and /(f, ft"1^, 1))) = / (£ , 1).

Therefore the integrand in (5.5.3) is a bounded measurable function on ^ for every A in 7 = [0,1 ] and the integral in (5.5.3) exists for e a c h / e O ^ x ^) and for each Ae7. Therefore u can be regarded as a linear operator

from CC&xV) into B(I), where B(I) denotes the space of all bounded complex-valued functions on I with the usual sup-norm. Clearly \\u\\ = 1 and ul<#x<# = l j . We shall show that the range of u is contained in the space C(I), which may be regarded as a closed linear subspace of B{I). Since u is a bounded linear operator, it is enough to establish that ufeC(I) for every / in a certain linearly dense subset W of 0 ( ^ x ) .

For Cetf let x% denote the characteristic function of the set {$'eft: 0 < £' < I}. For (!, £)etfx tf let xs®Xc De t h e function defined by

(Z9<8>ztu?,n = xs(n-xc(n ((?,n*vxv). Let W be the set of all functions / = Xs®Xt f° r (f > QeAxA, where

A = {| = ( | , ) e ^ : l i m ^ = l } . i

Observe that if fe/1, then { I ' e ^ : 0 < £' < £} is a closed and open subset of V. Therefore ^eC(^ ) for £eA. Hence W c z C ( ^ x ^ ) . Furthermore if £, £ j J? j c are in #, then

Zl O ft' Xr, ® X* = Zmin(|,,) ® Zmin(f,a) •

Hence the set of all linear combinations of functions which belong to W is a subalgebra of C{^ x ^) (because if f and ?? are in 4 , then min(£,»?) e/1). This subalgebra separates the points of ^ x ^ and contains the constant functions. Therefore, by the Stone-Weierstrass theorem, W is linearly dense in C(VxV).

If / = xt®Xc belongs to W, then, by (5.5.3) and (5.5.1),

(«/)W = J xMxc(KVh(v)+^-Mv))])dv i

= / xt^AxAKY'+W-*)])**-o

Since Xeih^s) = X[oMt)] for almost all s in I (where #[a6] denotes the characteristic function of the interval [a, &]), we get

i

(«/)W = JZ[o)ft(i)](«)Z[o)ft(f)](2s + ^(l — 8))ds for Ael.

0

The last integral expresses the lenght of the orthogonal projection onto s-axis of the intersection of the interval [(s, t): t — (A2 — A)s + A; 0 < 8 < 1] with the rectangle [0 < s < h(£); 0 < t < fe(£)]. An easy computation shows that for (£, £)e/lx A

(5.5.4) («/)(A)=u(jfe®jfc)(*)

rM£) for 0<A<fe(C) ,

A—MO * ( f ) J Z I T f o r M C ) < A < y ( f , C ) ,

0 for y / ( | , 0 < ^ < l .

Since O4A, (5.5.2) implies that ¥*(£, f) > 0 for (£, C ) e 4 x A Therefore

^•p(?,c)-oL A — / 2 J

because W{£, C) is the positive root of the equation (A2 — A)ft(|) +A — — h(C) = 0. This together with (5.5.4) show that ufeC(I) for every/eTT.

Hence w is a regular operator from C{^x^) into C{I). Now, let /e«F°[C(Z)],i.e./ = gro !F for some ^eO(I).Then (£, C)ecf *(A) (equivalently h(C) = (A-A2)ft(£) + ^) implies that / (£ ,£) = flf(A) and for almost all £

Hence the integrand in (5.5.3) is equal to </(A) for almost all f. Thus (w/)(A) = flf(A). This shows that w is a regular averaging operator for y . That completes the proof.

5.6. THEOREM. The Cartesian product of an arbitrary family of compact metric spaces is a Milutin space.

Proof. According to Proposition 5.3 it is enough to show that every compact metric space is a Milutin space. First observe that Milutin's Lemma 5.5 and Proposition 5.3 imply that I*0 is a Milutin space. Hence, by Proposition 5.2 there exists a map 9?: <€ -> Js° which admits a regular averaging operator, say v. Let T0 be an arbitrary compact metric space. Then there exists a homeomorphic embedding ip2: T0 -> I s°. Let S0

= C?_1(v,2^1o) a n a < 1^ Vi k e the identical embedding of 80 into cś. Since there exists a retraction of ^ onto each of its non-empty closed subset (cf. Kuratowski [1], p. 169), tpx is a coretraction. Therefore, by Proposition 3.3, there exists a regular extension operator for y>i, say ux. Finally, let q>0 denote the restriction of 2. Since xpx is a coretraction, Propositions 3.4 and 4.4 imply that v0y% is a regular averaging operator for <p0y)v That completes the proof.

The foregoing proof implies immediately 5.7. COROLLARY. Every homeomorphic embedding of an arbitrary

compact metric space into the Eilbert cube has a regular extension operator. 5.8. PROPOSITION. Each compact absolute neighbourhood retract is

a Milutin space. Proof. According to Proposition 5.4 it is enough to show that each

absolute retract is a Milutin space. Let T be an absolute retract and let 9? be a homeomorphic embedding of T into some Tichonov cube I*1. Then <p is a coretraction. Therefore, by Proposition 3.4, <p° is a regular averaging operator for a retraction r corresponding to 9?. Since Im is (by Theorem 5.6)

a Milutin space, there is a regular averaging operator, say v, for an epi-morphism ip : Dm ->- Im. Hence, by Proposition 4.4, q>°v is a regular averaging operator for the epimorphism rxp : Dm -> T. This completes the proof.

We complete this section by showing that Milutin spaces have a special separation property. This allows us to prove that there are dyadic spaces which are not Milutin spaces (cf. Notes and Eemarks, Example 5).

5.9. DEFINITION. A compact space 8 is said to have BocJcstein Separation Property — [B.S.P.] if every pair of disjoint open subsets of 8 can be separated by open Fa sets.

Let us observe that [B.S.P.] is equivalent to the following property (5.9.1). For every pair (U0, TJX) of open disjoint subsets of 8 there is a non-

negative f in G(8) such that / - 1 (^) => U0 and /_1(0) r^ U1 = 0 . Indeed if 8 has [B.S.P.] and (U0, Ux) is a pair of (non empty) open

disjoint sets in 8, then there are disjoint open i^-sets, say V0 and VXJ separating UQ and Ux. Since Vx is an open jFff-set, there is a non-negative function / in C(8) such that /^(O) = 8\V1 => U0. Clearly / has the property required in (5.9.1).

Conversely suppose that a compact space 8 satisfies (5.9.1). Let (U0, TJX) be any pair of non-empty disjoint open sets in S. Let / be as in (5.9.1). Let us put Vx = 8\f~1(0). Then Vx is an open l^-set such that Vx => Ux and Vxr\ U0 = 0 . Now let us consider the pair (Vx, U0) and again by (5.9.1) let # be a positive function in C(8) such that #_1(0) => Vx and g-l{0) ^ U0 = 0 . We put V0 = 8\g~1(0).

5.10. PROPOSITION. Let a map cp: 8 -> T admit a regular linear exave, say u. Let either 8 or T have [B.S.P.]. Then <j>S has [B.S.P.].

Proof. Let 8 have [B.S.P.]. Let (?70, TJX) be an arbitrary pair of nonempty open (in the relative topology) disjoint sets in yS. Let us put Vj = (p-1 TJj (j = 0,1). Then V0 r^ Vx = 0 . Therefore, by (5.9.1), there exists a non-negative / in C{8) such that /_ 1(0) •=> V0 and/ - 1(0) ^ Vx = 0. Since u is regular, Proposition 4.1 implies that

(«/)(<) = / MM**) for U<P(S)

where /ut is a positive normalized non-negative measure concentrated on cp~l(t) (teyS). Hence, if teUo, then <p_1(ź) <= V0 c / _ 1 (0 ) and therefore (uf)(t) = 0. UteUu then qT1^) c Vx. Therefore/(s) > 0 for every se<p~l{t). Hence the integral representation implies that if teUi, then {uf)(t) > 0. Thus if g is the restriction of uf to <p8, then </_1(0) => U0 and #-1(0) ^ Ux = 0 . Hence <pS satisfies (5.9.1) and therefore it has [B.S.P.].

Now, let T have [B.S.P.] and let (U0, TJX) be an arbitrary pair of open disjoint subsets of cpS. Clearly to show that 9?$ has [B.S.P.] it is Dissertationes Mathematicae LVIII 3

enough to find open disjoint subsets of T, say V0 and Vx, such that Yj => JJj for j = 0 , 1 .

Let us fix 6, with 0 < 6 < 2~l. Let us set for j = 0 , 1

A, = {geC(<pS) : - 2 + j < 0 < j ; 0 ^ g-\-2+j) cz ff,; flf^-j) => <P^Uj},

Ą- = {fcC(S) : / = gocp; geAj}, V, = U { ^ ' : (*/)(«) < - 2 + j + «5}.

Clearly F7- are open. Furthermore JJj c Vj for j = 0 , 1 . Indeed if teUj, then there is geAj such that — g(t) = \\g\\ = 2—j. Therefore, since u is a linear exave, u(go<p) is an extension of g on T (this is nothing else but <p°u<p° = <p°). Therefore

u(go<p)(t) =g(t) = 2 + j < - 2 + j + ó .

Hence, if teJJj, then /eFy (because gcxpeBj). Finally, let us suppose that the sets V0 and Vx are not disjoint. Let

t0eV0r^ Vx. Then there exists foeBj such that (ufi)(t0)< -2+j+d for j = 0 , l .

Thus ll*(fo+/i)ll > l(«/o)(«o) + «/i(<o)l > 3 - 2 d .

On other hand, if /, = ^099 for j = 0 , 1 , then

II/0+/1II = Il0o + 0ill = Bnp|flr0C*) + flri(<)l-U<pS

It follows from the definitions of Aj that if gjeAj for j = 0 , 1 , then l0o(O + 0i(«)l < 1 for tcctf. Hence H/0+/JI < 1.

This shows that if the sets V0 and Fx are not disjoint, then ||w|| > 3 — 2d. Hence, in particular, if ||w|| = 1, then V0 and Vx are disjoint. That completes the proof.

Kemark . Actually we proved that if a compact space T has [B.S.P.], then, for any map 99: 8 -> T such that <pS does not have [B.S.P.], the norm of every exave for 99 is ^ 3 .

5.11. COEOLLARY. Every Milutin space has [B.S.P.j. Proof. This is an immediate consequence of Proposition 5.10 and

the fact that Dm has [B.S.P.] (cf. Bockstein [1], K. A. Koss and A. H. Stone [1], R. Engelking [1]).

§ 6. DUGUNDJI SPACES

6.1. DEFINITION. A compact space S is a Dugundji space (resp. almost Dugundji space) if for every compact space T every homeo-morphic embedding T has a regular extension operator (resp. linear extension operator).

§ 6. Dugundji spaces 35

6.2. PROPOSITION. For every compact space 8 the following conditions are equivalent: (6.2.1) 8 is a Dugundji space (resp. an almost Dugundji space). (6.2.2) There exists a Dugundji space T (resp. an almost Dugundji space T)

and a homeomorphic embedding T which admits a regular extension operator (resp. linear extension operator).

(6.2.3) There is a homeomorphic embedding ip : 8 -> lm which admits a regular extension operator (resp. a linear extension operator).

Proof. (6.2.1) => (6.2.2). Put 8 = T and (6.2.3). This is an immediate consequence of the fact that

every compact space can be homeomorphically embedded into Im for some cardinal m and of the fact that the composition of two regular extension operators (resp. linear extension operators) is a regular extension operator (resp. linear extension operator) (cf. Proposition 4.4).

(6.2.3) => (6.2.1). Let v : C(8) -> 0(1™) be a regular extension operator (resp. linear extension operator) for ip : 8 -> Im. Let cp : 8 -> T be a homeomorphic embedding of 8 into an arbitrary compact space T. Since P" is an absolute retract, there exists an extension ^ : T ->- Im

of tp such that the diagram

8 • 7 1 "

 •

This shows that u is a linear exave for the homeomorphic embedding cp. Thus u is a linear extension operator. Clearly u is regular, whenever v has the same property. That completes the proof.

E e m a r k . Actually the proof of the implication (6.2.3) => (6.2.1) shows that if 8 is a compact space with the property that there is a homeomorphic embedding ip: 8 -> Im which admits a linear extension operator of norm < a, then for every compact space T, every homeomorphic embedding of 8 into T admits a linear extension operator of norm < a.

A compact space 8 is said to be a locally Dugundji space (locally almost Dugundji space) if each s in 8 has a closed neighbourhood which is a a Dugundji spa-ce (almost Dugundji space).

6.3. PEOPOSITION. Every locally Dugundji space (resp. locally almost Dugundji space) is a Dugundji space [resp. an almost Dugundji space).

Proof. Let cp: 8 -> I™ be a homeomorphic embedding of a locally Dugundji space S (resp. almost locally Dugundji space) into the Tichonov cube lm. For each s in S let 88 denote the closed neighbourhood of s which is a Dugundji space (an almost Dugundji space). Let us set T8 = lm and let <p8: 88-> Ts denote the restriction of <p to S8. Clearly there are regular extension operators (resp. linear extension operator) for <ps because 88 are Dugundji spaces (resp. almost Dugundji spaces). Hence, by Localization Principle 3.6, there is a regular extension operator (resp. linear extension operator) for <p. Therefore 8 satisfies (6.2.3). That completes the proof.

6.4. COEOLLAEY. Every compact absolute neighbourhood retract is a Dugundji space.

This is an immediate consequence of Proposition 6.3 and the fact that every absolute retract is a Dugundji space.

