CONVEXITY THEORY AND ITS APPLICATIONS IN FUNCTIONAL ANALYSIS
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CONVEXITY THEORY AND ITS APPLICATIONS IN FUNCTIONAL ANALYSIS
L.M.S. MONOGRAPHS Editors: D. EDWARDS and H. HALBERSTAM
1. Surgery on Compact Manifolds by C. T. C. Wall, F.R.S.
2. Free Rings and Their Relations by P. M. Cohn 3. Abelian Categories with Applications to Rings and Modules by N. Popescu
4. Sieve Methods by H. Halberstam and
Richert
5. Maximal Orders by I. Reiner 6. On Numbers and Games by J. H. Conway 7. An Introduction to Semigroup Theory by J. M. Howie
8. Matroid Theory by D. J. A. Welsh 9. Subharmonic Functions, Volume 1 by W. K. Hayman and P. B. Kennedy
10. Topos Theory by P. T. Johnstone 11. Extremal Graph Theory by B. Bollobás 12. Spectral Theory of Linear Operators by H. R. Dowson 13. Rational Quadratic Forms by J. W. S. Cassels, F.R.S. 14. 15.
Algebras and their Automorphism Groups by G. K. Pedersen Semigroups by E. B. Davies
16. Convexity Theory and its Applications in Functional Analysis by L. Asimow and A. J. Ellis
Published for the London Mathematical Society by Academic Press Inc. (London) Ltd.
CONVEXITY THEORY AND ITS APPLICATIONS IN FUNCTIONAL ANALYSIS L. AsIMow University of Wyoming, Laramie, Wyoming, USA
A. J. ELLIS University College of Swansea,
Swansea, Wales, UK
1980
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Copyright © 1980 by ACADEMIC PRESS INC. (LONDON) LTD
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British Library Cataloguing in Publication Data Asimow, L Convexity theory and its applications in functional analysis. —(London Mathematical Society. Monographs; 16 ISSN 0076—0560).
1. Convex sets 2. Functional analysis I Title II. Ellis, A I III. Series 517'.7 80—40648 QA640 ISBN 0—12—065340—0
PRINTED IN GREAT BRITAIN BY J. W. ARROWSMITI4 LTD. W1NTERSTOKE ROAD, BRISTOL
To Marilyn Asimow and Jennifer Ellis
Preface
With the appearance in 1966 of Lectures on Choquet Theory, Phelps [172], the representation theory of compact convex sets became accessible to a large, and as it develops, enthusiastic audience. The geometric appeal of the Choquet simplex is irresistible; it has led quite naturally to an exploration of related geometric structure in more general state spaces. This enterprise, while somewhat interesting in its own right, would be of little note but for the remarkable insights it has provided in appreciating various analytical aspects of the underlying function spaces. This feature was already apparent in the comprehensive treatment in 1971 of Choquet Theory in the book by Alfsen [5]. Since that time the geometric study of compact convex sets has rewarded
its devotees not only with generalizations of individual theorems in functional analysis, but with the prospect of a unified geometric theory that yields a clearer understanding of a reasonable variety of classical results.
Our object here is to promote this geometric perspective. We hasten to make the standard disclaimer concerning the lack of totality in our selection of topics. Granting the necessarily idiosyncratic nature of our subject matter
we have attempted to illustrate the means by which a fairly elementary geometric theory, based on partially ordered Banach spaces and duality, can
be applied in a systematic fashion to concrete function spaces (real and complex) and unital Banach algebras.
In Chapter 1 we have gathered together the functional analytic minaries that constitute our basic tools. The reader with only a basic knowledge of abstract real analysis will find there statements of the main results as well as short but self treatments of the Krein—Sinulyan Theorem, the basic Choquet Theory and the Bishop—Phelps Theorem. Chapter 2 gives the basic duality results, lattice theory and concrete representation theorems for order unit spaces and Banach lattices of type M and L. Since much of the book treats subspaces of continuous functions on
viii
PREFACE
compact Hausdorif spaces we felt the classical representation theorems of abstract ordered Banach spaces were especially pertinent. We continue the treatment of real affine function spaces in Chapter 3 by examining in detail the case where the state space is a Choquet simplex. In Chapter 4 we show how the study of real A(K) spaces can be employed even for complexvalued function spaces by means of a complex state space. Chapter 5 gives a survey of the application of the theory to the study of (non-commutative) Banach algebras.
Our thanks are due to Caroline Johnson and Paula Melcher for their expert typing of the manuscript. We are very grateful to Dr K. F. Ng for reading the manuscript and making many helpful suggestions. We have also
benefited from useful discussions of parts of the book with Dr T. B. Andersen. The second author wishes to thank the University College of Swansea for sabbatical leave during the Autumn Term of 1976, enabling him to visit the University of Wyoming.
July 1980
L. Asimow A. J. Ellis
Pref ace
V
1. PrelimInaries 1. Separation and Polar Calculus 2. Krein—Smulyan Theorem and Corollaries 3. Gauge Lemma and Completeness 4. Subspaces of C(X) and Aøine Functions 5. Representing Measures 6. Maximal Measures and the Choquet—Bishop-de Leeuw Theorem 7. Bishop—Phelps Theorems
I
2. Duality in Ordered Banach Spaces 1. Positive Generation and Normality 2. Order Unit and Base Norm Spaces 3. Directedness and Additivity 4. Homogeneous Functionals on Cones and the Decomposition Property 5. Banach Lattices and the Riesz Interpolation Property 6. Abstract L and M Spaces 7. Choquet Simplexes 8. Decomposition of Dual Cones 9. Stability of Split Sets and the Characterization of Complemented
1
4 7 11 15
18
26 30 31 35 39
44 48 57 69 74 83
Subcones
10. Application to A(K) 11. The Riesz Norm 3. Simplex Spaces 1. The Facial Topology for K and the Centre of A(K) 2. Facial Characterizations of Simplexes
91
99 102 102 108
x
CONTENTS
3. Ideals in Simplex Spaces 4. Stone—Weierstrass Theorems for Simplex Spaces 5. Prime Simplexes and Anti-Lattices
117 122 124
6. Topological Characterization of the Extreme Boundary of a 126 Compact Simplex 7. Poulsen's Simplex—its Uniqueness and Universality 8. Non-Compact Simplexes and Convex Sets
130 138
4. Complex Function Spaces 0. The Complex State Space 1. Complex Representing Measures 2. Interpolation Sets 3. Gauge Dominated Extensions and Complex State Space 4. Peak Sets and Al-sets 5. Dominated Interpolation 6. Decomposability and Exact Interpolation 7. M-Hulls and Function Algebras 8. Facial Topologies and Decompositions 9. Lindenstrauss Spaces
208 219
5. Convexity Theory for 1. The Structure Topology and Primitive Ideas in A(K)
224 224
2. The Ideal Structure of a Unital
and the Facial 226
Structure of its State Space 3. Commutativity and Order in
4. The Centre of a
146 146 148 154 159 164 169 179 193
232
Weakly Central Algebras, Prime 234
Algebras 5. Unit Traces for
6. The State Spaces of Jordan Operator Algebras and of
239 242
Algebras
References
252
Index of Definitions
261
Subject Index
263
CHAPTER 1
Preliminaries
We bring together in this chapter most of the general material in functional analysis that will be relied on in subsequent chapters. We have simply stated some of the standard results and have provided complete proofs of others. The distinction is based mainly on which theorems are pretty much standard in a graduate level analysis course (say, for example Royden [185] or Rudin [187]) and which are left for more advanced courses.
Thus we omit proofs of the Separation Theorem and the bipolar calculations (which are used repeatedly throughout the remainder of the text)
and the Krein—Milman Theorem. We do prove the Krein—Smulyan Theorem (also an invaluable tool in our later study) in Section 2. In Section 3
we present a technical lemma which incorporates most of the iteration procedures in Banach spaces (including the Open Mapping Theorem) that arise later on. In Sections 4—6 we introduce state spaces and prove the de Leeuw Theorem on representing measures. We conclude with a proof of the Bishop—Phelps Theorem.
1. SEPARATION AND POLAR CALCULUS Let (E, F) be a pair of (real or complex) linear spaces in duality, with each total over the other; i.e. if (a, x) =0 for all x E F then a 0 and if (a, x) =0 for all a E E then x =0. Let (E, F) be endowed with the weakest topologies for which the linear functionals and (a, .) are continuous on E and F for each a and each x. Then E and F are locally convex linear topological spaces with each being the (dual) space of continuous linear functionals over the other. 1.1. THEOREM (SEPARATION THEOREM). If A and B are disjoint convex subsets of E with A compact, and B closed then there exists an x E Fsuch that
max{re(a, x): a eA}0 there is
If a is
c{x: re (a, x) l}. Then taking polars an AcE with A finite and A° in E** gives (since A°° = co b(A) c çb(E)) x
Hence x e norm-cl j(E).
co
GAUGE LEMMA AND COMPLETENESS
7
E with 1kb (a,,)-- all 0. For r >0 let X For the converse let (a,,) and let A C(X), taking b Then 0 is an isomorphism onto a subspace of C(X) and lObil=rllbil. Hence SUPx b(a,,)—al=rlkb(a,,)-—alI— of a. 0 so that a E C(X). Thus Corollary 2.5 yields the
If E isa Banach space then the following are equivalent: linear functional on
2.6.
(i) a is a
(ii) a€bE, (iii) a is THEOREM).
Let F be a Banach
space and let K be a convex subset of E*. Then K is r 0. K
if and only if
2.7. COROLLARY
Let V = KC so that the condition implies V
is relatively Thus K is and hence is an intersection of hw*_closed half-spaces. But the previous corollary guarantees these half-spaces are in fact and therefore, K is Proo/
w*..open for all r0 and hence V is
3. GAUGE LEMMA AND COMPLETENESS It is convenient to formulate here a general iteration scheme in Banach spaces, which will allow us to conclude a variety of duality results later on. This is expressed in terms of gauges, i.e. Minkowski functionals of convex sets containing 0 and possessing certain convergence properties.