6.5. PEOPOSITION. The Cartesian product of an arbitrary family of Dugundji spaces is a Dugundji space.

Proof. Let (8a)aej. be a family of Dugundji spaces. Let <pa: 8a-> Ima

be homeomorphic embeddings, where cardinal numbers ma depend on Sa. Then each <pa has a regular extension operator, say ua (aeA). Hence, by Proposition 4.7, ®wa is a regular extension operator for the map Pcpa : PSa -> P/™". Since the product Plm" is homeomorphic to the Tichonov cube Im for m = max (A, sup ma), the space P8a satisfies (6.2.3).

OtA

That completes the proof.

6.6. THEOEEM. The Cartesian product of an arbitrary family of compact metric spaces is a Dugundji space.

Proof. Every compact metric space is a Dugundji space. This is a particular case of Borsuk-Dugundji theorem (cf. Borsuk [1], Dugundji [1], Michael [1]). Alternatively it follows from Corollary 5.7 and Proposition 6.2. The assertion of the theorem follows immediately from the previous remark and Proposition 6.5.

The next corollary is an analogue of Corollary 5.11.

6.7. COEOLLAEY. Every Dugundji space has [B.S.P.]. This follows immediately from Propositions 6.2, 5.10, and the fact

that the Tichonov cube F1 has [B.S.P.] (cf. Bockstein [1], K. A. Eoss and A. H. Stone [1]).

§ 7. EXAVES AND TOPOLOGICAL GROUPS

7.1. DEFINITION. A topological group G acts on a space 8 provided to each pair (g, s) in Gx S there corresponds a point y(g, s) in 8 such that the following conditions are satisfied:

(7.1.1) The transformation function y:Gx8-^8 is continuous; (7.1.2) y(g, •) = yg : 8 -> 8 is a homeomorphism of 8 onto itself (geG), (7.1.3) y„x = ygoyax; y9_x = {ya)~

l for g, gx in G.

The operator ag = (yff)° : C(#) -> C(#), with <rff(/) =foyg iovfeC(S), will be called the shift operator.

A map T is said to be G-invariant if

(7.1.4) G acts on 8 and on T with transformation functions y': G x 8 -» # and y" :GxT ->T respectively;

(7.1.5) ?>oy^ = y'g'cxp for each # in G.

A linear exave u for a (^-invariant map cp : 8 -> T is said to be (r-invariant if

(7.1.6) wo = a'g u for #€#.

7.2. PROPOSITION. Let G be a compact topological group and let v be a linear exave (a regular exave) for a G-invariant map cp: 8 -> T. Then there is a G-invariant linear exave {regular exave) for <p.

Proof. Let us set

(7.2.1) ug = OgVGg_x for g*G,

(7.2.2) uf= jugfdg for /«C(fl), o

where the integral in (7.2.2) is taken with respect to the normalized Haar measure of G.

I t follows from (7.1.1) that g -> ugf is a continuous function from G into C(T) for every fixed / i n C(S). Thus the compactness of G implies that the integral in (7.2.2) exists. Therefore u is a linear operator from C{8) into C{T) with ||u|| < sup||i*a|| < ||tf||. Furthermore for every g in G

g*G (7.1.3), (7.1.5), and the identity <p°v<p° = (p° imply

o o o u t o ' o o " i o II I I o o V(p Og_X = CgC) Og_X = OgCg^fp = C? .

Thus for every / i n C(T) we get

((p°itq>°) (/) = <p° f (ug<p°)(f)dg = f(<p°Ug(p0)(f)dg G G

= f<P°fdg = <p°f Jdg = <p°f. O G

Hence u is a linear exave for 9?. Moreover, if v is regular, then u has the same property.

Finally, let us fix h in G. Then (7.1.3) and (7.2.1) imply that for every g in G,

Ug<fh = Ga Vahg-l = Oft Otok-lWtyfc-1)-1 = °ft Ufa-1,

because for any shift operator a, ag0l = agpag for g, gt in G. Hence the invariantness under translations of the Haar measure implies that

(uo'h)(f) = juga'hfdg = f oń ugh-ifdg o o

= o'h fugh~ifdg = a'h j ugfdg = (<ri'w)(/) a a

for every / in C{8) and for every h in G. Thus ua'h = c'h'u for every heG. That completes the proof.

7.3. DEFINITION. Let G be a compact topological group and let if be a closed subgroup of G. We shall denote by GjH the left coset-space of G modulo H (cf. Montgomery and Zippin [1], p. 26) and by y>H the natural map from G onto G/H, i.e. ipHg = [gH] for geG where [gH] = {gxeG : g~1g1eH}. Clearly the map y)H:G->G/H is 6r-invariant (the transformation functions y' :GxG ~^G and y" :GxG/H -> G/H are defined by y'(g, gx) = g-g, and y"(g, [flfxH]) = [g-giH] for g, gx in G).

7.4. PROPOSITION. There exists the unique G-invariant regular operator for yjfI. This operator is defined by

(7.4.1) (uf)([gH]) = ff(gh)dh for feC(G) and for geG, H

where the integral in (7.4.1) is taken with respect to the normalized Haar measure of H.

Proof. Let us suppose that u is a regular (r-invariant averaging operator for yH. Then, by Proposition 4.1, there is a normalized non-negative measure pH in 31(G) such that

(uf)([H]) = jf(g)tzH(dg) = ff(g)/iH(dg) for feC(G). H O

(We regard here H = xpJil{[H}) as a subset of G). Since every / ' in C(H) can be extended to an / in C(G) and since u is ^-invariant, we have

jfih^ftnidh) = ff(h)m(dh) = Jf'(h)[tH(dh) G G H

for f'eC(H) and hxeH.

Hence regarding /uH as an element of M(H) we infer that /J,H is a non-negative normalized measure in M{H) which is invariant under transla-


tions by elements of H. Thus (JLH is the Haar measure of E. Therefore

(«/)([#])= JfWdh for feC(G). H

Finally, since u is (x-invariant, (7.1.6) implies that

m([gH]) = (o'g'uf)([H]) = (ue'gf)([H]) = $(ogf)(h)dh H

= jf(gh)dh (geG). H

This completes the proof of the uniqueness part of the Proposition. We leave to the reader the simple checking that (7.4.1) defines a 6r-invariant regular averaging operator for ipH.

7.5. THEOREM. Every coset-space of a compact topological group (less generally, every compact topological group) is a Milutin space.

The proof of this theorem is based on the next proposition.

7.6. PROPOSITION. Every compact topological group is homeomorpMc to the quotient group of a Cartesian product of a family of compact metric groups.

Proof. We repeat with a slight modification the arguments of Kuzminov [1].

Let G be a compact topological group and let G0 be the component of unit. Then G0 is a connected invariant subgroup of G such that the quotient group G/G0 is zero-dimensional. Hence, by a result of Mostert ([1], Theorem 8), G is homeomorphic to G0xG/G0. Since every infinite zero-dimensional compact topological group is homeomorphic to Dn

for some cardinal number n (cf. Hulanicki [1], Kuzminov [1], Hewitt [1], Hewitt and Eoss [1], pp. 95-98), G is homeomorphic either to G0X [N] (where [N] denotes the finite discrete space consisting of exactly N points), or to G0x Dn.

Now, 6r0, as a compact connected topological group, is isomorphic(l) to the quotient group (A x E*)/Z, where A is a compact Abelian group, Z* is a Cartesian product of a family of simple and simply connected compact Lie groups and Z is a closed subgroup of A x Z* which is contained in the centrum of Ax E* (Weil [1], pp. 89-93). To complete the proof it is enough to apply the following facts:

(a) Every compact Abelian group is isomorphic to a Cartesian product of a family of compact metric groups which are either finite, or are iso-

(*) For topological groups an isomorphism means always an algebraic isomorphism which is a homeomorphism,


morphic to the group of all rotations of the unit circle (cf. Weil [1], p. 89-93).

((3) Every compact Lie group is metrizable. (y) If Hx and H2 are (topological) groups, Z2 is an invariant subgroup

of H2 and {e} is the subgroup of Hx consisting of the unit of Hx, then the group HxxE.2fZ2 is isomorphic to (H1xH2)/Z, where Z = {e}xZ2.

Proof of Theorem 7.5. I t follows immediately from Propositions 7.4 and 4.4 that if a compact group is a Milutin space, then every of its coset-spaces has the same property.

Now, by Proposition 7.6 and Theorem 5.6, every compact group is homeomorphic to a coset-space of a topological group which is a Milutin space. Hence every topological group, and therefore every of its coset-spaces, is a Milutin space.

§ 8. APPLICATION TO LINEAR TOPOLOGICAL CLASSIFICATION OF SPACES OF CONTINUOUS FUNCTIONS

8.1. DEFINITION. A Banach space Y is said to be a, factor of a Banach space X (in symbols Y\X) if there is a Banach space Z such that X is linearly homeomorphic to YxZ, or equivalently if Y is linearly homeomorphic to a complemented subspace of X.

The next proposition is an immediate consequence of Corollaries 2.3 and 2.4.

8.2. PROPOSITION. If an epimorphism q>:S->T has a linear averaging operator, then C(T)\C(S).

If a homeomorphic embedding q>: 8 ->T has a linear extension operator, then C{8)\C(T).

8.3. PROPOSITION. Let a Banach space Y be a factor of G(Dn) and let C(Dn) be a factor of Y. Then Y is linearly homeomorphic to C(Dn).

Proof. First we need some notation. Let " ^ " denote the relation "is linearly homeomorphic to". If E is a Banach space and 8 a compact space, then C(8, E) denote the space of all continuous functions on 8 with values on E. The symbol (ExEx...)Co denotes the Banach space of all sequences (xn) such that xneE (n = l,2,...) and lim||a?n|| = 0. We

n

admit ||(a?B)|| = sup||a?n||. By c0 we denote the space of all scalar valued n

sequences convergent to zero. Let us observe that

(8.3.1) (0(Dn)xC(D , ,)x. . .)C o —C(Dn) for n > K0.

Indeed, let [co] denote the one-point compactificatioh of a countable discrete space and let n > K0* Then Dn is homeomorphic to Dn x [co],

§ 8. Application to linear topological classification of spaces 41

because Dn is homeomorphic to Dn x <€ and ^ is homeomorphic to ^ x [co] (as a zero-dimensional compact perfect space). Hence C(Dn)~C(Dn x [a>]). The space C(Dnx [co]) can be identified with C(Dn, 0fl>])). Since C([o>]) can be identified with the space c of all convergent sequences of scalars, we obtain C(Dn) ~ C{Dn, c). Since c ~ c 0 (cf. Banach [1], p. 182-184), the definition of C(8,E) implies that C(Bn, c) ~ C(Dn, cQ). Finally the space C{JDn,e0) can be identified with (C(Dn) x C(Dn) x ...)Co by assigning to every f(')eC(Dn, c0) the sequence of its coordinates (/«(•))• This proves (8.3.1).

Now, the assumptions of the proposition imply

(8.3.2) Y^C(Bn)xZ1 and C ( D n ) ~ Y x Z 2

for some Banach spaces Zt and Z2. Thus if n ^ K0 then (cf. Pełczyński [3], Bessaga [1]) we get

Y ~ C(Dn) x Z t ~ (C(Dn) x C{Dn) x .. .)Co X Z, ~Zxx C(Dn) x (C(Dtt) X C(Dn) X ...)Co

- Yx(C(Dn)xC(Dn)x...)Co

~Yx((YxZ 2 )x (YxZ 2 )x . . . ) C o

~Yx(Yx Yx...)Cox(Z2xZ2x...)Co

~(YxYx...)CoX(Z1xZ2x...)Co~((YxZ2)x(YxZ2)x...)Co

~(<7(Dn) x G(Bn) x ...)Co - C(JDn).

Finally if n < K0, then (8.3.2) implies that Y ~C(Dn) (because the spaces C(Dn) and Y, in this case, are of the same finite dimension).

8.4. PROPOSITION. Let 8 be an infinite compact space satisfying the following conditions: (8.4.1) 8 is either an almost Milutin space, or an almost Dugundji space, (8.4.2) 8 contains a subset homeomorphic to Dn, where n is the topological

weight of 8. Then C(8) is linearly homeomorphic to C(Dn). Proof. According to Proposition 8.3 it is enough to prove the impli

cations (8.4.1) =>C(8)\C(I)n), (8.4.2) => C{Bn)\C(8).

If S is an almost Milutin space, then, by Proposition 8.2, C(8) \C(Dn). If 8 is an almost Dugundji space, then there is a map <p: S -> I"

which admits a linear extension operator. Thus, by Proposition 8.2, C(S)\C(r). Since ln is a Milutin space (by Theorem 5.6), C(In)|C(Dn). Hence the transitivity of the relation "to be a factor" implies that C(S)\C(Dn). This completes the proof of the first implication.


The second implication is an immediate consequence of Proposition 8.2 and the fact that D" is a Dugundji space (by Theorem 6.6).

8.5. THEOREM (Milutin). Let 8 be an uncountable compact metric space. Then C{8) is linearly homeomorphic to C{^).

Proof. It follows immediately from Theorem 5.6 that every compact metric space satisfies (8.4.1).

Since 8 is an uncountable compact metric space, it contains a subset homeomorphic to ^ = Ds° (cf. Hausdorff [1], p. 136-138). Since every compact metric space is separable, its topological weight is K0. Thus 8 satisfies (8.4.2). To complete the proof we apply Proposition 8.4.

Let us recall (cf. Kelley [2] p. 42) that the first derived set #(1) of a topological space 8 is the set of all non-isolated points of 8. For ordinals a > 1 the a-th derived set of 8, denoted by $(a), is defined inductively. If a = j8 + l , then #(a) = (#(/?))(1); if a is a limit ordinal number, then

8.6. DEFINITION. Let us assign to every compact space 8 the ordinal number %(#) as follows:

if #(ct) is non-empty for all ordinals a > 1, then %(S) = 0, if 8 is finite, equivalently if 8(l) = 0 , then #(#) = the number of

elements of 8, if S(1) ^ 0, but S(a) = 0 for some a > 1, then *(#) = jS", where 0

is the smallest ordinal such that 8^ = 0 and to denotes the first infinite ordinal number.

The next corollary is an immediate consequence of Theorem 8.5, Theorem 2 of Bessaga and Pełczynski [1], and the fact that if 8 is metric and countable, then %{S) ^ 0 (cf. Bessaga and Pełczynski [1], the proof of Theorem 3).

8.7. COROLLARY. Let 8 and 8X be compact metric spaces. Then C{8) is linearly homeomorphic to (7($x) if and only if #(#) = x(^i)-

The problem of linear topological classification of spaces of continuous functions on non-metrizable compact spaces seems to be much more complicated than the metric case. There is rather narrow class of those compact spaces 8 for which C(8) is linearly homeomorphic to C(Dn) (cf. Propositions 8.11 and 8.13). However, products of compact metric spaces and compact topological groups belong to this class.

8.8. THEOREM. Let (8a)aeA be an infinite family of compact metric spaces each of which contains at least two points. Then C(P8a) is linearly homeomorphic to C(Dn) where tt = A.

Proof. According to Theorem 5.6 the space C(P8a) satisfies (8.4.1). Let Da be a fixed two-point subset of #a (aeA). Then PDa = DA can be

CleA

in a natural way homeomorphically embedded in P8a. Thus if n is the aeA

topological weight of PSa, then n > A because A is the topological weight of Dx. Since every Sa is a continuous image of Z>s° (aeA), the product PSa is a continuous image DN°Z. Therefore n < #0A = A and n = A.

aeA This shows that PSa satisfies (8.4.2). To complete the proof we apply

aeA Proposition 8.4.

8.9. THEOREM. Let G be an infinite compact group. Then C(G) is linearly homeomorphic to C{Bn), where n is the topological weight of G.

Proof. According to Proposition 8.4 and Theorem 7.5 it is enough to prove that G satisfies (8.4.2). This is shown in the next Proposition.

8.10. PROPOSITION. If G is an infinite compact group of topological weight n, then G contains a subset homeomorphic to Dn.

Proof. Our arguments are similar to those of Ivanovskii [1]. We use the following result of Pontryagin ([1], p. 327).

(P) Let G be an infinite compact group of topological weight tt. Then there exist a transfinite sequence of compact groups {Ga)a<& and group-epimorphisms <pi:Gp-> Ga such that (8.10.1) Gx is a Lie group which has at least two different points, and

G$ = G. (8.10.2) cpp

acpl = cpl for 0 < a < ($ < y < ft. (8.10.3) If 6 < ft is a limit ordinal number, then Q K6

a = {e}, where a<6

Kda = ker^f and {e} is the neutral element of Gd.

(8.10.4) kerc£+1 = K„+l is a Lie group which has at least two different points.

(8.10.5) ft is the smallest ordinal number of the power n. Let (D|)0 < |< d be a family of two-point spaces. We shall con

struct by transfinite induction homeomorphic embeddings xpa: P D% -> Ga

(0 < a < ft) such that

(8.10.6) <piyp = Wya (0 < a < 0 < ft),

where p« : P D% -> P Df is the natural projection defined by f</3 S<a

pix = (xs)s<aePDs for x = ( ą ) f < ^ P Df.

By (8.10.1) there is a homeomorphic embedding ipx: D0->Gy. Let us suppose that for 1 < £ < 6 < ft the homeomorphisms ys have been defined in such a way that (8.10.6) is satisfied for 0 < a < /? < d. We shall define \pd. Let us consider two cases.

1° d is a limit ordinal number. First, let us observe that in this case the intersection C\{<p\)~l{y$y\%} is a one-point set for each %ePDs.

Indeed, using (8.10.2) and (8.10.6) we get for 0 < a < 0 < d,

Thus since (c?J)-1 {f^pôc} are closed subsets of a compact group Gd, the intersection n(94)-1{v*2,ea0' ^s non-empty. Now, if yx and #2

a r e m

f l f a e ) - 1 ^ ^ } * then q>iyi = <pd(y2 = VsPi30 f o r £ < <5. Hence y ^ 1

«Zf = kerc?f for £ < <5. Thus (8.10.3) implies that ^ = y2. We define ip6x as the unique element of the intersection p) (^) _ 1 {$&*%}

for xePDc. Clearly q>daip6x = tpap

6ax for xePl)s and for 0 < a < 6. Let

f<<5 !<(5

#' and a?" be in PDS. If a?' x", then for some a < d, p6ax =£ $!#" and

therefore v'aâ^' ^ VaPW (because % is a homeomorphic embedding). Thus ipdx

f =fc ipdx". Finally, let y= ydxeGd. Let V be an open neighbourhood of y. I t follows from (8.10.3) that H Vî = M- Since (8.10.2) implies

K6a => Kp for 0 < a < § < 6, there is an a0 such that yK*0 c V. Thus,

since the operation of multiplication in a compact group is a continuous function of two variables, there is a neighbourhood Vx of y such that VxK

óa{i c V. Let us set

u = (%0póaor

1{<p6aov1K*ao},

Since (p*0 is a group epimorphism acting between two compact groups, <Pa0 is an open map (Hewitt and Eoss [1]). Thus U is open. Since ipao°Pa0

= <pdaoy>d and since VxK

daQ = ( A r ' ^ I F r A ) } , we get

This proves the continuity of ipd. Therefore ipd is a continuous one-to-one map from one compact space into another. Hence ipd is a homeomorphic embedding.

2° 6 = a + 1. Since K°+1 is a Lie group, it follows from a result of Gleason (cf. e.g. Montgomery and Zippin [1], p. 221) that the epimorphism <p°+1: Ga+1 -^ Ga is a local projection, i.e. for every yeGa there is a neighbourhood Vy of y and a homeomorphic embedding ry: Fy x X l ^ + 1 - ^ G a + 1 such that (Pa+1rv(y',1c) = y' for fccZ^1 and y ' e7 y . Since y a : PD$ ->Ga is a homeomorphic embedding, tpa( PDS) is a ZerO-dimensional compact subset of Ga. Therefore there exist in Ga a finite open cover {7<}£x of mutually disjoint sets, and homeomorphic embeddings

n: Vi x Kaa

+1 -> Ga+1, with (paa

+1n(y,Jc) =y for JceKaa+l and yeVt

(* = 1, 2 , . . . , N). Let us set

r(y,k)=Ti(y,Jc) for JceKZ+1 and yeVi (i = 1, 2 , . . . , JV).

Clearly T is a homeomorphic embedding from {J F* x if°+ 1 into Ga+i.

By (8.10.4) Z° + 1 has at least two different points, say k± and Tc2. We shall identify the set Da with the set {kx, Tc2}. Let us set

Va+\® = r(ipapaa

+1x, Jcv) for x = (oo^)s<a€ P Ds with xa = fc„ (v = 1, 2).

I t is easily seen that ya+1 is a homeomorphic embedding from P Ds

into Ga+i such that <p"+1y>a+i — V>aPl+1- By the inductive hypothesis <PeV>a = WsPi for 0 < £ < a. Therefore (8.10.2) implies <$+1 ya+1 = c) |^+ 1 Va+i = 9>iV«f£+1 = VW»Stf+1 = V«P?+1 f o r 0 < £ < a.

To complete the proof of the Proposition let us observe that tp& is, by (8.10.5) and (8.10.1), a homeomorphic embedding of PD$ — Dn into G& = G. ą<&

E e m a r k . Proposition 8.10 and Theorem 8.9 remain valid for arbitrary coset space of a compact group. The proofs are analogous to those for the group. We use a modified version of (P) in which the groups Ga are replaced by coset spaces and group epimorphisms by coset-space-epimorphisms which are defined as follows. Let G' and G" be compact groups, H', H" its closed subgroups. Let <p:G' -+ G" be a group epi-morphism and let <p(H') = H". Then the coset-space-epimorphism 0 : G' \R' -»»G" \R induced by (p is the unique map such that ip"cp = <Żty', where y': G' -+G'/H' and y>" : G" ^G"\R" are natural maps (cf. Definition 7.3). We modify (P) as follows. Let G be a compact group and let B. be its closed subgroup. Let {Ga)1<a<d and (gjf)i<a</»<* be as in (P). Let us set Ha = <p$(H) for 1 < a < 0, and H& = H. I t follows from (8.10.2) that Ea = (pp

a(Sp) for 1 < a GajEa induced by rf are well defined for 0 < a < (} < #. We withdraw from the sequence (GaIHa)1<asą& the intervals a < y < /? such that 0f are one-to-one. The remaining coset spaces we enumerate by succesive ordinal numbers from 1 to some #' in such a way that we preserve the initial order. Similarly we enumerate coset-space-epimorphisms. For the new sequences of coset spaces and coset-space-epimorphisms we repeat the same transfinite construction as in the proof of Proposition 8.10.

The next results indicate that for a non-metrizable compact space 8, in general, C(S) is not linearly homeomorphic to C(Dm), where m is the topological weight of S.

Let X be a Banach space. Let us consider the following two properties.

(a) If a linear subspace Y of X is linearly homeomorphic to c0, then Y is complemented in X,

(b) if W is a set in X with the property that every sequence in W contains a weak Cauchy subsequence, then W is separable.

8.11. PROPOSITION. If a Banach space X is linearly homeomorphic to a closed linear subspace of C(Dm), then X possesses both properties (a) and (b).

Proof. First remark that the properties (a) and (b) are linear homeo-morphism invariants and are hereditary in the sense that if a space possesses one of these properties, then each of its closed linear subspaces possesses the same property. Therefore it is enough to show that for every cardinal number tn the space C(Dm) possesses the properties (a) and (b). For the property (a) this is shown in Engelking and Pełczyński [1], Lemma 6. We shall show here that C(Dm) possesses the property (b). Let L2(Dm) denote the Hilbert space of all square integrable functions with respect to the usual product measure X on Dm. Let I: C(Dm) -> L2(Dm) denote the natural embedding (i.e. 1(f) is the A-equivalence class of/). I t follows immediately from the characterization of the weak convergence in G(8) (cf. Dunford and Schwartz [1], p. 265) and from the Lebesgue dominated convergence principle that I takes weak Cauchy sequences in C{Bm) into Cauchy sequences in the norm topology of L2(Dm). Hence, if W is a subset of C(Dm) with the property described in (b), then IW is compact in L2(Bm) and therefore IW is separable. To complete the proof it is enough to use the following consequence of the Peter-Weil theorem applied to the group Dm (cf. Pontryagin [1], p. 23, Weil [1], p. 74-76). If G is a compact topological group, then every separable set B in L2(G) belongs to the smallest closed subspace of L2(G) generated by a sequence of finite dimensional subspaces (En); each JEn is spanned by entiers c$${ •) of some irreducible representation of G (n = 1, 2, . . . ) . Moreover, if / is continuous and feB, then / is in the uniform closure of the linear subspace spanned by the sequence (En).

8.12. COROLLARY. If 8 is either a dyadic space (less generally, if 8 is an almost Milutin space), or if 8 is an almost Dugundji space, then C(8) in linearly homeomorphic to a subspace of C(Dm), where m is the topological weight of 8. Hence G(S) has both properties (a) and (b).

Proof. I t follows from a result of Sanin [1] (cf. Engelking and Pełczyński [1], Theorem 1) that for every dyadic space 8 of topological weight m there is an epimorphism C(Dm) is an isometric embedding.

If 8 is an almost Dugundji space of topological weight m, then there is a homeomorphic embedding xp: 8 -> Im which has a linear extension operator. Hence, by Proposition 8.2 and Theorem 8.8, G(8) is linearly homeomorphic to a complemented subspace of C(Dm).

8.13. COROLLARY. Let 8 be an non-metrizable compact space having one of the following properties (8.13.1) 8 is extremally disconnected (cf. Kelley [2], p. 216), (8.13.2) 8 is scattered (clairseme in French, KuratowsM [1], p. 95), (8.13.3) S does not satisfy the a-chain condition, i.e. there is in 8 an uncount

able family of open pairwise disjoint non-empty sets. Then C{S) is not linearly homeomorphic to any subspace of C(Dm). Proof. If 8 is non-metrizable and extremally disconnected, then

C(S) does not have property (a). (For the proof see Engclking and Peł-czyński [1], p. 61.)

If 8 is a non-metrizable scattered space, then the nnit ball of C(8) is not separable but every sequence in the unit ball contains a weak Cauchy subsequence (cf. Pełczyński and Semadeni [1], p. 214). Hence C{S) does not have property (b).

If 8 contains an uncountably family (Ua)afA of open non-empty and pairwise disjoint subsets, then one can construct a family (fa)aeA of continuous functions on 8 such that fa(s) = 0 for S€S\Ua and ||/a|| = 1 (aeA). Let W = \J {fa}. Then W is an non-separable subset of C{S), be-

aeA cause if a =£ b, then \\fa—fb\\ = 1- On the other hand every sequence of different elements of W weakly converges to zero. Thus C(8) does not have property (b).

8.14. COROLLARY. If S is a non-metrizable compact space having one of the properties (8.13.1)-(8.13.3), then 8 is neither dyadic, nor an almost Dugundji space.

§ 9 . LINEAR AVERAGING OPERATORS AND PROJECTIONS ONTO SPACES OF CONTINUOUS FUNCTIONS

9.1. DEFINITION. Let A > 1. A Banach space X (a separable Banach space X) is called a tyx space (resp. a ^ space) if for every Banach space Y (separable Banach space Y resp.) and every isometric embedding u : X -> Y there is a projection from Y onto uX of norm ^ X.

It is well known (cf. Day [1], p. 95, Kelley [3], Cohen [1], Hasumi [1]) that C(S) is a cp1 space if and only if the compact space 8 is extremally disconnected.