If p is a positively homogeneous, sub-additive functional on a normed linear space E such that p(x,,) < implies x = exists and p(x) then p is called a gauge. It is the Minkowski functional of the set
Definition.
B ={x€E: p(x) 1}. If p has the weaker property that p(x)
p(xn)
whenever x
exists
then p is called a pre-gauge. We say the convex set B (containing 0)is a gauge set, or a pre-gauge set if PB has the corresponding property. It may happen that B is a proper subset of {x:
8
3.1. PRoposiTioN. Let A and B be closed convex sets containing 0 in the normed linear space E and let T : E space F.
Fbe a hounded linear map to the normed
sets. (i) A and B are (ii) If B is complete and bounded then B is a gauge set. (iii) If B is a gauge set then A + B and Co (A u B) are (iv) If B is a gauge set then T(B) is a pre-gauge set.
Proof.
sets.
Note first that the Minkowski functional PA is lower-semi-
continuous if and only if A is closed, since
{xEE:pA(x)_ ii if and only if p,(f) v(f) for all f E 0(K). It follows from Theorem 6. 1(viii) that > ii implies p.(f) p(f)
for fe Q(K). 6.4. THEOREM. (i)
(ii) For each ii there is a
which is maximal with respect to > and
(iii) 1ff C(K) and p Mt (K) then there isa > vsuch
>v. v(f).
If a A(K) then ±a 0(K) so that p.(a) = v(a). The second part of (i) follows from Proposition 6.2. To show (ii). we show every chain with respect to> is bounded above. If fl is such a chain is a net in Mt (K) and hence there is a subnet, which to 1ff E Q(K)and i.' we denote (va) which converges then for each 8>0 we can choose > ii and (vo— i-'a)(f)I 0 then g = and g/v(g) E A. Since
A°,
p(g) v(g). But then g —0(K) implies v(g + c)>0 for c a sufficiently on 0(K) so large constant and = v(c) = c, since c A(K). Thus that
> v.
PRELIMINARIES
24
We can now characterize the maximal measures by their preservation of integral values for envelopes of functions. (K), then the following are equivalent: 6.5. THEOREM. Let E (i) is maximal; (ii) for allfE C(K); = = for alifE Q(K). (iii) p;(J) = That (i) implies the first equality in (ii) is immediate from Theorem 6.4(iii). The second equality follows from f= —(--f). If (iii) holds for and Proof.
then fE Q(K) implies fL(f) v(f). But —f€ 0(K) and Theorem 6.1(viii) shows
v(—f), or
Thus v(f) t'(f)
so that
> v. Hence ,u. is maximal.
If fE C(K) and B1 = {x : 1(x) = f(x)} =
fl {x : f(x) —f(x) < 1/nj
set containing ext K. On the other hand, if
then B1 is a
is maximal then
Theorem 6.5(u) shows p(K\B1) = 0 for all fE C(K). Moreover, ext K = {B1: fE C(K)} by Theorem 6.3. It would appear then that maximal measures live mainly on ext K. We next make some precise statements of this observation. First, if K is metrizable then K is separable and we can find a sequence
fl
of non-negative functions in A(K)1 which separates the points of K. Hence the function
is continuous and strictly convex, meaning h(Ax +(1
—A)h(y)
whenever 0 0).
Then for each 0< 0 be given. (i) For each x A* there is a support pair (a, y) with IIy — xli < e. (ii) For each a in the boundary of A there is a support pafr (b, x) with iIb—all<e. Take x 0 and apply Theorem 7.1 with any a A to x/iixII with any For (ii) choose A and llc — aIl €/2. Then there is an x A* y (lixil = 1) with
Proof.
(c,
Thus 7=
(a, x) + (c — a,
x)>pA(x). x) + €/2 and Theorem 7.1 applies with
BISHOP PHELPS THEOREMS
29
Notes
Some general references for the background material discussed in this chapter are Day [75], Holmes [131] and Semadeni [191]. In particular Holmes [131; Section 17] has a discussion of completeness that incorporates some of the material in Section 3 concerning the gauge lemma and regularity of domain. The embedding of a compact convex set K as the state space of A(K) (Theorem 4.7) is due to D. A. Edwards [87]. The standard text sources for the material in Sections 4—6 are Phelps [169] and Alfsen [5]. The latter has an excellent bibliography that traces the historical development of the
Choquet theory. We merely note some of the highlights. The metrizable version, Theorem 6.6, was proved by Choquet [60] in 1956. The general case, Theorem 6.8, is in Bishop and de Leeuw [40]. The ordering> or close variations of it are found in Bishop and de Leeuw [40] and Choquet [61]. Theorem 6.4(iii), the main development in showing the existence of maximal representing measure, occurs in Bonsall [45] and Choquet—Meyer [63]. The characterization of maximal measures in Theorem 6.5 is due to Mokobodski [161]. The Bishop—Phelps Theorem of Section 7 was proved by them in 1961 [41]. See also Bishop and Phelps [42]. An updated treatment with applications may be found in Phelps [170].
CHAPTER 2
Duality in Ordered Banach Spaces
We now take up the study of real Banach spaces that are partially ordered by means of a closed convex cone. We say the set P is a convex cone if (i) AP C P
for all A 0 and
(ii) P+PcP.
If P is a closed convex cone in the normed linear space E then (E, F) is partially ordered by a
•
b
if and only b — a
P.
We note that properties (1) and (ii) of P imply that (i)' a b and c E E implies a + c b + c, (ii)' a b and A 0 implies Aa Ab, (iii)'
Conversely, if (iY, (ii)', (iii)' hold for a partial ordering "s" then P = {a: a 0} is the convex cone yielding "and P is referred to as the positive
cone of "s ". Our principal source of examples for ordered Banach spaces consists of
the A(K) spaces introduced in Chapter 1 Section 4 (or equivalently, containing constant functions). The ordering is the subspaces of natural one, induced by taking F as the cone of non-negative functions. The cone P in A(K) has the additional feature of having non-empty interior (indeed, 1 mt F). This will not be the case in our general study. Finally, observe that (E, F) induces a natural ordering on E* with dual positive cone F" = —P° = {x E*: (a, x) 0 for all a P}. Thus, for example, A(K)* is ordered by F", where, using Proposition 1.4.2 and the fact that K is the state space of A(K),
A 0}. 30
pOSrFIVE GENERATION AND NORMALITY
31
Notes Some general reference books on ordered vector space theory are Jameson [134], Peressini [168], Vulikh [213] and Wong and Ng [217].
1. POSITIVE GENERATION AND NORMALITY Every a e A(K) can be written as
a0 (i=1,2).
a=a1—a2; Indeed, if a1 is any element
a =a1—(a1—a).
The fact that a is bounded above on K guarantees the presence of many such
ais. The ordered space (E, P) is positively generated if each a E E can be written as Definition.
(i=1,2).
a=al—a2; Equivalently, E = P — P.
If E is a Banach space then we shall see that if E = P — P then in fact one can find a bound (depending only on flail) for the norms of the a (i = 1, 2). Accordingly, we say E is aE
a,()
a=a1—a2;
(i=l,2)
and ha ill + 11a211
hail.
1.1. PRoPosITIoN.
The following are equivalent: (i) (E, P) is a..generated; (ii) E1 c a co (P1 u —P1).
Proof.
a = a1 —a2EE1 with a1, a2O and
Let
0< r = IIaj II + hla2hl a hail a. Then (with 0/0 taken to be 0)
a=
r[(flalfI/r)al/hlalhh+(hla2hh/r)(—a2/Iha2Ih)]E a co (P1 Li
Conversely, given b
0 let a = b/hibhl. Then (ii) implies
a = a(Aai +(1 —A)(—a2)) (a, E P1) so that, with b1 b=
hlbhlaAai, b2 =hlbhla(1 —A)a2, b1 —b2;
b
0
and
hhbiIl+
ailbhl.
DUALITY IN ORDERED BANACH SPACES
32
If E is a normed space then we say a new norm p is equivalent to 1' if there are positive constants a and b such that
onE. Of course equivalent norms induce the same topology on E. 1.2. THEOREM. Let (E, P) be an ordered Banach space which is positively generated. Then there is an a >0 such that Eisa-generated. Moreover there is a new norm p, equivalent to such that
(i) p = II. II on P; (ii) (E, p, P) is A-generated for all A > 1.
Let B = Co (P1 u —P1). Then, by Proposition 1.3.1(iii), B is a pregauge set. But Proof.
E=P-P=IJ nB=IJ ni so Corollary 1.3.4 yields an a such that C a'B
for all a' > a.
Thus E is a'-generating for all a'> a. Since B C E1 we have
BcEica'B foralla'>a. Thus if p = p(B) (the Minkowski functional of B) then p is a norm (since B is symmetric and absorbing) equivalent to 11. If a E P then a B if and only if a E E1 sop = on P. That (E, p, P) is a'-generating for any a'> I now
follows from Proposition 1.1 together with the fact that B is a regular pre-gauge set (p(B) = p(B), 1.3.3).
Example 1.6 below shows that Theorem 1.2(u) is not the same as being 1-generated. Our next goal is to determine when an element x E (E, can be decomposed as the difference of two non-negative functionals, that is, when is (E*, p*) positively generated? If we take the polar of the statement in Proposition 1.1(11), using some polar calculus we get the property
+p*)CE* whiCh is dual to a-generation in E. This leads us to the following definition. Definition.
The ordered space (E, P) is a-normal if
(El-P)rl(El÷P)cEa.
POSITIVE GENERATION AND NORMALITY
33
The next proposition gives the usual characterization of normality. The proof is straightforward and is omitted. 1.3. PRoPOsITION.