9.2. DEFINITION. A map <p: A -> B is called irreducible if q>(A) = B and <p{F) ^ B for every proper closed subset F of A. We recall (cf. Gleason [1]) that for every compact space T there is an irreducible map cpT : GT -> T from an extremally disconnected compact space GT onto T. The map yT

will be called a Gleason epimorphism onto T. I t is unique up to a homeo-morphism of GT. Precisely, if cp'T : G'T -> T and yi '> G'T -> T" are Gleason epimorphisms onto T, then there is a homeomorphism y> from G'T onto G'T such that c?y = y)cpTip~l.

9.3. PROPOSITION. Let X > 1. Then for every compact space T the following conditions are equivalent: (9.3.1) Every epimorphism T has a linear averaging

operator of norm < X. (9.3.3) C(T) is a >A space.

Proof. (9.3.1) => (9.3.2). This implication is trivial. (9.3.2) => (9.3.3). Combining (9.3.2) with Corollary 2.3 we infer that

there exists a projection of norm < X from C(GT) onto its subspace <pr[C(T)] isometric to C{T). Since GT is extremally disconnected, C{GT) is a °Pa space. Therefore C(T) is a °PA space, because it is isometric to the range of a projection of norm < X from a ^ space (cf. Goodner [1], Day [1], p. 99).

(9.3.3) => (9.3.1). Let y: 8 -> T be an epimorphism. Then C(T) is isometric to the subspace <p°[C(T)] of C(8). Therefore (9.3.3) implies that there is a projection from C(S) onto (p°\C{T)] of norm < X. To complete the proof we use Corollary 2.3.

9.4. COBOLLARY. A compact space T is extremally disconnected if and only if every epimorphism from an arbitrary compact space onto T admits a regular averaging operator.

I t is well known (cf. Grothendieck [2], Amir [1], [2], Pełczynski [3], Pełczynski and Sudakov [1], Semadeni [1], Lindenstrauss [1], Nachbin [1]) that in general C{T) is not a ^ space for any X (1 < X < +oo). For such a compact space T the Gleason epimorphism q>T'.GT~>T does not possesses linear averaging operators.

9.5. DEFINITION. A map <p: 8 ->T is said to be of order n, in symbols o(c>) = n, if n is the least integer (if such an integer exists) such that for every t in T the inverse image (p~l(t) consists of at most n points.

Isbell and Semadeni [1] (cf. also Amir [1]) examined the relationships between the order of the Gleason epimorphism onto T and the projection constant,

PC{T) = inf {X 1 : CAT) is a ^ space}.

Combining their Theorem 1 with Proposition 9.3 and Corollary 9.4 we get(2)

(2) Isbell and Semadeni considered the spaces of real-valued functions. Combining their result with Proposition 2.9 we get the same result for complex-valued functions.

§ 9. Linear .averaging operators and projections 49

9.6. Let the Gleason epimorphism <pT: QT ~>T have a linear averaging operator, say u. Then (9.6.1) if I N K 3, then o{<pT) < 2 ( 3 - \\u\\)-1; (9.6.2) if \\u\\ < 2, then there exists a regular averaging operator for (pT;

equivalently T is extremally disconnected. Our next step will be Proposition 9.8 which enables us to give an

alternative proof of Amir's characterization of those compact metric spaces T for which C(T) are ^ spaces for some I > 1 (cf. Amir [1], [2]). We shall consider a class of epimorphisms of order 2 which do not have linear averaging operators. The typical example of such an epimorphism is the Cantor map h :& -> I (see Lemma 5.5 for the definition).

9.7. DEFINITION. Let 8 and T be compact metric spaces and let d(-, •) denote the metric of S. A map <p: S -> T is said to be of Cantor type if (9.7.1) (p is an epimorphism; (9.7.2) o(c>)=2; (9.7.3) if ó > 0 , then the set F9(6) is finite, where

F9(d) = {se8: d(s, s') > 6 and (p(s) = <p(s') for some s' in S}.

E e m a r k . One can restate (9.7.3) in purely topological terms (by eliminating the metric d(', •) from the definition of F9(d)) as follows: (9.7.4) if % = {Ua}aeA is an open cover of a compact space S, then the

set F6{<%) is finite, where

FsW = {seS: there is s'eS such that <p(s) = (p{s') but both s and s' do not belong to the same Ua for any aeA}.

Eeplacing (9.7.3) by (9.7.4) one can extend Definition 9.7 to the case of an arbitrary compact spaces.

9.8. PROPOSITION. Let S and T be metric spaces and let cp:8->T be of Cantor type. Let

Fl:]=F0

F[k] = {seFv : there is s'eFv such that s ^ s', <p{s) = q>{s'), both s and s' are limit points of Flk~^} (k = 1, 2, ...).

If for some positive integer m the set F[m^ is non-empty and if u: C(8) -> C(T) is a linear averaging operator for <p, then \\u\\ > m.

The proof of Proposition 9.8 requires some preliminary results. We begin with a generalization of the Stone-Weierstrass theorem (Proposition 9.9) which may be of interest for other purposes. In fact we shall need only Corollary 9.10 which is a very particular case of Proposition 9.9 and which can be proved directly, more simply than the general case.

9.9. PROPOSITION. Let S and T be compact spaces and let 9?: 8 -> T be an epimorpMsm. Then for every f in C{S)

(9.9.1) Q(f,<P°[C(T)]) = inf 1|/—flro H = supr(t), geC(T) UT

where r(t) is the radius of the smallest circle in the complex plane circumscribed on the set /(^ -1(i)).

Proof. Throughout this proof / is a fixed function in C(S). Let

B = {(*eM(S) : \\fi\\ < 1, /A<p°g) = 0 for geC(T)}.

Then B is a weak-star compact and convex subset of M(S). Therefore, by the Krein-Milman theorem (cf. Dunford and Schwartz [1], p. 440), B is the weak-star closure of the convex hull of the set Be of all extreme points of B. ITence sup|//(/)| = sup| /a(/)|. Combining this identity with the IEahn-Ba-

nach extension principle (cf. Dunford and Schwartz [1], pp. 62-65) we get

(9.9.2) Q(f,<P°[C(T)-]) = *vp\p(f)\. HtBe

Next we shall show that if peB6, then /u, is concentrated on <p~l{t) for some t in T. Indeed, let //eJ5 and let there exist in T two different points, say t0 and tlt such that the carrier of \x meets both sets 9?~1(<0) and 9?_1(*i). Let us choose g in C{T) such that 0 < g < 1 and g(t) = j for t in some neighbourhood, say Uj, of t,- (j = 0,1). Let V, = cp~1{Uj). Let 11 be the unique non-negative element in M(8) such that \/i\ = hfi for some unimodular Borel function h on S (for any Borel function Jc, kjj, denotes the usual product of function and measure; as a functional on C(S), kp(f) = fk(8)f(8)p(d8) for / in C(S)). Clearly |^|(F7) > 0, because

Vj r^ <p~l{tj) ^ 0 and |^| is concentrated on the same set as fj,. Let fi0

= {ls — (p0g)(* and ^ = <p°gfi. Then p = /*0 + /*i; \\/ij\\ > \[*\(Vj) > 0; /ujeB (j = 0,1). Therefore \i does not belong to Be. Hence for each ^ in Be there is a unique t^ in T such that ^ is concentrated on cp~ 1(tfi).

Now, for e > 0 let us choose //e in I?e such that

(9.9.3) A i e ( / )>sup | / u( / ) | -e .

Let tB = tM and let ze and r£ denote the centre and the radius respectively of the smallest circle in the complex plane circumscribed on the set f[cp~l (te)). It is easily seen that for every non-empty set in the plane there exists the unique circle with that property. Since fie(ls) — 0 and \\fie\\ = 1 (because [ieeBe) and since /ue is concentrated on <p~l{te), we have

(9.9.4) \fie(f)\ = I M / - V l s ) l = j J (m-se)fi,(d8)\

< \\f*e\\ sup |/(s) —ze| =re.

§ 9. Linear averaging operators and projections 51

Thus, letting e tend to zero, and combining (9.9.2) with (9.9.3) and (9.9.4) we get

(9.9.5) Q(f,<P°[C(T)])^mVr(t). UT

Now, let us fix i in T and let geC(T). Then the definition of r(t) implies that there is s in cp~l{t) (depending on g and t) such that

II/-V0II >\m-(<p°g)(s)\ = \f(s)-g(t)\>r(t). Hence

(9.9.6) Q(f, <p°[C(T)]) = inf \\f-<p°g\\ > supr(*). geC(T) UT

Clearly (9.9.5) and (9.9.6) imply (9.9.1). That completes the proof. 9.10. COROLLARY. If y: S -> T is an epimorphism and o(cp) = 2 ,

then for every f in G(S)

(9.10.1) Q(f, <p°[C(T)]) = s u p i \f(s')-f(s")\,

where the supremum is taken over all pairs (s', s") in SxS such that <P(S') = <p{8").

9.11. LEMMA. Let cp: S ->T he of Cantor type and let the set Fę be infinite. Then the quotient space C(S)j<p°[C{T)'] is isometric to the space c0. The isometry w: (C (S) f<p° [C (T)]) -> c0 is defined by

(9.11.1) w([f]) = if{S2k~l)~f{S2k)) for [J]*C(S)l<p0[C(T)] \ * /&=1,2,...

where [/] denotes the coset class of feC(S), and the sequence (sk) consists of all elements of Fę ordered in such a way that <p(s2k-i) = <p(s2k) for & = 1, 2, ...

Proof. It follows from (9.7.3) that the set F9 (defined in the statement of Proposition 9.8) is at most countable. Since Fv is infinite, we can order all elements of Fv in an infinite sequence (s&)&=i,2,... • Moreover, by (9.7.2), it is possible to order it in such a way that <p(s2k^i) = <p(hk) for Jc = 1,2, ... Let us observe that for every/ in C(S) the right side of (9.11.1) depends only on the coset class [/] of / . Indeed, if we choose /x and / 2 in C(S) such that [/x] = [/2], then fx—f2 = <p°g for some geC(T). Hence

( / l - / 8 ) ( * 2 * - l ) = ( / l - / 2 ) ( * 2 * ) = 9(h),

where tk = 9>(s2*-i) = <p(*ik) (& = 1, 2 , . . . ) . Therefore

/ l(*2*-l) —/l(*«*) = Afefc- l ) —/2(*2ft) (* = 1 , 2 , . . .) .

Next we will check that w([f]) belongs to c0 for every feC{8). Indeed, since 8 is compact, every / in C(8) is uniformly continuous. Therefore for a given T\ > 0 there is d = d{rj,f) > 0 such that if d(s',s") < 6, then |/(*')—/(*") I < f] for any s' and *" in 8. Since by (9.7.3) the set Fv(d) is finite, there is an index N such that if n>N, then sn4F<p(S). Therefore, if k>N, then d(8ik_lf s2k) < d, and consequently |/(s2&-i) —

f(s2k-i) —f{s2k) —fis2k)\ < V- Hence lim = 0, equivalently w([f])cc0.

k 2 Next we will establish that w is a linear isometry. By the definition

of the norm in the quotient space, we have

ll[/]| |= inf \\f-<P°g\\ = Q(f,<P°[C(T)]).

Thus, by Corollary 9.10,

ll[f]||= SUp lr\f(8*_1)-f(8*)\ = \\u>([f])\\.

Hence w is an isometry. The linearity of w is obvious. Finally we will show that the range of w is the whole space c0. For

©ach pair (k, n) of positive integers we define f[n) in C(8) such that

>

Clearly

and

ffi^Sm) = 0 for m < n and m ^ 2& — 1 , m ^2k.

Htón)])ll=i

1 , v , x (0 for m ^ k, lim - [/?> (s^) -fp (s2m)] =

» ^ 1 for m = k.

Thus, by the characterization of weak convergence in e0 (cf. Dunford-Schwartz [1], p. 339), the sequence (w([./iB)]))~_i weakly converges in c0

to the fc-th unit vector, say ek (k = 1, 2 , . . . ) . Since w is a linear isometry, the range of w is a closed convex subset of e0. Consequently it is weakly closed. Therefore ek belongs to the range of w (k = 1, 2, . . . ) . Thus the range of w contains the smallest linear subspace spanned by all unit vectors, i.e. the whole space c0. That completes the proof.

Proof of P r o p o s i t i o n 9.8. By Corollary 2.3 it is enough to show that if P is a projection from C(8) onto <p°[C(T)l then ||P|| ^ m.

We observe first that if P is a projection from a Banach space X onto its subspace Y and if v denotes the restriction to kerP of the quotient

§ 9. Linear averaging opeartors and projections 53

map x -> [x] from X onto X/Y, then v is a linear homeomorphism of kerP onto X/Y such that

(9.8.1) INI > Ml X l + HPIirMNI f o r #*kerP.

The left-hand side inequality is clear, because ||#|| > ||[a?]|| = \\vx\\ for all x in ker P . Now, f or e > 0 and for ^ekerP let x' e[x] be choosen in such a way that ||aj'|| < ||[a?]|| + e = ||w»|| + £. Clearly x' — xeY and Pa? = 0. Hence x = x'-Px'. Therefore INI < ||a?'|| + ||P|| ||a?'||. Thus (1 + HPIir'INI <\\x'\\ < ||t«p|| + £. Since this is true for all e > 0, we get the right-hand side inequality of (9.8.1).