The following are equivalent:
(i) (E, P) is a-normal; (ii) then lixilsa max{IIcII, flbII}. 1.4. THEOREM. LetE be an ordered normed linear space. The following are equivalent: (1) E* is positively generated;
(ii) E* is a-generated for some a; (iii) E is a-normal. Proof. Since E* is a Banach space, (i) implies (ii) by Proposition 1.2, and (iii) follows as the polar of (ii). If (iii) holds then, since E1 — P and E1 + P are
regular pre-gauge sets, for any A >1 and hence
caEi=Ea which
by taking polars, gives (ii).
Theorem 1.4 establishes the duality between a-normality and a-generation in one direction. To complete the picture we investigate the situation when E* is a-normal. Theorem 1.4 yields that (E**, is a-generated but (E, P) does not quite inherit the full strength of this conclusion.
The space (E, P) is said to be approximately a-generated if (E, F) is a'-generated for all a'> a. Thus, the statement in Theorem 1 .2(u) could be rephrased as saying (E, p, F) is approximately 1-generated. Definition.
Let (E, P) be an ordered Banach space. The following are
1.5.
equivalent: (i)
E is approximately a-generated. Proof.
The dual E* is a-normal if and only if
(Er
+p*)CE*
are already we can take the polar of the Since the sets Er ± intersection and get the equivalent statement (iii) (P1 u —P1) E1 C a
DUALITY IN ORDERED BANACH SPACES
34
But then B = Co (P1 u —F1) is a regular pre-gauge set and hence (iii) Is equivalent to
Eica'co(P1u—P1) foralla'>a, which is precisely (ii).
The following example of an approximately 1 -generated space which is not 1-generated shows that Theorem 1.5 is the best conclusion. 1.6, EXAMPLE.
denote the Banach space of sequences x =
Let
=0 with
such that
lix ii = max
n = 1,
2,. . .}.
Let
E={xEco: Xl+X2 Then E is (1, 1, —1/2,
where x
E
a closed subspace (the zero set of the linear functional . Moreover, E is positively generated, 1' . .) 0. Let
0 means each
B = co (P1 u —F1) in E. We show x = (1/2, '-1/2,0,0,. . .)e AB\B for all A
y, z e P1 and a + (3
>1. If x = ay —(3z;
1, then
1/2 =
—f)zi
—1/2 = ay2—(3Z2. Thus, 1
But Yi
1, z2—z1
1, and y', z, 0(i =
1,
2). It follows that Yi =
1,
=0 and a = 1/2 = j3. But then
Y2 =
E
implies
=
and
1
= 1 = Zn+2 for all ii. Since y, z E
If 0 1.
ORDER UNIT AND BASE NORM SPACES
35
It can be shown in similar fashion that Pi — P1 in the above example fails to be closed. In fact
(1, —1, 0,0,. . . ,)eA(P1 —P1)\(P1 —P1) for all A >1.
2. ORDER UNIT AND BASE NORM SPACES We observed in the introduction that if E = A(K) then the function 1 E mt P. Indeed, we have E1
which clearly implies 1-normality. In general we say u is an order unit for the ordered norm space (E, P) if
E1 =(u—P)r'i(u+P).
(1)
In this case we call (E, F) an order unit space. We shall see shortly that the presence of an order unit essentially characterizes A(K) spaces. We say the ordering in a linear space is proper if P —P = {0}. This just
means that ±a 0 if and only if a =0. Let (E, F) be an order unit space. Then (i) E is 1-normal; (ii) u +E1 c P (hence u E mt P);
2.1. PROPOSITION.
(iii) the ordering in (E, F) is proper; (iv) if f is a non-negative (on F) linear functional on E then fE E* and Il/il
=f(u).
Proof. Parts (i) and (ii) are immediate from equation (1) above. If ±a 0 for all A and hence, a =0. For then —u Aa < u for all A >Oso that Aa
(iv) we have for any a E E1,
Hence f 0 on P implies If(a)I 11111
f(u)
Therefore IIfIIIIuII = IIfII.
With the aid of an order unit we can define the state space of E (in close analogy to Chapter 1, Section 4) by
S ={x EE*: x 0 on P and (u, x)= 1). From 2.1(iv) we have that x E S implies lxii = 1.
By an order isometry we mean an isometry between ordered normed spaces taking the positive cone of one onto that of the other.
DUALITY IN ORDERED BANACH SPACES
36
2.2. THEOREM. Let (E, P) be a normed order unit space. Then the state space S is -compact, convex and E is order isometric to a dense subspace of A(S). If E is a Banach space than E is order isometric to A(S). Proof.
From the definition of S we have that S is the intersection of a
hyperplane with the dual cone and is contained in Thus S is E implies and convex. Moreover ES. Since E is 1-normal, E* is 1-generating and hence
=co(Pr u—Pfl=co(Su—S). If 0: E
A(S) is the restriction map then IlOaII=sup{J(a,x)I: x ES}=sup{I(a,
Since Bu =
1
x
we have from Theorem 1.4.7 that OE is dense in, or if complete,
equal to A(S). Definition.
If P is a closed convex cone then a subset B of P is called a base
for P if (i) B is closed, convex and bounded, and (ii) for each 0 a E P there is a unique A >0 such that a/A E B. If (E, F) is an ordered normed space such that P has a base B and then (E, F) is called a base norm space. By Theorem 1.5 this implies E is approximately 1-generated so that in fact
E1= fl aco(Bu—B). We say (E*, is a dual base norm space if the base B is gives hence compact). In this case the
(and
=co(Bu-B). 2.3. THEOREM.
Let £ be a Banach space. Then the following are
equivalent: (i) (E, P) is an order unit space, (ii) (E*, is a dual base norm space. is the state space of E, E is order If these conditions hold then the base of isometric to A(S), and E* is order isometric to A(S)*. If (1) holds then the state space S of E is since fact a base for Proof.
S={xEP*: (u,x)=1}.
- compact, convex
and in
ORDER UNIT AND BASE NORM SPACES
37
Thus, the fact that
Er =co(Su—S) shows (ii). If (ii) holds and B is the base then the restriction map 9: E A(B) is an order isometry (into) since
Er =co(Bu—B). Moreover, if u is defined to be identically 1 on B and extended (uniquely) to
be linear on E* then Corollary 1.2.5 shows a E. Thus 0 is an order isometry onto A(B).
We now show that a linear space with an appropriately defined order unit can be made into a (normed) order unit space. This will provide a strictly order-theoretic characterization of (dense subspaces of) A(K) spaces. Let P be a proper convex cone in the linear space E. We say the ordering on £ induced by P is Arc hi:: if for a, b E,
Definition.
ra
b
for all r 0 implies a
0.
We say u is an Archiinedean order unit if for a e E (a) a E E implies there is an r 0 such that —ru a
ru, and
(b) ra a for all r 0 implies a .x1 =0}. The ordered normed space E is (cx, n)-directed if and
3.3. only if
Eica(Di -F). The proof is a simple verification and is omitted. 3.4. PROPOSITION.
(A)°
If A = L1 —P then
={
I E P*:
1, 1 = (xi,. .
.,
Proof.
p*
(A)° =
and
for alldEEi} (
=jIEE*:
fl
i—I
1
DUALITY IN ORDERED BANACH SPACES
42
3.5. THEOREM.
If E is an ordered Banach space then the following are
equivalent: (1) B is approximately (a, n)-directed,
(ii) E* is (a, n)-additive. Proof.
If (i) holds then
(A)°c fl
(iii)
= aEr.
But this says precisely that E* is (a, n)-additive. If (ii) holds then taking the polar of (iii) gives
E1ca cl(151—P). But then 131—P is E-regular so that Corollary 1.3.3 shows ca'(151—15)
for all a'>a
and hence (1) holds.
The most interesting case results when a = 3.6. COROLLARY.
1.
If E is an ordered Banach space then the norm is
additive on the dual cone
if and oniy if the open unit ball in E is directed.
Theorem 2.3 characterizes the dual base norm spaces (those with a compact base). We can now characterize the dual spaces that happen to be base norm spaces (with base not necessarily
Let (E, P) be an ordered Banach space. The following are equivalent: (1) E is 1-normal and approximately I -directed; (ii) E* is a base norm space. 3.7. COROLLARY.
Proof.
If (i) holds, the dual properties give
Er =co(P? u—Pr) and
is additive on
But then Proposition 3.1 says
B = {x E
Dxli = 1}
is a base. Hence (ii) holds. Conversely, Proposition 3.1 in the other direction gives E* is 1-additive. This together with the 1-generation of E* gives (i) by Theorems 3.5 and 1.4.
pIRECTEDNESS AND ADDITIVITY
43
To obtain the "dual" duality theorems the whole procedure is simply reversed. We take E as before but give E the 1' product norm, Dali
and
then E*
has
the
=
haul
norm,
liii = max
This direction is easier in that no approximate properties arise. We simply state the results. 3.8. THEoREM. equivalent:
If E is an ordered norm space then the following are
(i) Lis (a,n)-additive; (ii) E* is (a, n )-directed. The following are equivalent: (i) E is a base norm space;
3.9. COROLLARY.
(ii) E* is an order unit space. If these conditions hold and B is the base forP and u the order unit for E* then
B={aeP:(a,u)=l}. In the case when E = A(K)* it is easy to check that the order unit space E* coincides with the space Ab(K) of all bounded real-valued afline functions on K with the supremum norm.
3.10. EXAMPLE. We can construct an approximately 1-directed space along the same lines as Example 1.6. Let
E={i ec0: Let
a,1 v forces Ci = cz = c3 =1 and hence each If c E E1 and c a, b then c,, = 1, which is impossible if c e c0. Thus E fails to be 1-directed. It is not
difficult to check, however, that E is a -directed for all a >1.