We shall use the previous observation in the case where X = C(S) and Y = <p°[C(T)]. In this case, by Lemma 9.11, the quotient space XjY is isometric to c0. Let en denote the n-th unit vector e0. Let us set

fn = v-lw~len {n = 1,2,...),

where w is defined by (9.11.1). Then, by the definition of v, /MekerP. Furthermore, by (9.11.1),

(9.8.2) fn(S2n-l)-fn(s2n) = 2 (n = 1 , 2 , . . . ) ,

where the sequence (s»)n=i,2,... is defined in the statement of Lemma 9.11. Let us set

for |/(«')l > l/(«")l „ , ,„ e e . „,-. ({s', 8")eSxS: fcC{S)),

for |/(«")l >!/(«') I

<?(*',*",/) =

.£(*, d) = {s'eS:d{s,s') < 6} for d > 0 and for seS,

where d( •, •) denotes the metric of 8. Let e > 0. Let us choose an index nx in such a way that 8M cPjp (the

existence of nx follows from the assumption that P^m] is non-empty). Let qx = gfaaij-iy hn^fn^- Pick óx > 0 so that if d(s', s") < dlt then l/»!(*') —/»i(*")I < e/(w + l ) . Next define n2 so that

S e ^ - 1 ' ^ K(ql,2-1d1) and d ^ - i , s2n2) < 2"1*!.

(Such n2 exists, because q1eF[™] and therefore it is the limit of a sequence of different points belonging to F[™~1]. Thus in K{ql,2~1d1) there are infinitely many points belonging to F[™~1]. But only finite number of them belong to Pc,(2~1ó1) (by (9.7.3)). Next we pick a positive <52 < dt so that if d(s', s") < <52, then |/„2(*')-/na(*")l < e/(w + l) for (s', s")€SxS. We put q2 = q{s2n2-i, s2n2,fn2)- Repeating this procedure we define indue-

tively a finite sequences of indices nx, n2, . . . , nm+l and of positive numbers 6X > <52 > .. . > dm such that

(9.8.3) s2nveF^-v+V,

(9.8.4) if d(s', s") < dv, then \L (s')-L(s") | < m + 1

for (s',s") in SxS (v = 1, 2, . . . , ro), (9.8.5) d(«a%_i, *»>) < 2~1(5V_1 (v = 2 , 3 , . . . , m + 1),

(9.8.6) ^ e f l t f f o / . * 2 " 1 ^ ) for * = 2 , 3 , . . . , m + l ,

where #„ = g(s2^-i ,s2riv,fv) for y = 1, 2, . . . , m+1. It follows from (9.8.5) and (9.8.6) that qm+1eK(qv, 2 - 1 dv) for v =

1 ,2 , . . . , m. Hence by (9.8.4)

for v = 1 ,2 , . . . , m + 1.

Thus m+i

l/n,(?m+i) /n>(?v)l < * v m + 1

m+i m+i m+i

£ \fnv{qm+i)\ > £ \fn>(Qv)\-£ \fnp(qv)-faiCm+i)I > J ^ l / ^ ( & ) | - « -v=l v=l v=l v—1

On the other hand, combining (9.8.2) with the definition of the points q we infer that \f„v(qv)\ = max(|/„t((s2niF_I)|, \fn(s2np)\) > 1 (v = 1, 2, . . . , m + 1). Hence

m+i

£ l/«»(flV+i)l ^™ + l — e. v = l

Now we pick complex numbers av such that |d„| = 1 and avfnv(<lm+i) = \fn,(qm+i)\ (v = 1,2, . . . , m + l) . Then

m+i m+i m+i

(9.8.7) || J ^ avfrĄ > | ^ avfnj>{qm+1) | = ^ | / ^ ( f f „ + 1 ) | > m + 1 — e . y = l » = 1 y = l

m + 1

Let x = ]? avfnv> Since w is an isometry, the identity vfn = w - 16n v = l

(w = 1, 2, ...) together with the well-known propertes of the unit vectors in c0 imply

m+i m+i

1 = max|a„| = V avev\\ = w - 1 ! ^ 1 a»ev) i = 11 11-" v=l v=^l

On the other hand, combining (9.8.1) with (9.8.7) we get

i = wixi+iiPiir^Ni = 7+Hpir

9. Linear averaging operators and projections 55

Let £ tend to zero, and we get 1 + ||P|| > w + l . Hence ||P|| > m. That completes the proof.

9.12. COROLLARY. There is no linear averaging operator for the Cantor oo

map h-.tf^I where h = ^2~x ^ for £ = (&)€#. i=l

Proof. Clearly h is of Cantor type. The set Fh consists of all f = (&)e# such that either almost all & = 0, or almost all ff = 1, and 0 # £ ^ 1, (If we represent ^ as the subset of I consisting of all triodic fractions with figures equal either 0 or 2, then Fh consists of all end-points of all open intervals being components of 1\<S.) Therefore F[™] = Fh (m = 1,2, ...). Thus, according to Proposition 9.8, there is no linear averaging operator for h.

9.13. THEOREM (Amir [1], [2]). If S is an infinite compact metric space, then the following conditions are equivalent: (9.13.1) Some derived set of S (cf. Kelley [1], p. 42) of finite order is empty; (9.13.2) C(S) is linearly homeomorphic to the space c of all convergent se

quences ; (9.13.3) G{8) is a ^ space for some ). ^ 1.

Proof. We recall that if # is an ordinal, then [•&] denotes the space of all ordinals < # endowed with the usual order topology (cf. Kelley [2], pp. 57, 266-271).

(9.13.1) => (9.13.2). Let k be the first ordinal with the property that SSk) = 0 . Since S is infinite and satisfies (9.13.1), 1 < k < w. Therefore X(S) = Jc(° = 2m = %([>]). (For the definition of x{8) see 8.6). Thus, by Corollary 8.7, C(S) is linearly homeomorphic to <7([co]). Clearly O ([to]) can be identified with the space c.

(9.13.2) => (9.13.3). By a theorem of Sobczyk [1] (see also Pełczyński [3], p. 217, Dean [2]) the space c0of all scalar-valued sequences convergent to zero is a ^ space. Since c is linearly homeomorphic to c0 (cf. Banach [1], pp. 182-184), (9.13.2) implies that C(8) is linearly homeomorphic to c0. To complete the proof of the implication it is enough to use the following general observation (cf. Pełczyński [3], Proposition 1).

(*) If X is a ty'i space for some A > 1 and if Xx is linearly homeomorphic to X, then Xx is a ^ space for some /J, > 1.

Proof. We have to show that Xx regarded as a subspace of an arbi trary separable Banach space, say Yx, admits a projection nx : Yx -> Xx (onto) with \\nx\\ < fi where /J, depends only on Xx. Let 11 - j]x denote the original norm in Yx. By the assumption there is a linear homeomorphism, say v, from X onto Xx (which may be regarded also as a linear homeomorphic embedding of X into Yx). Hence there are positive constants a = H^-1!!-1 and b = \\v\\ such that a||#|| < \\vx\\x <6||a?|| for xeX. We

introduce in Yx the new norm || • || being the Minkowski functional of the convex body W, where

W = {zeYx: z — tvxJr(l — t)yx, where 0 0 : — eW\ for yeYx.

Let Y denote the space Yx under the new norm ||-||. Then Y is linearly homeomorphic to Yx. Precisely, one can easily check that the identity map i : Y -> Yx with iy = y for yeY is a linear homeomorphism satisfying the inequalities

fl|MKIl*y|li<&lly|| for y€Y.

Furthermore i~1v is an isometric embedding of X into Y. Since X is a %>[ space, there is a projection n : Y -> X (onto) with ||TC|| < X. One can easily check that nx = ini~l is the desired projection from Yx onto Xx with \\nx\\ ^Xba~l = X\\v\\ \\o~%

non (9.13.1) => non (9.13.3). This implication is an immediate consequence of the following three lemmas

9.14. LEMMA. If X is a ^C)[ space and if Z is a complemented subspace of X, then Z is a ^p[ space.

9.15. LEMMA. If 8 is a compact metric space which does not satisfy (9.13.1), i.e. if all derived sets 8(k) are non-empty for Tc < co, then C(S) contains a complemented subspace isometric to 0([a)t0]).

9.16. LEMMA. C([CO'°]) is not a ^ space for any X ^ 1. Proof of L e m m a 9.14. Let Zx be a complementary subspace to Z

in X, and let p and px be projections from X onto Z and onto Zx respectively such that kerp = Zx and k e r ^ = Z. Let u be an isometric embedding of Z into a separable Banach space Y. Then v, with vx — (upx, pxx) for xeX is a linear homeomorphism from X into Y x Zx. Since X is a ^ space, (*) implies (cf. the proof of the previous implication) that vX is a 'V'n space for [i = ||v|[||v_1||/l. Hence there is a projection n from YxZx

onto vX with \\JI\\ < jn. Now one can easily verify that qnj is the desired projection from Y onto uZ. where j is the natural isometric embedding of Y into YxZx (i.e. jy = (y, 0) for ye Y) and q is the natural projection from YxZx onto Y (i.e. q[(y, zx)] = y for (y, zx)eYxZx). That completes the proof.

Proof of Lemma 9.15. We observe first that it is enough to show that S contains a subset, say S0, homeomorphic to [a*03]. Indeed, that implies that there is a regular extension operator from C{S0) into C(S)

§ 9. Linear averaging operators and projections 57

(because [of3] being metrizable is, by Theorem 6.6, a Dugundji space). Thus, by Corollary 2.4, C([com]) is isometric to a complemented subspace of C{8).

Let us consider two cases. 1° 8 is uncountable. Then, since 8 is metric, it contains a homeomorphic copy of the Cantor discontinuum (cf. Hausdorff [1], p. 136-138) and therefore a homeomorphic copy of every zero-dimensional compact metric space, in particular a copy of [a)*0]. 2° S is countable. Then, by a result of Mazurkiewicz and Sierpiński (cf. Kuratowski [2], p. 58), Sh homeomorphic to the space [cif-n], where ft is the last countable ordinal number such that the derived set S^&) is nonempty and n is the number of elements in S^ (clearly 8{&) is finite). By the assumption of the Lemma, ft ^ co. Therefore af-n^af^ co**. Hence the space 8 homeomorphic to [co^-n] contains a component homeomorphic to [a/0]. That completes the proof.

Proof of L e m m a 9.16. First we shall define inductively maps <pn: [con-2~\ -> [con] of Cantor type such that

(9.16.1) F™ = {con}^{con-2}, (9.16.2) <pm{x) = <pn{x) for x < com-2 (1 < m < n; n = 1, 2, . . . ) .

Let us put 9>i(<*>) = 9>i(ft>'2) = co,

9?1(2fc — 1) = <px(2k) = 2k —1, 9>1(fl> + 2fc —1) = <p1(co + 2fc) = 2k (k = 1 ,2, . . . ) .

Clearly the map yx: [ w 2 ] -> [co] is of Cantor type and F^\ = {co} w {co -2}. Let us suppose that for 1 < m < n the maps of Cantor type satis

fying (9.16.1) and (9.16.2) have been already defined. Let us set

(pn(oj -2) = <pn(oj ) = co ,

(9.16.3) (pn[con-l{2k-2) + x) = (pn(con-1(2k~l) + x) ^con-1(2k-2) + <pn_1(x),

cpn(con+con-1(2k-2) + x) = <pn(con+con-1{2k-l) + x) = ojn-1(2k-l) + (pn_1(x)

for 0 <x ^o"-1 {k = 1 ,2 , . . . ) .

If follows from inductive hypothesis and (9.16.3) that OO 0 0

4 T 1 ] => U {fi)""1*} v U {a>nĄ-oon-lk}. k=l k=l

Therefore, by definition,

Observe that if cp: 8 -> T is any map of Cantor type, then Ąw] c 8^n), where #(w) denotes the n-th derived set of 8. Since the n-th derived set of the space [ojn-2] is exactly the two-point set {a/1} w {a/1 -2}, we have the inclusion ĄnJ c {of1} w {«/*-2}. Therefore P g = {a>n} w {con-2}. Clearly 9?w(#) = <pw-i(^) for 0 < x < con-1<2. Furthermore if o(9?n_i) = 2, then o(y>n) = 2. Finally, it is easy to verify that if <pn_x satisfies (9.7.3), then <pn does. Hence <pn is of Cantor type. That completes the induction.

Now, let us define the epimorphism <pa : [a/0] -> [cow] by

0) ) = 0) ,

<Pm(x) = <Pn(®) f ° r 0 < x ^ ojn-2 (n = 1,2, ...).

Since all cpn are of Cantor type, (9.16.2) implies that <pa is also of Cantor type. By (9.16.1), F[£] is not empty for n = 1,2, . . . Thus, by Proposition 9.8, the map <pm : [ojm] -> [of3] has no linear averaging operators. Equivalently (by Corollary 2.3) there is no projection from C([OJU>]) onto its subspace ri[C,([cott>])] isometric to C([a>m]). Therefore C([a/°]) is not a ^ space for any I ^ 1. That completes the proof.

NOTES AND REMARKS

Ad § 2. The existence of an extension for an arbitrary real valued continuous function defined on a closed subset of a metric space M was proved by Tietze [1] (cf. also Hausdorff [2]), and generalized by Urysohn [1] to the case of normal topological spaces. The case where M is an n-dimensional Euclidean space was treated earlier by several authors; L. E. J. Brouwer [1], § 4, [2], H. Bohr (see Carathóodory [1], § 541-542), Carathśodory [2], §§ 541-542, De la Valle Poussin [1] p. 319, and Le-besgue [1], p. 379.

The extension problem for continuous function is closely related to several topological theories. After Borsuk's papers [2], [3], it became clear that many extension problems for continuous functions can be restated in terms of the theory of retracts. We refer the reader to the books of Borsuk [4] and Hu [2] and to references in this books for further information. Borsuk's homotopy extension theorem and other connections between extension theory and homotopy theory can be found in Hu [1] and references given there. For the relation to obstruction theory we refer the reader to the pioneer paper of Eilenberg [1] and to the book of Steenrod [1]. In E. Michael [4], [5], [6], [7] the connection between extension theorems and selection theorems is discussed. Roughly speaking, every selection theorem contains as a special case some extension theorem. Several topological notions previously intrinsically defined are either equivalent to some extension or selection properties, e.g. the characterization of dimension (cf. Hurewicz and Wallman [1], p. 83 in the metric case and Nagata [1] in the more general case), collectionwise normality (cf. Dowker [1], Bing [1]). For other similar results see Arens [1], Hanner [1], [2], [3], Michael [1], [5].