DUALITY IN ORDERED BANACH SPACES
44
Notes The pioneering work in abstract ordered vector spaces was conducted by a Russian school led by Kantorovich in the mid to late 1930s. The first duality result of the type considered herein is due to Grosberg and Krein [122] who proved Theorem 1.4 in 1939. The reverse duality (Theorem 1.5) was shown by Andô [18] (without regard to the constant a) and, in its present form, by
Ellis [100]. The representations of order unit spaces and the results of Theorems 2.2 and 2.5 representing them as function spaces are essentially due to Kadison [136]. The notion of base-norm space and the duality of Theorem 2.3 is due to Ellis [100] and Edwards [87]. Theorem 2.6 on locally compact cones is shown by KIee [142]. The duality between additivity and directedness is established by Asimow [23]. The case a = 1 (Corollary 3.6)is proved in Asimow [22] and Ng [165]. Corollary 3.9 is shown in Ellis [100]. Example 3.10 appears in [22].
4. HOMOGENEOUS FUNCTIONALS ON CONES AND THE DECOMPOSITION PROPERTY
In this section we develop some techniques involving homogeneous functionals on cones which will be used in characterizing Banach lattices and discussing their duality properties.
We take E to be an ordered norm space with closed convex proper positive cone P. We say P has the decomposition property if whenever
(b1,b2O) there exist a1, a2 0 such that
a=a1+a2 and
a1_Obegiven. Letx E be such that each y x let = h(x) and
y X let be a neighbourhood of y on which such that if we can choose . .,
1)
= h(x) 1.For = h(y). For each > h — e. By compactness
•
then gx(x) = h(x) and g, > h — r. For each x let be a neighbourhood on which the (as above) satisfies > h + 6. Again, by compactness, we choose Xi,. . . , Xm such that if
g=gx1A then
h—e0 on X,, then
a= n-=1
is
then there is an
Q*{xEp*:(a,x)0}.
and U,,,
in
the required function in (M P)1.
L)(JAIITY IN ORDERED EiANACH SPACES
78
Such a Q* is called exposed and Corollary 8.7 shows that Q* is exposed if is semi-exposed if and and only if Q* is a semi-exposed G8. Moreover, only if Q* is the intersection of exposed sets of P*. Next, let Q* be a section by f in and consider the property in question (3). We say Q* has the positive extension property (PEP) if for each a F with a C) on Q* there is a b E F with
b()
on
and h=a on
in terms of Q and J we can characterize this quite easily.
8.8. PRoPosrrloN. If only if
isa section by fin
then
Q* has the PEP if and
Q=f+P. Combining this with Proposition 8.3 and Corollary 8.5 we have the following.
Let J be a closed subspace of F. Then J gives rise to a with the PEP if and only if (i) J + P is closed in E,
8.9. COROLLARY.
section Q* JO (ii)
If in addition (iii)
then Q* is a semi-exposed face of P* with the PEP. If q : F Elf is the quotient map then the adjoint q* (E/J)* E* (using Theorem 1.1.2(v)) identifies (E/J)* with .1° and is an isometry (where J° has unit ball J?). Moreover, properties (i) and (ii) in Corollary 8.9 are equivalent
to the cone qP being closed and proper in Elf. Thus, the dual cone Q* —(qP)° = (q*) l(p*) = p*
that (E/J)* is order-isometric to (J°, Q*), In particular, the quotient space (E/J, qP) is Archimedean if (i) and (ii) hold since (ii) implies qP is proper and rq(a)q(h) if and only if b—ra €P+f. so
Thus rq(a)q(b) for all r
0 if and only if
b/r—aeP+J forall
r0.
Then, letting we get —a P+J so that qa 0. A closed subspace J of F satisfying (i), (ii) and (iii) is called an Arc himedean order ideal, if in addition (iv)
DECOMPOSITION OF DUAl. CONES
79
(so that the statements of 8.1 also hold) then I is called strongly Archimedean.
Let J be a closed subspace of E and let Q* The following are equivalent:
8.10. COROLLARY,
Jfl
(i) I is Archimedean; (ii) Q* is a section of.P* by I such that for each a E Q there is an m E Jsuch
that
a +rn a
E J.
a is a E Since I is positively generated, choose m2 El
with m2a—b,0. Then Conversely, (ii) clearly implies I is positively generated with the PEP. Verifying these conditions can, in practice, be quite difficult. What we seek next are sufficient conditions of a geometric nature (involving Q* in p*) for various of these properties to hold.
To motivate the next definition we consider the case where Q* is a complementary face of with projection ir. Then ir is a bounded (by normality and positive generation) projection of E* onto N = Q* — Q*. Let denote the complementary projection with range L. Then both N and L are norm-closed subspaces. Let q:
E*/N (quotient map)
and
withp oq=o., Then p is an isomorphism onto L with Ip(qx)II = IkrxII = Ikr(x
so that p is a Banach space isomorphism. The point here is that the induced quotient norm on L is equivalent to the restriction norm. The quotient norm, IIqzII, is the distance d(N, Er )(z) (Chapter 1, Section 3) of z from N, and if Z EL, llqz liz Ii iIoli IIqz Ii.
In particular if x E
then
xy+z;
YEQ*, ZEP*
with =
DUALITY IN ORDERED BANACH SPACES
$0
Thus x can be decomposed into an element of Q* and an element of F" far (relative to lixil) from N.
Now let Q* be a section of P* by .1, so that N = J°. We can measure the distance from N in E* by taking any norm-closed convex neighbourhood W of 0 in E* and letting
d(N, W)(x) = p(N + W)(x) = inf {r 0: x EN + rW}.
For convenience we let the N be understood and denote this by dw(x). We define Q* to be a decomposable section of P* by J if there is a positive homogeneous map ir : F"' Q* satisfying (1) ir 1 (the identity map) on Q*, J)*, (ii) for each X E E (1 (iii) 11(1 — ir)xjj for some (norm) neighbourhood W of 0 in E*. Thus each x P"' can be decomposed, as in the case where Q* is complemented, but the set of elements z = x — irx need not, in general, be convex. Indeed Q* may be "surrounded" by (1— iT)P*, and in particular, need not be a face of 8.11. THEOREM. Let Q* be a decomposable section of F"' by J. Then is ,v*_closed in E*, (i)
(ii) P +J is norm-closed in E.
Proof. We can assume that the neighbourhood W in (iii) is
r>0, so that x
dw(x)r (i) If
for some
implies
if and only if
x €J°+Er.
+J°)1 then
u=y+z+n;
yEQ*,
n€f°,
ZEP*
and IIzII
(y + z)
r.
Thus
(p* +J°)1 =
which is
Hence, by the
(ii) The decomposability is equivalent to
Theorem, F" +J° is
DECOMPOSrJ ION OF DUAL CONES
81
which yields
in E. We now apply the Gauge Lemma (1.3.2) with
B+'J,,
A+*P,
If a E Q then J(a)- = ?(P, Jr) and
so that d(P, Jr)(a) = d(a)< co. Hence a E P+J, for all s > r. This shows
(l+P) = Q =J+P. 8.i2. COROLLARY.
If 0" is a decomposable section of
by land
q
then
in E*/J° and
is
(E*/J°, qP*)
(I, I
(order-isometry).
From Theorem 8.11(i) we have qP* is in E*/J° with the usual identification of E*/J° and J* (q is the adjoint of the inclusion map i : Ic E). Moreover, Proof.
qP* = (qp*)OO = (i '(—P))° = (I
If (I, I P) is positively generated then (Corollary 8.5) Q* is semi-exposed. This leads us to the notion of positive decomposability: we say 0* is a positively decomposable section of P* by I if there is an h E** with h on 1° and the set W in (iii) can be taken as
0
W={zEE*: h(z)1}. Then
dw(x)=inf{r0:xEJ°+rW}=max{h(x),O}. Thus on
we have X
= y + z;
y E Q*
and
IlzII h(x).
8.13. THEOREM. Let Q* be a positively decomposable section of by I. is an approximately a-directed Banach space (a = lihil). Then (I, I Proof.
From Corollary 8.12 and Theorem 3.5 it suffices to show the
quotient norm on Let IIqx1II
is a-additive. Let (xj)7..i c
be given with x =
z = x — x. Then IIz.II
= h (x) = h (x + n)
(any n E
J0)•
x1.
I)UALLTY IN ORDERED BANACH SPACES
82
Thus JJhIIJJqxII.
8.14. COROLLARY. If Q* is a positively decomposable section of P* by J then Jis an Arc himedean order ideal and Q* is a semi-exposed face of with
the PEP. 8.15. COROLLARY.
of E with
Let (E, F) have the DP and let J be a closed subspace The following are equivalent:
(i) Q* is a face of and a section off; (ii) J° is a positively generated order ideal in E*; (iii) J is a strongly Archimedean order ideal in E; (iv) f is a positively generated order ideal in E. Proof.
Since (E, P) has the DP Q* is a face of
if and only if Q* is
complemented. In this case Q* — Q* is and Q* is a positively decomposable section of Thus, since Q* is a section by J, (I) implies that Jo =0* — Q* and, since Q* is a face, that J° is an order ideal. If (ii) holds it follows that 0* is a facial section by J so that Corollary 8.14 shows J is an Archimedean (hence strongly Archimedean) order ideal in E. Then (iii) implies (iv) by definition. If (iv) holds then we observe first that 0* is a face since 0* {x P*: (a, x) =0 for all a J} = {x P*: (a, x) =0 for all a J P} and hence is an intersection of faces. To conclude (i) then, we must show
or equivalently =
0* _Q*
Clearly Q* Q* cJ0. If x e J° and a f P
since Q* Q* is then, using Theorem 5.5,
(a, x vO)=sup{(b, x): Ob since J is an order ideal. Since J is positively generated, we have x
Hence J°= Q*...Q* Notes Theorem 8.1, in the context of A(K) spaces, was shown by Edwards [87] and Alfsen [3]. In the latter the number r is related to the bound on the norm of
the extension function. This is discussed in greater detail in Chapter 4 in
sIABILrFY OF SPLIT SETS
83
connection with interpolation in complex function spaces. The notion of seif-determinacy is introduced in Alfsen [1]. An example of a non-selfdetermining face is given by Asimow [25]. Proposition 8.3 (the A(K) case) is found in Ellis [101]. Other related results characterizing order ideals whose annihilators determine faces of the dual cone are found in Jameson [134], Ellis [100], Bonsall [44] and Asimow [26]. In the years 1968—70 much work on A(K) spaces centred on the various
extension properties of functions in A(F), F a face of K. The two main properties are incorporated in the condition of Corollary 8.10(11). These are the positive extension property (PEP) and the directedness in A(K) of the
null functions on F. As Corollary 8.10 shows these are tantamount to the conditions of Corollary 8.9 and hence the usefulness of designating the
Archimedean ideals. This terminology and the applicability to A(K) is found in Størmer [206] and, in the case of strongly Archimedean, Alfsen [3].