Linear extension operators are usually called operators of simultaneous extension. The first result in the theory is due to Borsuk [1]. For further results see Kakutani [1], Dugundji [1], who proved the existence of regular linear operator of extension for arbitrary closed subset of metric space, Arens [1], Michael [1], Pełczynski [1], [2], Borges [1], Michael and Pełczynski [1], and Semadeni's expository papers [1], [2]. Clearly linear extension operators, as well as linear exaves, can be defined for

arbitrary topological spaces and for special classes of functions on these spaces (cf. Pełczyński [1]).

If cp : 8 -*• T is a homeomorphic embedding and S, T are non-metriz-able compact spaces, then in general there is no linear extension operators for cp (Arens [1], Day [2], Semadeni [1], Corson and Lindenstrauss [1], cf. also Corollary 8.14 in the present paper. The related examples are also given by Michael [1], Klee [1]).

Eecently Corson and Lindenstrauss [1] constructed for Jc — 1, 2, . . . a homeomorphic embedding <pk : Sk -> Tk (Sk, Tk — compact spaces) such that (pk has linear extension operators, but the norm of every such operator is ^ 21c — 1 (cf. also Example 6 in this Notes and Eemarks).

E x t e n s i o n s and l inea r e x t e n s i o n s of smoo th func t ions to smoo th func t ions . The basic results on extension of differentiable functions are due to Whitney [1] (cf. also Hestenes [1], Calderon [1]). The formulas of Whitney (cf. Whitney [1], [2]) define a linear extension in the case of finitely differentiable functions defined on an arbitrary closed subset of the Euclidean space EN but not in the case of infinitely differentiable functions.

Operators of linear extension for infinitely many time differentiable functions have been constructed recently by several authors under various assumptions on the sets of arguments (cf. Mitjagin [1], Seeley [1], Aron-szajn [1], Adams, Aronszajn and Smith [1], Ogrodzka [1]. Ogrodzka's method is essentially based upon some previously unpublished result of Eyll-Nardzewski). Some counterexamples are given in Mitjagin [1] and Adams, Aronszajn and Smith [1]. Ogrodzka [1] treated the case of linear extension operators in vector bundless. I t seems that the general theory of linear exaves can be extended to this case.

Extensions of continuous functions defined on a closed subset of a (compact) topological space T to the function belonging to a given closed linear subspace of C(T) are treated by Bishop [1], Glicksberg [2], Pełczyński [1], [2], Gamelin [1], Michael and Pełczyński [1]. These results generalized previous results concerning disc algebra due to Eudin [2] and Carleson [1].

An interesting result on extending a metric is due to Hausdorff [3]; see also Arens [1], Bing [2], Kuratowski [3].

Several authors have studied the problem of extending a function with given modulus of continuity (especially a Lipschitz function), which is initially defined on a subset of a given metric space, to a function with the same modulus of continuity defined on the whole space (cf. McShane

[1], Kirszbraun [1], Banach [2], Mickle [1], Valentine [1], [2], [3], Cipszer and Geher [1], Aronszajn and Panitchpakdi [1]). For more detailed information we refer to the expository paper by Danzer, Griinbaum and Klee [1] and Isbell [1].

There is rather unsatisfactory information as to whether there exist linear extension operators taking uniformly continuous functions into uniformly continuous functions or Lipschitzian functions into Lipschit-zian functions. The next Proposition and Corollaries show that for arbitrary metric spaces such operators need not exist. This is related to the recent results of Lindenstrauss [2] concerning non-linear projections (cf. Corollary D in this Notes and Eemarks).

PROPOSITION A. Let X be a Banach space and let Y be a closed linear subspace of X. Let CU{X) and CU(Y) denote spaces of all uniformly continuous real-valued functions on X and Y respectively. Let v : CU(Y) -> Gu(X) be a linear extension operator which is continuous provided GU(Y) and CU{X) both carry the same one of the following three topologies: topology of simple convergence, topology of compact convergence, topology of uniform convergence (cf. Bourbaki [2J, pp. 4-5). Then there is a linear operator w : CU(Y) -> X* such that the restriction of w to Y* is a linear homeomorphic embedding of the Banach space Y* into the Banach space X* such that (<»y*)(y) = y*y (y**Y*,yeY).

Proof. Let / . . . da denote an invariant mean on the space B(A) A

of all bounded real-valued functions on an Abelian group A. (cf. Hewitt and Eoss [1], pp. 230-245). Let us put

(wf)(z) = J(J[(vf)(x + y + z)-(vf)(x + y)]dy)dx, feCJY), zeX. X T

I t can be easily verify that the operator w has the desired properties. COROLLARY B. If a Banach space X together with its closed linear

subspace Y satisfy the assumption of Proposition A, then the pair (X, Y) satisfies each of the following equivalent conditions

(i) there is a bounded linear operator w : Y* -> X* such that (wy*) (y) = y*y for y*eY* and for yeY.

(ii) there is a bounded linear projection n from X* onto the annihilator YL = {x*eX* : x* (y) = 0 for yeY},

(iii) there is a bounded linear operator u: X ->• Y** such that uy = y for ycY( s).

(3) Here and in the next Corollaries we identify a Banach space E with its cannonical image in E**.


Proof. In view of Proposition A it as enough to show that for every pair consisting of a Banach space X and its subspace Y the conditions (i), (ii) and (iii) are equivalent each to other.

(i) => (ii). Let r : X* --> Y* be the restriction operator, i.e. (ry*){y) = y*y for every y in Y and for every y* in Y*. We put nx* = x* — wrx* for x*eX*.

(ii) => (i). Let a : Y* -> X* be an operator (not necessarily linear and continuous) such that y* is an extension of y* and \\ay*\\ = \\y*\\ for y* eY*. We put w?/* = ay* — jra?/* for y* e Y*. Clearly w is a bounded operator from Y* into X*. Let #* and #* be arbitrary linear functionals in Y*. Then a(yf + 2/*) — ay* — ay* €^± (because ay* is an extension of y*). Therefore

rc[a(yi+y2) —c^!—a&] = a(yl+y2)-ayl-ay2.

On the other hand, by linearity of n,

n[a(y*-)-yi) — ay* — ay*] = 7za(y* + y*) — nay* — 7tay*.

Combining those formulas with the definition of w we get

w(y* + yt) = wyt + wy*.

That proves the linearity of w. Clearly wy* is an extension of y*, because nay* e Y1- for every y* e Y*. That completes the proof of the implication.

(i) => (iii). Define u as the restriction to X of the adjoint operator w* . x** -> Y**.

(iii) => (i). Define w as the restriction to Y* of the adjoint operator „.* I / * * * -V*

it : Y -> JL . COROLLARY C. 1/ tffte pair (X, Y) satisfies the assumption of Propo

sition A and if there exists a bounded linear projection p form Y** onto Y, then there exists a bounded linear projection q from X onto Y.

Proof. Put q = pu, where u : X -> Y** is defined in (iii). COROLLARY D (cf. Lindenstrauss [2]). If there exists a uniform re

traction from a Banach space X onto its closed linear subspace Y, then the pair (X, Y) satisfies the equivalent conditions (i), (ii), and (iii). In particular if there exists a bounded linear projection from Y** onto Y, then there exists a bounded linear projection from X onto Y.

Proof. Observe that if 99 is a uniform retraction from X onto Y, then 9?0: Cu( Y) -> CU(X) is a linear extension operator satisfying the assumption of Proposition A. Then apply Corollaries B and C.

E em ark. Let Y be an infinite dimensional reflexive subspace of the space X = C(S), ^-arbitrary compact space. Clearly the identity operator Y -> Y can be regarded as a projection from Y** = Y onto Y. On the other hand there is no bounded linear projection from X — C{8) onto Y

Notes and Eemarks 63

(cf. Grothendieck [2], Pełczyński [3]). Therefore, by Corollary C, there is no linear extension operator from CU(Y) into CU(X) which is continuous provided both spaces CU(Y) and CU(X) carry one of the topologies described in Proposition A.

Proposition 2.2 is well known (cf. Borsuk [1], Bessaga and Pełczyński [1], Pełczyński [1], [3], Dean [1], Semadeni [1], [2]).

For a result similar to Proposition 2.6 cf. Dean [1] and Amir [3].

P r o b l e m 1. May one replace in (2.8.1) and (2.8.3) the terms "Banach space" by "normed linear space" or, more generally, by "locally convex linear space"? In the last case by C(S, E) we mean the space of all continuous mapping from a compact space S into a locally convex linear space E with the topology of uniform convergence induced by a base of all neighbourhoods of 0 of the form

r v = (f€C(S,E):f(s)eV)

where V is a neighbourhood of zero in E.

Proposition 2.9 has been observed by E. Michael (oral communication) and is published here with his permission.

Eegular averaging operators were introduced by Birkhoff [1] and have been investigated by Kelley [1], Davis [1], Wright [1], Lloyd [1], [2], Moy [1], G. C. Eota [1], Brainerd [1], [2], Michael [2], [3], Corson and Lindenstrauss [2]. For further information we refer the reader to the expository paper by Birkhoff [2]. Michael, Corson and Lindenstrauss used different terminology but considered exactly the same kind of operators (cf. Notes and Eemarks to § 3). Averaging operators are a "limit case" of Eeynolds operators which arose from the operator used by the British engineer, Osborne Eeynolds [1], in his classic calculation of the average stresses due to turbulent momentum convection. The mathematical investigation of Eeynolds operators was initiated by Kampe" de Ferriet [1], [2]. For further information we refer the reader to the paper of G. C. Eota [2] and the references in this paper.

Actually Birkhoff defined an averaging operator in C(8) as a regular linear operator A : C(S) ~» C(8) such that

(B) A(fAg)=Af-Ag for f,g<C(S).

I t can easily be shown that A is a projection whose range is a self-adjoint subalgebra of C(8). Hence there is a compact space T and an epimor-phism <p:S -+T such that <p°[C{T)] = A[C(S)]. Thus the projection A

with mil = 1 uniquely determines the regular averaging operator u : C(8) -> C(T) for C(T) is a regular averaging operator for an epimorphism T, then applying Corollary 2.3 and Proposition 4.1 one can easily show that A = <p°u: C{8) -^C(S) is a regular operator satisfying (B).

EXAMPLE 1. N o n - n o r m a l exave . Let 8 = T = {1} v {2} be the two-point discrete space and let T be defined by 99 (1) = 99(2) = 1. Let us set u(x{l), x{2)) = ((»(l) + a?(2))/2, x{l)) for x = (x{l), x(2)) e(7({l} w {2}). Then wis a non-normal regular exave for 99.

Let 99: 8 -> T be a map and let M: C(S) -> C(T) be a continuous operator (not necessary linear) such that M(fg) — M(f)-M{g) for / , g in G{8) and <p0Mcp° = <p°. Then M will be called a multiplicative exave for 99.

I t seems to be interesting to investigate what maps have multiplicative exaves.

P r o b l e m 2. Does there exists a multiplicative extension operator from C(dBn) into C(Bn), where Bn is the unit ball in w-dimensional Euclidean space and dBn is the boundary of Bn — the (n — l)-dimen-sional sphere?

We are able to show that there is no multiplicative extension operator from C{dB2) into C(B2).

Ad § 3. Proposition 3.3 is due to H. Yoshizawa [1]. Propositions 3.3 and 3.4 can easily be extended to the case where S and T are arbitrary completely regular spaces.

L o c a l i z a t i o n p r inc ip le . As far as we know the idea of localization was first used by Lichtenstein [1] and was applied by several authors in the case of smooth extensions (cf. McShane [2], Hestenes [1], Adams-Aronszajn and Smith [1], Ogrodzka [1]).

Ad § 4. Let u be a regular averaging operator for an epimorphism 99: S -^ T. Then, Proposition 4.1 implies that u satisfies the inequality (cf. Michael [2])

( + ) inf /(*) ^uf(t) CR(T) is a linear operator satisfying (+ ) , then uR is a regular averaging operator from CR{8) onto CR{T) and therefore, by Proposition 2.9, it can be extended to a regular averaging operator from C{8) onto C(T). Therefore a result of Michael [2], Theorem 1.1, may be restated as follows:

If S and T are compact metric spaces and if cp is an open map from S onto T, then cp admits a regular averaging operator.

Corson and Lindenstrauss [2], pp. 501-504, showed that the above result may fail for non-metrizable compact spaces. However they proved that for some special non-metrizable compact spaces the assertion of Michael's theorem still holds.

The map *F: ^ X ^ -> I constructed in Milutin Lemma 5.5 is not open, because every open map from zero-dimensional space has zero-dimensional image. On the other hand, by Lemma 5.5, W has a regular averaging operator.

P r o b l e m 3. Let 8 and T be compact metric spaces. Give a necessary and sufficient condition (in topological terms) in order that an epimorphism cp : S -> T admit a regular averaging operator.

EXAMPLE 2. Let S = I = [0 ,1] and let T be the unit circle on the complex plane. Let q>(s) = e2nls for s-el. Then there is no regular averaging operator for cp. Precisely, if u is a linear averaging operator for cp, then \\u\\ > 2.