Geometric conditions guaranteeing these extension properties were first noticed for simplexes where each closed face is split (Alfsen [3]) and also discussed in Effros [93] and Lazar [147]. Asimow [23] gives geometric conditions on Kand F (in terms of gauges) that assure the directedness of .L
In Asimow [25] the more restrictive but geometrically more appealing notion of positive decomposability is introduced for compact convex sets as a sufficient condition for the Archimedean extension properties of Corollary 8.10.
9. STABILITY OF SPLIT SETS AND THE CHARACTERIZATION OF COMPLEMENTED SUBCONES Let E be a Banach space with bounded complementary projections ir and whose ranges are the closed complementary subspaces N and L. Let S be a closed convex set containing zero with Minkowski functional p = Ps. We say S is split with respect to ir if
p=p C The next two results will clarify the geometric content of this definition. 9.1. PROPOSITION.
Let A and B be closed convex subsets of N and L
respectively, each containing zero. Let
S=cö(A, B). Then
(i) S = Co (A, B) + PA + PR where
PA={aEN:pA(a)=O} and Pfl={aEL:pB(a)=0}, (ii) Ps = PA ° IT +
0
84
DUALITY IN ORDERED BANACH SPACES
Proof.
Let c,,
c
with
c,, = A,,a,, +
a,, E A,
E B and
By passing to a subsequence we can assume A,,
A
A,, +
1.
and p.,, -> p.. Now
-ftc = urn irc,, = urn A,,a,,
so that PA °
ir(c) slim inf pA(A,,a,,)
Hence 'ftc E AA and similarly
A.
E p.B. Therefore
c = irc +crc
where we mean 1 for 0/0. Then both (i) and (ii) follow, since A + p. 1.
9.2. PRoposiTioN.
The following are equivalent: (i) S is split with respect to IT,
(ii) (iii)
Proof.
if ps=psoir+psoo.then Ps = PA ° IT + PB ° U,
where A = S N, B = S L so, using Proposition 9.1, (ii) holds. If (ii) holds then Proposition 9.1(i) shows -irS c S N, oS c S L. Thus equality holds and (iii) follows. Clearly (iii) implies (ii) and hence, by Proposition 9.1 (ii), (1).
9.3. COROLLARY. IfS and Tare split with respect to IT then so is
R=
(S, T)
and
,rR =
(ITS, ITT).
We now consider a Banach space E with a closed subspace M for which the
polar N in E* is the range of a bounded projection ir. We let L and o• denoted the complementary subspace and projection. We take A, B as closed convex subsets containing zero in E such that both and B° are split. The next result shows the preservation of the "splitness" of polar sets
under the formation of intersections in E. We formulate this wi iout reference to order properties in E for application in Chapter 4, although our immediate goal is to determine the order properties of subspaces in (E, P) giving rise to complemented subcones of Pa'. Let A, B be closed convex sets containing zero in are split with respect to IT, whose range, the Banach space E, such thatA°, N, is the polar of the closed subspace M c B. Let C = A n B. Then 9.4. THEOREM (ANDO).
STABILITY OF SPLIT SETS
85
(i) A + M and B + Mare closed in E, (A°, B°) in E* then C° is split and (A + M) r's (ii) if C° =11.
(B+M)=C+M,
(iii) in general, if flcrfl 1, where o = 1—ir, then (A + M)
(B + M) C A
(B+E5)+Mforalls>O. Proof. We use the Gauge Lemma to show A +M is closed, with A *-+A, D -*(A + M). We need to show B the notation of I Section 3)
(A
kd(A, E1)(x)
lull. Then p
Now, for any n EN, iio'xil =
liux +oizll = ilu(x + n)ii
kilx + nil.
Hence
II.1100.kp(N+Efl. Since A° is split we then have
pp(A°)o ir+p(A°)ou+rII.Ilocr = p(A°) ° ir + (p(A°) + ni' Ii) ° which says, by Proposition 9.1,
(A+Er)°),
(A+Micr)°
from which (*) follows by taking polars. Hence A + M is closed. If
C° =fl'
(A°, B°)
then, by Corollary 9.3, C° is split and so that
(C+M)=(C+M)=(A and hence (ii) is shown. Assume now that the complementary projection o' with range L in E* has 1. Let B' = B + E5 and CS = A B5. We shall apply the Gauge Lemma IIcrII
DUALITY IN ORDERED BANACH SPACES
with
As above, we require (**)
(A
To prove (**) we first evaluate the support functional, p(C5 +M,), of Cs+M,. Now p(Mr) =
o
(since 11o11 1).
Next p(C5) =
since (B')° is
(A°,
=
(Theorem 1.1.5). Thus,
=p(B)+slIi1p(B) ° ir+(p(B)+sII'II)oo since B° is split and 11o11
1. Hence
so that
=ëö
c
(D°, (C')° L),
where Hence
p(C5)p(D)°
o
and
p(CS +M,) = p(C5)+p(Mr) p(D) °ir +p(C5 + rh'
II) °
Therefore, (CS +
c
(D°, (C5 +
so that (**) holds, and hence (iii) is shown. complemented subcone of P* under the Assume now that Q* is a projection ir with N = Q* — Q*• Since N is the range of a projection N is Combining this with closed and hence, using Theorem 8.1, N is Corollary 8.9, we have that in particular 0* is a section of P* by the strongly Archimedean ideal J = N° in E. If Q = (_Q*)O then J =0 —Q, and
Q=J+P.
87
STABILITY OF SPLIT SETS
We have (E/J, qP) is order isomorphic to A0(Q*) and
a0b0 inAo(Q*) if and only if there are a, b e E with
P +J = Q, in Ao(Q*) if and only if a —b €J. Thus we say a b(Q) if qa = a0,
and a0 = b0
and
qb = b0
b— a
qa qb, or equivalently aIQabIQ*. We write a b + s if there is a z E Ee with a b + z.
Since E is positively generated we can, by renorming E, assume E* is i-normal, so that the norm in E* is monotonic on This guarantees that are less than or equal to I so the hypothesis of Theorem 9.4(iii) is flirfl and satisfied.
Let Q* be a let a, bj(i = 1,. . ., n ; j = 1,.. . (i) all 9.5. THEOREM.
,.
complemented subcone of m) c, c0 be elements of E such that
and
(ii)
Then for each g >0 there exists m e J such that
(In short, if the b's and a's can be interpolated in E and c0 interpolates the b's then c0 has an extension in E = A o(P1c ) that interpolates in E.) and a's on Proof.
By translating we can assume c =0 in E. let
A={c€E: Then the support functional p(A,) just equals a1 on
(since
0 a,) and
clearly
OnP*
so that
is split. If
then the support functional p(A) equals
inf{a1,. .. ,a,,} which we shall denote as p. But the definition of j5 (Section 4) shows that fl is
also the support functional of
{cEE**:cai,.
.
in(E**,P**)}
DUALITY IN ORDERED BANACH SPACES
88
and hence the Minkowski functional of A2).
This shows the w*..and fi. Il-closed convex hulls of A°1,. . ., Hence
coincide.
.
Thus
c0a1,. ..,a,,(Q) if and only if c0 "extends" to
c+ma1,. .
mEJ.
.,a,,,(P);
Similarly
bi,...,bpnco(Q) if and only if for some m'€J
b1,.. .,bmco+m'. Thus hypothesis (ii) says
c0E(A+M)tTh(B+M) which, by Theorem 9.4(iii) gives Co E
A (B
+M.
We now present properties in the ordered Banach space (E, F) that characterize the complemented subcones of P*. Let Q be a closed convex cone containing Pin E and let M = Q —Q be the annihilator of the
dual subcone Q* of P*. As above we write a b(Q) for the ordering induced by Q. In the next theorem, part (ii) characterizes complemented subcones in terms of the polar cones in E and part (iii) characterizes the annihilators of complemented subcones. The following are equivalent: complemented subcone of (ii) if a, b1, b2 are given in E with
9.6. THEOREM.
(i) Q* is a
Ob1, b2 then for
each e >0 there is an in E Msuch that
—e a+m
b2,
STABILITY OF SPLIT SETS
89
(iii) (a) if a1, a2 EM, b E E with a1, a2 b then for each 8>0 there is an
a €Msuch that
a1, a2a (b) if a EM, b1, b2 E P with a b1 + b2 then for each e>0 there exist
ai, a2EMsuch that
a=a1+a2 and a,b,+e
(i=1,2).
If J is a closed subspace of E satisfying (iii)(a), (b) then .1° Proof.
(i) implies (ii) is immediate from 9.5. To show (iii)(a) we have
0b—a1, so
is a
with annihilatorf.
complemented subcone of
b—a2
b—a2(Q)
and
that (i) gives a EM with b—a2
or —b—s 0. This shows is a semi-exposed face. We show next that I = Thus let
J=
and e>0 be given. Then
a=m1+p1+z1=m2—p2+z2;
mel,
p,€P and
IIziII<e.
DUALITY IN ORDERED BANACFI SPACES
90
Hence
m=rn2—m1=p1+p2+w,
flwll2e.
Since E is positively generated, for some constant a, we have
and
llpllae.
Hence
(11,2).
a1EJ and
rn=a1+a2; Now
rn—a1 —a2_0, Ai2 and hence a is an Archimedean order unit for A(K)/J (as defined in Section 2). Thus (A(K)/J, z2) is a normed order unit space with unit ball Proof.
1={a:±aAa} and positive cone P = qP, by Theorem 2.5. Hence we have qE1
I
and so a + qE1
a + Ic P
so that mt P (quotient norm) is non-empty. Since P is linearly closed with non-empty interior, P is closed in the quotient norm, and hence I +P is closed. Then
I = (I +P) (I —P) of follows as well, since P is proper. This shows (iii). Now, the (iii) and (iv) follows from Corollary 8.9. If (iii) holds and b for all r 0 then
forallrO.
APPLICATION TO A(K)
93
Thus ra e 1+ J — P for all r 0
so that
aE(J—P)=J—P and hence
E —P. Then
shows P is proper so (i) holds.
This allows us to characterize the Archimedean and strongly Archimedean order ideals of A(K), using Corollary 8.10. K. 10.2. COROLLARY. Let J be a closed subspace of A(K) with F = (1) The following are equivalent: (i) A(K)/J is Archimedean ordered with proper cone P and .Tis positively generated, (ii) Fis a section ofKbyJsatisfying: foreach a E A(K), a 0 on F there is a b E A(K) such that
bIF=aIF and (iii) J is an Archimedean order ideal. (2) The following are equivalent: (i) .1 is a strongly Archimedean order ideal, (ii) property 10.2(1)(ii) is satisfied and a E A(F) implies there is b E A(K) with
We say F is a decomposable section of K by J if the cone Q* spanned by F is
a decomposable section of P* by J. The notion of positive decomposability is defined similarly.
10.3. THEOREM. LetFbe a section of Kby J. (1) The following are equivalent: (i) F is a decomposable section of K, (ii) there is a 8>0 such that each x e K can be written as a convex bination of y and z where
y F and
z e K with liz — J°II
8.
(2) The following are equivalent: (i) F is a positively decomposable section of K, (ii) there is an h E A(K)** such that h 0 on J° and each x K can be written as a convex combination of y and z where
y F and z K with h(z)> 1.
DUALiTY TN ORDERED I3ANACH SPACES
94
If (1)(i) holds then for some r y E Q* x = y ÷ z;
Proof.
1 each x e and
lizil
can be written nix —J°fl.
Then, using the additivity of the norm, (ii) follows easily (with ô = l/r). The
converse is also straightforward, as is the proof of (2), using the remark preceding Theorem 8.13. Let F be a positively decomposable section of K by J.
10.4. COROLLARY.
Then J is an Archimedean order ideal. Thus the extension property of Corollary 10.2(1)(ii) holds.
We now consider the case where F is a split face of K. Since the norm is additive on P" we can use 4.4(u) to obtain a strengthening of 9.5. 10.5. THEOREM. Let F be a split face of K and let —f, g be continuous convex functions on K.
(i)If and a E A(F) with
then for each s >0 there is a b A(K) such that
biF=a and (ii) If
g00
x E Q* implies
f(x) = (flo*Y SO
that = (Af)°.
(A1)°
By taking polars of this we have A1 + J = A1.
Thus (i) follows from Theorem 9.4(iii) and (ii) follows from Theorem 9.4(u) since in the case 0 e. Put = e/4a and choose a compact set E with E' E c i3K, /L(E)> 1— and
v(E)> I —13;letf€ 1 an extension off satisfying ajif II we obtain This contradiction gives ji = JEf dji f dv
v,
v(g) and and hence K
is a simplex. (ii) Let be a boundary measure in A (K)1 and let E be a compact subset we have FiFE A(K)', and so ji(E)=0. Since K is of aK. Then, if
standard and p. is regular it follows that p. =0, that is, K is a simplex. The next result shows that K being standard is necessary in 2.3. 2.4. THEOREM. There exists a compact convex setK which is nota simplex but which has the following properties:
Ill
FACIAL CHARACTERIZATIONS OF SIMPLEXES
(i) .3K is a Bore! subset of K; E is a split face of Kfor all compact subsets E of .3K.
(ii)
Let V = U { Va: a [0, 1]}, where the disjoint sets Ya each consist of three points se,, ta}. Topologize Y so that each and ta is isolated and such that each sa has a neighbourhood base consisting of the sets {sa} U 00. Then V is a compact Hausdorif space, each Ya is discrete in the relative topology and a s(1 defines a homeomorphism between [0, 1] and {sa: 0 a 1 }. We will identify [0, 1] and {sa: 0 a 1} in the remainder of the proof. Let A denote Lebesgue measure on [0, 1] and Jet A1, A2 be the restrictions 1] respectively. Define L to be the closed linear subspace of A to [0, and for all a [0, 1], of CR( Y) given by L = {f€ CR( Y): f(Sa) = dA2(a)}. Certainly L contains the constant functions. Sf(sa) dA 1(a) = To show that L separates the points of Y we first note that the functions =0 for all a, f,3(ra) = for a /3 and belong to L, where Therefore each and each is a peak point for L, and =1= so we need only show that L separates the points of {sa: 0 a 1}. But a /3 there exists anf in dA1 =5fdA2, and we can extend f to a function in L by putting f(rA) = f(s.,) = f(t.d,) for all 'y in Proof.
[0, 1].
L": 4(1) = = 1} of L then we may If K denotes the state space identify L with A(K) in the natural way (see 1.4). The Choquet boundary for L is identified with .3K, and this clearly equals U {ra, ta: 0 a 1}. Therefore the closure .3Kof .3K identifies with the Silov boundary V for L. In particular we observe that .3K, being the complement in V of the compact set {sa: 0 a 1}, is a Borel subset of Y. Define two measures on V by the equations v,(f) = ff(Sa) dA1(a), Y). Let B denote the family of functions of the form g + h, j = 1,2,1 E where g, h E Y), g = C) except at finitely many of the points ra, ta and = h(ta) for all a in [0, 1]. Then B forms a subalgebra of CR(Y) h(ra) =
containing constants and separating points of Y, so that B is dense CR( Y) by the Stone—Weierstrass Theorem. The mapping a
in
+ h)(ta) = + h)(ra) g takes only finitely many values. Consequently the mapping a is also measurable for f€ CR(Y), using the density result which has just been established. We therefore obtain, for / = 1, 2,
J
+h)(ra) + (g
and hence for all f in
dA1(a)
=J
dA,(a)
=
J (g + h)
Y)
j'
(1)
SIMPLEX SPACES
112
Clearly the measure
We now wish to find all measures in
to A(K)1 M(Y), and the definition of L implies that is the linear span of and N, where N is the 0 a 1} in M( Y). Suppose linear span of the measures — to in N, where that {'q a)} is a net =
—
belongs
=
in A(K), Then for each a [0, 1], we can find we see that converges pointwise to = 0. Using the fact that there is a g(, A(K) with ga(ra)= 1 = we see that = discrete part fld of rj has the form
1,
such that fl({sa}) +
=0 except for and hence the
00
has no — qd, belong to A(K)1. Since — 1} and {ta: 0 a 1} are discrete it atoms and since the sets 0 a follows from regularity that — is concentrated on {sa: 0 a 1}. But then — rld)(h) =0 for all h E CR[0, 1], and so q = This description of N shows that the only possible measures in aM(K) A(K)1 are the multiples of because no maximal measure can have an atom lying outside then E must be Now if E is a compact subset of
Consequently fld, and also
finite, and it is easy to construct a function in A(K) which peaks on a face F
with aF = E. Since p(E) = 0 for a finite set E we have p.'jE E whenever and hence isa split face of K. To complete the proof we need to show that K is not a simplex, and this will be the case if we show that is a boundary measure. We first note that + is the only maximal probability measure on K with resultant se,.
Indeed, if p.' is another such measure then
+ Eta) — p.' E t9M(K)
+ + bp. for some real number b. Since p.' is a A(K)1 and so p.' = probability measure and p. has no atoms we can deduce that b =0. The measures p1 will be maximal measures if and only if v,(f) = v1(f) for
each continuous convex function f. Using (i) we have =
f
= j'
dAj(a) dA1(a) =1/1(/),
+
where the second equality follows from 1.6.2 and the fact that
the only maximal measure representing sa. Therefore p. = boundary measure.
—
is is a
FACIAL CHARACTERIZATIONS OF SIMPLEXES
113
If F is a face of K then the complementary set F' consists of the union of all faces of K which are disjoint from F. Each x in K may be decomposed F, z E F' (see [2, Proposition 2.6.5]). If x Ày +(1 —A)z, with 0 A 1, y F' is itself a face of K and if the constant A is uniquely determined by x then F is called a parallel face of K (see 2.10). A closed face F of K is parallel if 8M(K), or equivalently if is an affine =0 whenever E
function, where for any bounded real-valued function f on K we write gf}, x€K. f(x)=inf{g(x): A closed subset D of K is said to be dilated if whenever is a maximal probability measure on K with resultant x E D then is supported by D. Evidently every closed face of K is dilated, and every compact subset of 8K
is dilated. If D is a dilated subset of K then a continuous real-valued function f on D is called affine if p(f) =f(x) whenever is a maximal probability measure on K with resultant x €D. If D is a compact subset of 8K then it is clear that every continuous function on D is affine. It is easy to verify that if D is a closed face of K then a continuous function on D is affine in the preceding sense precisely if f E A(D). In the case when K is standard, Theorem 2.3(i) shows that K is a simplex if there is a constant C such that whenever E is a compact subset of 8K then
E is a simplex and a face of K and each f This result can be rephrased to say that K is a with simplex if and only if every continuous positive function f on E has an with extension g Cu/h, where C is a constant independent of E. Theorem 2.4 shows that the hypothesis that K is standard may not be dropped in this result of Rogalski. E
We present below an analogue of Theorem 2.3(1) which is valid for all sets
K. In the case when K is standard the compact subsets of 8K play a role similar to that played by the dilated sets when K is not standard; this is because when K is standard the supports of the maximal measures on K can be approximated by compact subsets of 8K. A"(K) stands for the space of bounded affine functions on K. 2.5. THEOREM. vex sets K.