Proof. According to Corollary 2.3 it is enough to show that if p is a projection from C(I) onto its subspace E = <p°[C{T)~\, then \\p\\ > 2. Since E is a maximal hyperplane of C(I), p is of the from, p(f) = f—ju(f)g for / in C(I), where (x is a linear functional on C(I) with ker^ = E and geC{I) is picked such that pt{g) — 1. Thus, replacing (if necessary) pi by apt and g by a~lg where a is a suitable complex number, we may assume that pi — %(dl—dQ), where di denotes the unit point mass at the point i (• = 0,1). Hence p(f) = / - | [ / ( l ) - / ( 0 ) ] < / for feC(I), where i(g(l)-— 0(0)) = ^{d1— d0)(g) = 1. Now for a positive e < 1 we pick a point s0

in the open interval (0,1) so that g(s0) > \\g\\ — e. Clearly there exists fe

in C(I) such that ||/,|| = ||flf||;/.(0) = flf(0); / . ( l ) = </(l); fe(s0) = -flf(«0). Then \\p(fe)\\ > |p(/.)(«„)I =2|flf(«0)| >2| |<, | | -2 e . Thus ||p|| > 2(||flr||-6)/||flr||, because ||/e|| = \\g\\. Let e tend to zero, and we have \\p\\ ^ 2 . That completes the proof.

Observe that for [p0(/)](«) = / ( « ) - * ( / ( l ) - / ( 0 ) ) ( 2 « - l ) for sel and/e(7(I) we have \\p0\\ = 2. Therefore cp has a linear averaging operator of norm 2.

P r o b l e m 4. Let a map cp: S -> T has a linear exave of norm < 2. Does there exist a regular exave for c>?

Proposition 4.2 is due to Arens [1] (see also Corson and Lindenstrauss [1]).

EXAMPLE 3. Let W: Vx *€ -> I be the map defined by (5.5.2) in the proof of Milutin Lemma 5.5. Let cp : <$ -> x # be the homeomorphic Dissertationes Mathematicae LVIII 5

embedding defined by <p| = (£, 0) for £e<£\ Then it is easily seen that W I is the Cantor map. Therefore, by Corollary 9.12, there is no linear averaging operator for h. On the other hand cp, being a core-traction has a regular extension operator, and, by Milutin Lemma 5.5, W has a regular averaging operator. This example shows that

1° In Proposition 4.3 the assumption <piQ = <p~1T0 cannot be dropped. 2° Proposition 4.4 fails if we change the order in which we compose

a linear exave with linear extension operator (resp. with linear averaging operator).

EXAMPLE 4. Let Q = {c, d}, S = {I, II, III, IV}, T = {a, b} be discrete four and two point spaces. Let ^(c) = I,q>x{d) =11; q>(I) = <p(H) = a, (p(III) = <p{IV) = b. Then cpx is a homeomorphic embedding of Q into S, and <p is an epimorphism from S onto T. Clearly cp^Q) = cp~l{{a}). Let us set

*(/) = {f(c)J(d),m, 2/(<*)) for / = (f(c),f(d))eC(Q), , , lg(I) + g(II) , . „_ . .__. g(III) + g(IV)\

Hg) =1 g Vg{Hi)-g{iV), 1 for g = (g(I),g(II),g(III),g(IV))eC(8).

I t is easily to verify that u is a linear extension operator for <pt and v is a linear averaging operator for cp. However, uv is not a linear exave for 959?!, because

( W l ) °w( W l ) ° ( l T ) = ( 2 , 2 ) ^ {qxpiflr = 1 0 = (1, 1).

This example shows that in Proposition 4.3 the assumption of regularity of linear exaves u and v is essential.

Ad § 5. Dyadic spaces have been introduced by P. S. Alexandroff [1] and have been investigated by Sanin [1], Esenin-Volpin [1], Marczewski (Szpilrajn) [1], Ivanovskii [1], Kuzminov [1], Engelking and Pełczyński [1], Efimov and Engelking [1], Engelking [1], Efimov [1], [2], [3], Mar-deśic and Papic [1], Alexandroff and Ponomarev [1] and Ponomarev [1].

Lemma 5.5 was proved by Milutin in his thesis in 1952 but was published only recently (cf. Milutin [1], [2]).

The first example of a dyadic space which does not have [B.S.P.] is due to Engelking [1J. The example given below is due to H. H. Corson and is published here with his permission.

EXAMPLE 5. Let Tm be the space obtained from Dm by identification of two different points of Dm, say r\ and f. Let = i\{r\] w {£} be the natural epimorphism from Dm onto Tm. Then, according to the remark

after Proposition 2.6, for every e > 0 there exists a linear averaging operator for \) of norm < 2 + e. Hence Tm is an almost Milutin space. However, if m > 8 0 j then Tm does not have [B.S.P.] and therefore, by Corollary 5.11, Tm is not a Milutin space.

Proof. For £eDm let | 0 denote that coordinate which is different for rj and £. Without loss of generality we may assume that r/Q = 0 and Co = 1. Let us put for j = 0 , 1 ,

^ = {|eZ>«<:£0=j},

Vi = ^Aj\{p}

where p = f)rj = fy£-Clearly Ux r^ U2 = 0 . Since At are closed and since C^ = Tm\$A0

and ?70 = Tm\^A15 Z7?- are open. The pair (U0, UJ cannot be separated by open JPa-sets. This is an immediate consequence of the following facts

1° if m ^ K0, then At is homeomorphic to Dm (j = 0,1), 2° the restriction of j? to Aj is a homeomorphism (j = 0,1), 3° if m > X0 and f is an arbitrary point of Dm, then Dm\{£} is not

an .Fa.

The proof of 1° and 2° is trivial. For 3° see e.g. Corson [1].

The proof of Proposition 5.10 in the form presented in this paper has been obtained by the author jointly with H. H. Corson.

P r o b l e m 5. Let X0. Is it true that \\u\\ > 2 for every linear averaging operator for 99? (cf. Problem 4).

P r o b l e m 6. Give an example of a dyadic space which is not an almost Milutin space? We conjecture that the product P Tjf for

2<iV<+oo X > m0 possesses the properties in question. For the definition of T^ see Notes and Eemarks to § 6.

P r o b l e m 7. Construct for every a > 1 an almost Milutin space Sa

with the property that every linear averaging operator for an arbitrary epimorphism 9?: Dm -> Sa is of norm > a.

P r o b l e m 8. Give a topological characterization of Milutin spaces.

Ad § 6. The fact that every compact metric space is a Dugundji space has been observed by Arens (cf. Arens [1], Theorem 5.2).

EXAMPLE 6. The space Tm defined in Example 5 is for m > x0 an almost Dugundji space, but it is not a Dugundji space. Precisely

(a) if ip: Tm -> Im is a homeomorphic embedding, then every linear extension operator for ip is of norm > 3,

(b) if (p: Tm-^T is an arbitrary homeomorphic embedding of Tm into a compact space T, then there exists a linear extension operator for 93 of norm < 3.

Proof, (a) follows from the Remark to Proposition 5.10 and the fact that r" has [B.S.P.] but Tm does not have [B.S.P.] (cf. Example 5).

According to the Remark to Proposition 6.2 to prove (b) it is enough to show that every homeomorphic embedding ip : Tm -> lm admits a linear extension operator of norm < 3.

Let the epimorphism \): Dm -> Tm and the sets Aj (j = 0,1) have the same meaning as in Example 5 and let p = §A0 r^ \)AX be the unit point of Tm such that ^~1{p) is a two-point set. Let ipj be the restriction of ip to §Aj (j = 0,1). First we show that there is a regular extension operator for ipf, say ui} such that

(%/)(*) =f(P) for /eC(M/) and for tĄA^,

where j * = (j + l)mod2 (j = 0,1). Let if = iftAf denote the natural epimorphism from I™ onto the

space T^fyAj obtained by glueing together of all points of \)Aj. I t is easily seen that <pj = i}-*o ipj is a homeomorphic embedding of \)Aj into lmftAj: Since §Aj- is homeomorphic to Dm and since Dm is a Dugundji space, the map <pj has a regular extension operator, say %• Let Uj — (ijnfu'j. I t can easily be verified that Uj\ C{\)Aj) -> C(Im) has the desired property.

Finally we put

« / = ttJo + Ni/i —%[(V°«o/o)i] f o r f€C(Tm) where for geC(Tm) by gj we denote the restriction of g to \)Aj (j = 0,1).

One can easily check then that u is a linear extension operator for ip with ||w|l < ||«oll + IMI+ llwill llv°ll IKH = 3. That completes the proof.

Example 6 admits the following generalization obtained jointly by H. H. Corson and the author.

Let N be an integer ^ 2. Denote by ĄiV) the space obtained from Dm

by identification of the points of some JV -point subset of Dm. Clearly m m(2)

If m > K0, then (a(N)) if ip: T^ -> Im is a homeomorphic embedding, then every

linear extension operator for ip is of norm ^ 2N — 1; (b(N)) if <p: T^ -> T is an arbitrary homeomorphic embedding of

T^ into a compact space T, then there exists a linear extension operator for 9? of norm 2 ^ — 1.

P r o b l e m 9. Let S be an almost Dugundji space of topological weight m. Is it true that there exists an odd integer rj(8) such that

1° for every compact space T and for every homeomorphic embedding cp: 8 -> T there is a linear extension operator of norm <??(#),

2° for every homeomorphic embedding y>: 8 -> lm every linear extension for ip is of norm > r}{8)%

P r o b l e m 10. Show that if m > N0, then every averaging operator for arbitrary epimorphism from Dm onto T j ^ is of norm ^ JV + 1.

P r o b l e m 11 (cf. Corson and Lindenstrauss [1]). Let cp : 8 ->• T be a homeomorphic embedding (8, T compact spaces). Let r)(<p) denote the g.l.b. of the set of norms of all linear extension operators for cp. Is 77(9?) an odd integer?

P r o b l e m 12. Let cp : 8 -^ T be an epimorphism. Let C(<p) denote the g.l.b. of the set of all linear averaging operators for cp. Is £(<p) an integer %

P r o b l e m 13. Give a topological characterization of Dugundji spaces. P r o b l e m 14. Is every Dugundji space (every almost Dugundji

space) dyadic? All known examples of compact non-dyadic spaces are not Dugundji

spaces (cf. Proposition 8.11 and Corollary 8.14). On the other hand, if m > Ko> then the product P Tjf > is a dyadic space, but it is not an

2<iV<+oo

almost Dugundji space (because of (a(N)). P r o b l e m 15. Does the class of all Milutin spaces (almost Milutin

spaces) coincide with the class of all Dugundji spaces (almost Dugundji spaces) ?

P r o b l e m 16. Let 8 be a compact zero-dimensional space. Are the following conditions equivalent?

(i) 8 is a retract of Dm, (ii) 8 is a dyadic space and S has [B.S.P.],

(iii) S is a Milutin space, (iv) 8 is a Dugundji space. Condition (i) implies each of the remaining conditions. The impli

cation (i) => (ii) was first proved directly in Engelking [1]. The implication (iii) => (ii) follows from Corollary 5.11.

The next problem is related to Problem 16. P r o b l e m 17. Let a homeomorphic embedding 97: 8 -> T of a com

pact space S into a zero-dimensional compact space T admit a regular extension operator. Is cp a coretraction, i.e. is cp8 a retract of T%

The affirmative answer to Problem 17 implies (by Proposition 3.3) that the existence of a regular extension operator for a homeomorphic


embedding of zero-dimensional compact spaces is equivalent to the existence of linear multiplicative extension operator. This is somewhat related to the works of Phelps [1] and Bonsall, Lindenstrauss and Phelps [1]

P r o b l e m 18. Let S be a Dugundji space (an almost Dugundji space). Is it true that for each s in 8 there is a closed neighbourhood Fa which is a Dugundji space (an almost Dugundji space)?

Ad § 7 . Proposition 7.2 is closely related to the result of Eudin [1] on the existence of the invariant projection (see also Glicksberg [1] and Eosenthal [1]). It is also related to the results of A. lonescu Tulcea [1] concerning liftings commuting with a group of maps (see also C. lonescu Tulcea [1]).

P r o b l e m 19. Does the conclusion of Proposition 7.2 remain true under the assumption that G is an arbitrary locally compact group ?

Probably using the notion of invariant mean as did de Leeuw (presented in Glicksberg [1]), one can generalize Proposition 7.2 to the case where G is an arbitrary locally compact abelian group.

Theorem 7.5 can also be proved by using the Pontryagin representation of a compact topological group as an inverse limit of a Lie series (cf. Pontryagin [1], p. 327, Ivanovskii [1]; see also the proof of Proposition 8.10).

Perhaps the same idea may be useful in solving the following problems. P r o b l e m 20. Is every compact coset-space of a locally compact

topological group a Milutin space (an almost Milutin space)? P r o b l e m 21. Is every compact topological group, or more gener

ally, every compact coset-space of a locally compact topological group, a Dugundji space (an almost Dugundji space) ?

The results of this paragraph can be partially generalized to the case of principal fibre spaces using the existence (in certain special situations) of local cross sections (see e.g. Serre [1], Borel [1], Mostert [1], Sklyarenko [1], Michael [4]).

Ad § 8. The proof of Proposition 8.3 uses a special case of the abstract decomposition scheme (described in Bessaga [1], p. 283). The decomposition method was discovered in Borsuk [1] and has been developed for various purposes in Pełczyński [3], [4], Bessaga and Pełczyński [2], [3] and Kadec and Levin [1].

Theorem 8.5 is due to Milutin [1]. (cf. also Milutin [2]). This settled a question of Banach (cf. Banach [1, p. 185]). Linear topological classification of spaces of continuos functionus on countable compact spaces is given in Bessaga and Pełczyński [1].

Notes and Kemarks 71

P r o b l e m 22. Let 8 be a compact space of weight tt. Assume

(*) 8 cannot be represented as a countable union of its closed subsets of topological weights less than the topological weight of 8.

Also let 8 have one of the following properties: (i) S is a Dugundji space (an almost Dugundji space).

(ii) 8 is a dyadic space, less generally S is a Milutin space. (iii) S is an absolute retract. Does 8 contain a closed subset homeomorphic to D"? Is C{8) linearly homeomorphic to C(Dn)% Clearly if the topological weight of 8, say n, cannot be represented as

a countable sum of cardinals less than n, then (*) is satisfied automatically. The next example shows that assumption (*) is, in general, essential.