The following statements are equivalent for compact con-
(i) K is a simplex.
(ii) If D is a dilated subset of K and if f0 is continuous and affine moreover for each closed on D then f has an extension g e face F of K, inf {a(x): a E Ab(K), 0 a 1, aIF = 1} =0 for all x inF'. (lii) The convex hull of any finite number of closed faces of K is a face of K, and for each closed faceFof K, inf {a(x): a A"(K), 0 a 1, alP =
1}=Oforallx inF'. (iv) Every closed face of K is parallel.
SIMPLEX SPACES
114
The implication (1) implies (iv) holds because every split face is parallel. (i) implies (ii). The second statement in (ii) holds trivially for split faces, so it is sufficient to prove the first statement of (ii). D) then If D is a dilated set then D is a face of K. Indeed if x e x eD, so that the maximal representing measure for x is supported by D. But then = by the extremity of x, so that x 9K. Lemma 1.6 now shows that ë6 D is a face of K. To complete the proof it will be sufficient to show that every function f which is continuous and affine on D has an extension in whenever and are probThis will follow if = D. ability measures on D with the same resultant y D let denote the unique maximal probability measure For each x D representing x. Given 6 >0 there exists a continuous concave on + 6 = A.,(f) + 6. Since f is affine on D function u f such that = ü(x) (see 1.6.2), and because ü is we have, for x in D, f(x) = convex we obtain Proof.
/Li(Ü)
It follows that obtain =
i(f)
=
AV(f) + 6.
and applying the same argument to —f we
= /h2(/) as required. (ii) implies (iii). It is sufficient to prove the first part of (iii) for two closed
faces F and G of K. If x belongs to aK\(F G) then it is evident that F u G u{x} is a dilated subset of K and that the function is affine and = 1 and =0 on F u G. Therecontinuous on F u G u {x}, where The set H = fore there exists on extension A(KY of
fl
(0): x E
u G)} is a closed face of K containing F u G, whose
extreme points belong to F u G. Therefore H = co (F G) and (iii)
is
proved. (iii) implies (i). To prove that K is a simplex we will show that A(K)* is a vector lattice. Using the Bishop—Phelps Theorem it will be sufficient to prove that p exists whenever p A(K)* attains its norm at some f E A(K) with exists in A(K)* for n 1, and if p in norm, then 11tH = 1. (Indeed if p in norm (see [108]).) If is a representing measure for p on A(K) with , and so it = Dpil then Il/Lu = V ÷D,L1 =P(f) =ffd —ft follows that supp + F, supp /h c G where F and G are the closed faces of K given by F = {x K: f(x) = 1}, G {x K: f(x) = —1}. Without loss of 0. Define o-(g) =f g 0, for generality we can assume that g A(K), so that clearly if p, 0. To show that o' = p + we need to prove that
f if whenever / p, 0 in A(K)*. By the Bishop—Phelps Theorem, for A(K)* which attains its norm and is each natural number n there exists with such that jifr,1 — (/ — o')lJ < 1/n. If A?, is a representing measure for of K such that then there are disjoint closed faces H, = EA(K)* is defined by If suppA
OF SJMPLEXES
FACiAL
for g E A(K), then we can decompose EG
H, y,, E (G
115
+ (1---
=
)'. We will show that
where
-.0 as n
If (1— 0 then E H, and hence belongs to the complementary set for the face co u G). Therefore, by (iii), we can find an A"(K) with 0 on co u G) and i—i/n. Identifying A(K)** with A"(K) as usual, we have 0 (4)
= 1
1
I
1\ 1 ——) +.
— It follows that in general we have A"(K) with 0 =0 Ofl Co (F u 1,
—*0. SimiLarly we can find
1—1/n, and
hence
-÷0, and consequently
We obtain
/1
1
1
——
-.0. It now follows that if
h A(KY then = urn { J h dA —
(4) —u)(h) = urn
0,
and hence 4) o as required. (iv) implies (iii). Let F and G be closed, and hence parallel, faces of K and let H = co (F u G). To show that H is a face of K we need to show that if A is a maximal probability measure on K with resultant x E H then supp A c H. v be We have x = ty + (1— t)z, where 0 t 1, y F and z E G. Let maximal measures on K with resultant y, z respectively. We have = A —
faces
— (1— t)v
=
E A(K)1 aM(K), and since F, G and F G are parallel =
G) = 0. Therefore
A(FuG)=A(F)+A(G)—A(FrG)=
1,
because and ii are supported by the faces F and G respectively, so that supp A is contained in H. The second condition of (iii) is satisfied since, for any closed parallel face F, the function with =1 on F and XF =0 on F'. belongs to The following result gives two ways in which facial structure distinguishes l3auer simplexes amongst simplexes. We will say that a compact convex set
SIMPLEX SPACES
116
is a closed split face of K whenever K satisfies Størmer's axiom if (Ua {F4j is any family of closed split faces of K.
2.6. THEOREM. Let K be a compact simplex. Then K is a Bauer simplex if and only if either of the two following conditions hold. (i) The closure of each face of K is a face of K; (ii) K satisfies Størmer's axiom. Proof.
The necessity of the two conditions is trivial. (i) implies (ii): Let {Fa}
be closed faces of K, and hence split faces of K. Since the convex hull of finitely many of the is a face of K it is easy to check that co (Ua Fa) is a face of K. Consequently Fa) is a face of K; and so K satisfies StØrmer's axiom. Now assume that K satisfies Størmer's axiom, and let E c i.iK be relatively closed. Then there exists a compact set G K such that E = G and Milman's Theorem shows that E) c G. Now each x E is a (split) face of K, and hence E is a face of K because of Størmer's axiom. But then E) G aK, so that E = E). Therefore E is facially closed and it follows from the proof of 1.2 that K is a Bauer simplex. Although faces of simplexes are simplexes it is far from true that crosssections of simplexes are simplexes, as the following result amply illustrates.
2.7. THEOREM. Let S be the w*(l', c)-compact simplex {y 11: IlyIl = 1, y,, 0, for all n}, and let K be any metrizable compact convex subset of a locally convex Hausdorff space. Then there exists a linear subspace M of 11 such that M S is affinely homeomorphic to K. Proof. Let be a sequence which is dense in 0 for all n, and let n 1, and go = 1 g,,. Define a linear operator T: c -+ A(K) by Tx = n 0, belongs to c. Let M be x,..g,, where x =
the range of T* in 1'. Since T(c) is dense in A(K), T* is one-to-one; also T* is positive since T is positive. If K and if e = {1}E c, then we have = (Te)(4)
= 1.
=
Therefore T* maps K into M S. Suppose that eM S. Then we have 'I'(l) = 'V(Te) =
= 1,
and =
=
0,
n 1,
E A(K)* and that
IDEALS (N SIMPLEX SPACES
117
0 for all n, where is the nth coordinate vector in c. Hence we have ensures that 'I' 0 and consequently that and the density of {f,,} in 'If
K. Therefore T* is an affine homeomorphism of K onto M S.
Notes
Examples with similar properties to those of 2.1 and 2.2 have been given by Rogalski [181], to whom Theorem 2.3 is due. It appears to be unknown whether, in the standard case, the condition that E is a simplex can be removed from 2.3(i). Theorem 2.4 is due to Ellis and Roy [110] and it involves the Bishop—de
Leeuw porcupine topology construction [40]; it is related to examples studied by Alfsen [5, 1.4, 2.3]. The fact that condition (i) of 2.5 implies the first statement of (ii) is due to Lifros [94,3.3; 95,2.3]. The equivalence of (i) and (iv) of 2.5 is due to Briem [54]. Theorem 2.5, in the form given here, is due to Ellis and Roy [110]. The first statement of 2.5(u) certainly does not characterize simplexes since it
does not imply uniform boundedness of the affine extensions involved. Similarly the first statement of 2.5(iii) does not characterize simplexes; an example of this phenomenon is given by McDonald [159, 1.9]. In the duality between A(K)* and A"(K) the set F4 = 1} = 0} is called the quasi-complement {x E K: inf{a(x): a E A"(K), XF a of the face F (see Alfsen and Shultz [10, p. 15] and also Section 5.6 (below). The second statement of 2.5(u) and 2.5(iii) can be re-written F* = F'. This condition does not imply that F' is convex, but it does imply that for each x E K the constant A E [0, 1] in the decomposition x = Ày + (1— A)z, y F, z e F' is unique. Theorem 2.6 is due to Størmer [206]. Of course every finite-dimensional compact convex set satisfies condition 2.6(i) since every face is closed. It is not know precisely which infinite-dimensional compact convex sets satisfy
this condition. Some results of a local nature on this topic have been obtained by Roy [184]. Theorem 2.7 is due to Lazar [146]. In the same paper he shows that every infinite-dimensional Banach space contains a compact set which is contained in no bounded simplex lying in the space.
3. IDEALS IN SIMPLEX SPACES
Recall that a subspace I of an ordered space (E, P) is an order ideal c El implies b El) if and only if P is a face of P. It is (a b convenient in the context of A(K) spaces to also require that I be positively and generated. Thus we will say I is an ideal in A(K) if is a face of 1= The first result is a direct reformulation of 2.8.15.