EXAMPLE 7. Let (n„)^ be an increasing sequence of cardinals. 00

Let n = sup nv. Let 8 be a compact space such that 8 = {J Sv, 8V 0 < y < + o o v=0

are closed in 8 and n, is the topological weight of 8V (v = 0 , 1 , . . . ) . Then (o) no closed subset of 8 is homeomorphic to Dn,

(oo) C(8) is not linearly homeomorphic to C(Bn) Proof. We shall show that there is no linear homeomorphism from

C(Dn) into C(8). This fact clearly implies (oo), and it also implies (o) because, if Dn were homeomorphic to a subset of 8, then C{8) would be a factor of C(8) (cf. Proposition 8.4).

Let u : C(Dn) -> C(8) be an arbitrary linear map. Let E be the closed linear subspace of C(Dn) spanned by the family of functions (fa)aeA, where /0(£) = 2 | a — 1 for £ = (ib)bcA and A is a set of indices of cardinality n. We shall show that even the restriction of u to E is not a linear homeomorphism. First we observe that E is linearly homeomorphic to the space lx{A) of all scalar valued functions t = {ta)aeA such that ||t|| = £ \ta\ < +°° -

aeA

This follows from the inequality

i— m

V2 y i > — 2J 1**1 f o r t = (kWiW and 1=1 i=\ i=\ 2Xi> Sta'fa'

for every finite subset {ax, a2, . . . , am} cz A (m = 1, 2, . . . ) .

n

Let Rn: C(8) -> 0 ( U 8V) be the operation of restriction of functions i>=0

n

to [JSV and let un = Bnu (n = 0 , 1 , . . . ) . Since the topological weight v=0

n n n

of U ®v is ]£ n„ which is less than n, the density character of C({J8V) v=0 v=0 v™0

is less than the density character of li(A) which is equal to tt. Thus no un is a linear homeomorphism. Hence there are gn in E such that

n i 1 | | 0 » | | = 1 ; S U p { * 6 ( J # , \\(ugn)(s)\} = | |Wn^n||< ~ T (n = 0, 1, ...) . v=o n-\-l

Therefore the sequence {ugn) weakly converges to 0 in C(S) (because oo

\\ugn\\ ^ llwll (n = 0,1, ...) and ]im(ugn)(s) = 0 for seS = U #„). Since łl v=0

||flrB|| = 1 (w = 0 , 1 , . . . ) , the sequence (gn) does not converge weakly to 0 in E. (We use the fact in lx(A), and therefore in E, weak and norm convergences of sequences are equivalent (cf. Day [1], p. 33). That shows that the restriction of u to E is not a linear homeomorphism.

Let us specify Example 7 as follows. Let 8 be the one-point compacti-OO 00

fication of the discrete sum [J D"". Then 8 = U 8„, where 80 is the one-v=l v=0

point set consisting of "the point at infinity" and 8V = Dn" for v = 1,2, ... Clearly 8 satisfies the assumption of Example 7 while being a Milutin space and a Dugundji space.

Problem 22 is a particular case of the following: P r o b l e m 23. Let S be a compact space satisfying (*) and such that

C(S)\C{Dn). Is C{8) linearly homeomorphic to C(Dn) (n is the topological weight of S)1

P r o b l e m 24. Does there exist a compact space 8 such that C(S)\C(Dn), in particular C(8) is linearly homeomorphic to C(Dn), but 8 is not dyadic?

The next problem seems to be a proper generalization of the results of Bessaga and Pełczyński [1].

P r o b l e m 25. Give a complete linear topological classification of 00

C{8) spaces such that S = [J 8V, where 8V is closed in 8 and homeo-

morphic to D"" (v = 1, 2, ...), 8 — compact.

P r o b l e m 26. Generalize Theorem 8.9 and Proposition 8.10 to the case where 8 is a compact coset space of a locally compact topological group.

Some results of Hulanicki [2], [3] and Jones [1] (see also Hewitt and Eoss [1], p. 79) on cardinality and metrizability of compact topological groups can easily be deduced from our Proposition 8.10.

P r o b l e m 27. Give an example of a compact non-dyadic space 8 such that. C(S) possesses the properties (a) and (b) stated before Proposition 8.11.

Following Banach [1], p. 242, we assign to each pair (X, Y) of normed linear spaces a real number a(X, Y) with 1 < a(X, Y) < +oo as follows.

a(X, Y) = +oo whenever X is not linearly homeomorphic to Y, a(X, Y) =mi\\u\\\\u~1\\ if X is linearly homeomorphic to Y; the

infimum is taken over all linear homeomorphisms u from X onto Y. Let Si, S2 be compact spaces. We put

a(8lt8t)= 0(0(8^ tC{8t)).

P r o b l e m 28. Is it true that if a(Sx, S2) < +oo, then <*(#!, S2) is an integer?

P r o b l e m 29. Compute the following numbers i° a(i, n 2° ($(<£) = sup{a(#, <V)\8 compact metric uncountable}, 3° /8* = sup{a(#u #2)l#i> $2 compact metric uncountable}. Let us note that recently Amir [4] proved that if a(Sr, S2) < 2,

then St and S2 are homeomorphic. On the other hand it follows from the analysis of the proof of Theorem 8.5 that 0(V) <12. Hence 2 < 0* < 144.

Ad § 9. Proposition 9.9 was conjectured by Z. Semadeni. The proof given in the present paper is based upon de Branges [1] proof of the Stone-Weierstrass approximation theorem (cf. also Glicksberg [2]). The real analogue of Proposition 9.9 has been considered by several authors. Clearly for feCR(S) formula (9.9.1) gives

(*) Q(f,<P°[CR(T)])=mV sup 2-1\f(s1)-f(s2)\.

The formula (*) was announced without proof in Pełczyński [5]. An elegant proof is due to S. Mazur (cf. Semadeni [2], p. 20). The same problem has been considered independently by Kripke and Holmes [1].

Eecently Olech [1] gave an alternative proof of Proposition 9.9 in the complex case as well as in the case of some vector valued functions. He also showed that for each feC(S) there is a g in C(T) such that Q{fi <P°W(T)']) = ||/— q>°g\\. The same result for the real case is due to Mazur.

Corollary 9.12 has been proved by M. I. Kadec in an unpublished part of his thesis (about 1949) and independently in a recent paper by Foia§ and Singer [1] who restated it as follows:

There is no projection of D(I) onto its subspace C(I), where D{I) is the space of all bounded functions on I continuous at each non-dyadic point, continuous on the right and on the left at each point and having for every e > 0 only a finite number of "jumps" greater than e. This is related to Example 2 in Corson [2], p. 13.

Another example of an epimorphism y: S -> T, S and T compact metric spaces, which has no linear averaging operators is given in Arens [2], Theorem 3.5.

P r o b l e m 30. Let 8 be an infinite compact metric space. Are the conditions (9.13.1)-(9.13.3) equivalent to the following conditions?

(9.13.4). There is an epimorphism T which does not have linear averaging operators.

(9.13.5) There is a compact space T and an epimorphism <p: S -> T which does not have linear averaging operators.

The construction described in Lemma 9.16 together with Proposition 9.8 show that C([<wn]) is not a ' K space for X < n. One can show that C([wn]) is a ^ + 2 space. Modyfying the proof of Proposition 9.8 and the construction of Lemma 9.16, one could probably show that w + 2 is the exact constant, i.e. that C{[con]) is not a *pi space for X < n + 2 (for n = 1, cf. McWilliams [1]).

APPENDIX: CATEGORY-THEORETICAL APPROACH

The basic notions of the theory of categories can be found in the monographs by Brinkmann and Puppe [1] and Mitchell [1].

Let & and J5f be two categories. Objects of 9~ (denoted by S, T, Q,...) will be called "spaces" and morphisms of 9" (denoted by <p, ip, ...) will be called "maps". Morphisms of J5f (denoted by u,v,w,...) will be called "operators".

Let J ' be a fixed contravariant functor from 9 into S£. DEFINITION 1. An operator u:F(S)-+F(T) is called an F-exave

for a map y : S -> T if

(1) F(<p)uF(<p)=F((p).

Let us observe that if F(<p) is an epimorphism(4), then (1) is equivalent to the condition

(2) F((p)u = idF{S);

i F((p) is a monomorphism, then (1) is equivalent to the condition

(3) uF{cp) = idF(T).

DEFINITION 2. An operator u satisfying (2) is called an F-extension operator for q>. An operator u satisfying (3) is called an F-averaging operator for cp.

In the most important examples the contravariant functor F has some additional properties.

DEFINITION 3. A contravariant functor F from J' into t£ is said to be of Banaeh-Stone type if the following conditions are satisfied:

(4) Let us recall that a morphism a in a category #f is said to be an epimorphism (resp. a monomorphism) if for arbitrary morphisms j8 and y oi s/ the condition jffa = ya implies /? = y (a/? = ay implies /? = y).

If A is an object of s/, then id^ denotes the identity morphism of A.

& •

1

2

3

4

5

6

Symbol

Comp.

>»

i t

>>

»»

Tot. disc.

Comp.

object

compact Hausdorff spaces

>»

>>

~-~ >>

»>

totally disconnected compact Hausdorff spaces

morphisms

continuous transformations

>»

>»

>>

»>

continuous transformations

symbol

a) C(Comp.) b) CR(Comp) c) C+(Comp)

a) reg. C(Comp) b) reg. Cj2(Comp) c) reg. C+(Comp)

a) mult. ad. C(Comp) b) mult. ad. Cij(Comp) c) mult. ad. C+ (Comp)

a) mult. C(Comp) b) mult. Cjj(Comp) c) mult. C+(Comp)

Co (Comp)

a) C int(t-d. Comp.) (J) mult. ad. Cint(t-d.

Comp.) y) mult. ad. Cint(t-d.

Comp.)

JS?

objects

spaces of all continuous a) complex valued b) real valued c) non-negative

functions on compact Hausdorff spaces with the topology of uniform convergence

the same as in 1 a), b), c)



topological groups of all continuous maps from a compact Hausdorff space into a given topological group O with pointwise multiplication as group operation and with the topology of uniform convergence.

topological groups of all continuous integer valued functions on totally disconnected compact Hausdorff spaces with the topology of uniform convergence

morphisms

a) and b) bounded linear operators

c) continuous affine operators

a) and b) regular operators

c) regular affine operators

a) and b) linear multiplicative operators

c) affine multiplicative operators

continuous multiplicative operators

continuous group homomor-phisms

continuous operators, a) additive P) multiplicative y) multiplicative-additive

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(1) There is a faithful (5) functor a from 3~ into the category Ens of sets as objects and with arbitrary mappings as morphisms.

(2) There is a set M depending only on «£? such that for every object T of 3~ there is the unique object of if, denoted by F(T) and being a set of functions from a{T) into M. Conversely if X is an object of JSf, then X = F{T) for some object T of F.

(3) For all objects F(8) and F(T) the set [F(T), F(S)]<r of all morphisms of JS? from F(T) to F(8) consists of some transformations from F(T) into F{8). If <p: 8 ~> T is a morphism of ^", then <p°e[.F(T), F(S)]&, where <p° denotes the induced operator defined by cp°g = goa{cp) for g : a{T) -> M.

(4) F assigns to each object 8 of &" the object F{8) of & and to each map (p: # ->- T which is a morphism of 9~ the induced operator <p° (i.e. F(<p) = <p°).

We list in the table on pp. 76-77 various examples of functors of Banach-Stone type. Clearly in each case it is enough to describe categories^" and SP, because the functor of Banach-Stone type from S' into if, if it exists, is uniquely determined by the given categories & and if, and by the functor a which in all these examples assigns to the topological space T the set of its elements.

Let us complete the list of examples by the following functor which is not exactly of Banach-Stone type.

Let^" = Euclid. Vect. Bund, be the category whose objects are vector bundles with compact bases, with fibres being finite dimensional Euclidean spaces and with Q = lim ind Qn — the infinite orthogonal group as the transformations group. Morphisms of Euclid. Vect. Bund, are continuous transformations from one bundle to another which map bases into bases, fibres into fibres and which acts on fibres as partial isometries (composition of unitary transformation with orthogonal projection). For a given bundle B (being an object of Euclid. Vect. Bund.), let Sec(U) denote the space of all global sections of B with natural (i.e. pointwise) operation of addition and multiplication by scalars and with the topology of uniform convergence. Under an admissible norm Sec(J5) may be regarded as a Banach space. Let if = Sec (Euclid. Vect. Bund.) be the category whose objects are Sec (B) for B being objects of Euclid. Vect. Bund., and whose morphisms are bounded linear operations. The functor F assigns to each bundle B the space Sec (B) and to each morphism <p: Bx -> B2 the induced operator <p°: Sec(2?2) ->Sec (BJ defined by <p°7i — noy for :rceSec(2?2).

(5) A functor K from a category si into a category 38 is said to be faithful (cf. Mitchell [1], p . 51) if for every objects A and B in sć the function [A, B]s/ -> [K(A), K{B)]£, induced by K, is univalent. (By {A, B]s/ we denote the set of all morphisms in a category s/ from A to B.)


One can easily define the analogous functor from the category of Zc-times differentiable bundles into the category of their ft-times differentiate global sections.

For a given functor F from 5" into 3? we can develop the theory of .F-exaves along the same line as it is done in the present paper for the Banach-Stone functors from the category Comp into the categories C(Comp), reg C(Comp) and mult. ad. C(Comp). The analogies are most significant in the case where &~ is a subcategory (6) of the category of topological spaces and S£ is a subcategory of the category of linear topological spaces.

(6) A category @ is said to be a subcategory of a category si if every object of SS is an object of si and every morphism of & is a morphism of si and the rules of composition of morphisms in si and in 38 are the same.

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7 eng.pdfcontents introduction 5 preliminaries 9 § 1. kegular operators and their products 11 § 2....

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