SIMPLEX SPACES
118
3.1. THEOREM.
Let K be a compact simplex and let I be a closed linear
subspace of A(K). Then I is an ideal in A(K) if and only if I = FL {fE A(K): f(x) =0, Vx E F} for some closed face Fof K. It is worth noting that if I is a (not necessarily closed) ideal that I' = j0 K is still a closed face F and hence F1 = J00 = I is a closed ideal. If {4} is a family of closed ideals in A(K) then, if Fa = we have fla 'a Ua Fcc)1. Hence if K is a Bauer simplex, so that Størmer's axiom holds, then fla is a closed ideal in A(K). This fails in general, as the following example shows.
L = {fE ca[—1, 1]: f(0) = 3.2. EXAMPLE. Let K {4i E L*: 4(1) = = 1} with the w *topology, and let 1,, = {f E A(K): f(1/n) = 0}. Then A(K) is a simplex space and is closed ideal for each n, while fl1 I,, is not positively generated.
An interesting problem is whether Bauer simplexes are characterized among simplexes K by the property that the intersection of an arbitrary family of closed ideals in A(K) is again an ideal. In the metrizable case, at least, the answer is affirmative, and we now discuss this result. We first consider a theorem which leads to the construction of many interesting examples of simplex spaces. c3Mq (K) will denote the maximal probability measures on K with resultant q. 3.3. THEOREM. Let K be a compact metrizable simplex and let q 8K\8K. Suppose that Y is a closed subset of 9K such that Y (aK\aK) = {q} and Then A = {f€A(K): f(y)= 0, where {ir}=
ir(f), Vy E Y} is a simplex space A(H) where aH can be identified with aK\ Y. Proof.
If a = ir(Y) then 0< ir(ÔK\ Y)
1—a. Let 7r(f), Vfe Y}
L ={f€ and put
X is homeomorphic to Y. To each naturally a measure A M( Y) so that (A
M(X) there corresponds
—A(Y)ir)(f)
for each f CR(ik). Therefore, if N={A —A(Y)ir: A €M(Y)},
IDEALS IN SiMPLEX SPACES
119
by approximating A by point measures we have N un X). If i9K) we have A — A (Y)ir E N then (for each Borel set B (B
A
=
( Y)ir(B), so that for each Borel set A c Y we obtain
A(A)= ,1(A)+A(Y)ir(A) Hence A is uniquely determined by hAil
—a) and also
p. 1(Y) 4 ir( Y)Ip.l( Y)(1
—
a)
1
It follows easily now that N is norm-closed, and hence generated) so that N = L1. since N is We can write
(1.3.5
p.(f)
={p.
VfE
some v E
=J
whcrc (i9K) for x E K; in fact = 0 when fE A(K)IaK, — satisfiesf h dv = the function h(x) = while if p. E A(K)' and f o (cf. Alfsen [2] Proposition 2.3.14). Since A =
L A(K), A' equals the
of A(K)'+L', and A(K) A EM(Y), Vf€ CR(aK)};
+(A
(*)
if and only if it is norm-closed (1.3.5). In the represen-
this set is
tation of p. E A(K)' + L' we may, by transferring an atom if necessary, assume that A =0; hence for any Borel subset C of Y\{q0} we have
p.(C)= v(C p.
and consequently
=
aK\oK
and
We obtain the inequality IIAIIlIpll(1 +2a/(1 —a)),
and as before, it follows that A(K) + L' is norm-closed.
120
SIMPLEX
Let r denote the quotient map of into M(aK)/A ',the dual space of A. Define, for p E M(aK), A (ii) E M( Y) by A (v)({qo}) = 0,
A (v)(A)
dv + v(A) + ir(A)A (v)( Y)
= JaKVIK
and let be as in (*) so that (v)(B) = v(B) for each Bore! subset of
for each Borel subset A of Y\{qo}, r(JL (ii)) =0. It is easily verified that (aK\aK) u Y. Therefore
Y) = 0,
—
and
r(v —p(v)) =
If r(ii1)
and
T(v),
Y) =0
lvi
then
Y)=0} and so there is a A v2(B)
M(Y) with
ii2(B
dv2+ A (B
Y)— A(Y)ir(B)
aK\aK
for each Bore! subset B of obtain
(reasoning as above). Since v2(aK\aK) =0 we
v2(B)
A(B
Y)—A(Y)ir(B)
putting B = Y we have A (Y) =0; this together with the fact that P21 Y =0 gives A =0 and consequently P2=O. This argument shows that the A* -. S, such that /(i(v)) = p — map is well defined. Clearly is linear and onto, and if çb(r(vj)) = #('T(V2)) then i.'1 —p.(vi) = and
—
so that r(vi)=r(v2).
Letv 0 belong to M(9K) B c (aK\aK) u V then
and let B be r(p)(B) =0, and if B
a Borel subset of
If
V we have
(v)( Y)ir(B) 0, ôKVIK
so that b(r(v))0. We have, for r(v)EA*, lir(v)ll = 'nf {liv
= liv —
For p M(aK) and B a Borel subset of
II:
r(v ) =
V we see from the definition of
A(v)(Y) that =
-f
aK\iiK
—v(Y\qo)ir(B)(1 _a)_1.
dv
IDEALS IN SIMPLEX SPACES
121
Therefore Y)
kb(r(v))li = I
ivj(aK\ Y) +
Y)j'
Y) +
uK \uK
Y) ( Y))d
Y) + ivl(OK\aK) + l'H( Y\q0) so
+ vl( Y\q0) I
lvii,
that and 4' is an isometry. We have proved that A * is isometrically order isomorphic to S. It is
evident that S is an L-space and that the extreme points of the state space of S can be identified with 3K\(aK\aK) Y) = 3K\ Y. 3.4. COROLLARY. Let K be a compact metrizahie simplex such that whenever is a sequence of closed ideals in A(K) their intersection is also
an ideal. Then K is a Bauer simplex. Proof.
if K is not a Bauer simplex then there is a q e aK\aK such that
supp (ir) has at least two distinct points
and P2, where fir) = Mq (3K). Let
be a sequence in 3K\{p,, P2) which converges to q, and let Y u{q}. If A is defined as in 3.3 then A is a simplex space A(H) with = c)K\Y. Since co (p3. P2) is a closed face of H we can find g,, g2 E A(H) i, j = 1, 2 and gj 0. We must have g1(q)> 0 since Pi, P2 C with gj(pj) supp (ir) and sowe can definef = —(g,(q)/g2(q))g2 c A. Sincef(q) = 0 and fc A we have f(x,1) =0 for all n. Put 4 = in A(K) so that is a closed
ideal and fE I,, g f, 0, and so g(q) =
I. Because I is an ideal there exists a g €1 with =0. Therefore Pi belongs to the closed
face g '(0) of K which contains q, so that 0= g(p1)f(p1) =
1.
This
contradiction shows that K is a Bauer simplex. if K is a compact simplex with 0 c aK then A(K): f(0) = 0) is a = partially ordered Banach space with normal, generating cone and such that Ao(K)* is a vector lattice. The space A0(K) is also called a simplex space (without a unit) and Theorem 3.1 has a complete analogue for A0(K).
Notes Theorem 3.1 is due to Efiros [93]. Examples of simplex spaces for which the family of closed ideals is not closed under arbitrary intersections have been given by Bunce [58] and Perdrizet [167]. The example presented here in 3.2 is due to A. Gleit. Gleit [116, 117] proved Theorem 3.3, and also proved Corollary 3.4 for simplex space A0(K), with K metrizable, showing that A0(K) is an M-space
SIMPLEX SPACES
122
if and only if the intersection of an arbitrary family of closed ideals in A0(K) is a closed ideal. are in one-to-one correspondence with the The maximal ideals in points of aK\{O}; in fact an ideal I is maximal if and only if I = {X}j for some x E aK\{0}. Effros [93] developed a hull-kernel topology for the maximal
ideal space of A0(K) and also of A(K) which is essentially the facial topology for aK\{O}.
4. STONE-WEIERSTRASS THEOREMS FOR SIMPLEX SPACES Let L be a simplex space and a closed linear subspace of A(K) containing the constants, with Choquet boundary äK. Then for each 4.1. THEOREM. x E öK the sets
Q(x) —={y c K: f(x) =f(y), Vf€L} are closed faces of K, and
L ={fE A(K): f is constant on Q(x), Vx aK}. Consequently, if L separates the points of Proof.
then L = A(K).
Let x E ÔK and suppose that Ày +(1 —A)z E Q(x) for some y, z e
K and Og(x)=g(Ay+(1—A)z)Af(y)A, and since e >0 was arbitrary we obtain a contradiction; so Q(x) is a closed
face of K.
Let fE A(K) be constant on 0(x) for all x in inf {g(y): g L, g f} for each y
and define = K. We evidently have and also for
x€K
(by 1.6.2)
1(x=sup{,i(f): where P. is the set of probability measures on K with L-
resultant x. Therefore if x E
and
we have supp
contained in the face Q(x),
so that f(x)=f(x). Let F denote the family
{g€L: g>f}, F and write u for the pointwise minimum of g1 and g2. Since L is a let g, simplex space the family
F1={v€L: vf on K. Consequently for and so attains its infimum on F1 with >f(x). A simple compactness each x E K there is a
argument together with the fact that L has the Riesz separation property now shows that there is a v E L with u > v >1 on K, and hence F is directed downwards. It follows that / is affine and upper semi-continuous and hence that f = on K. Therefore f is a uniform limit of a sequence of functions in F, so that
/
f E L.
Finally, if L separates the points of =
then, since ={x},
for each x OK, we obtain L = A(K). The classical Stone—Weierstrass Theorem corresponds to the special case of
Theorem 4.1 where K is a Bauer simplex and L is a closed sublattice of A(K) containing the constants; in the case L is a lattice ideal in L for each x E OK, and hence the Choquet boundary O,,K of L equals OK. Simple finite-dimensional examples show that the conditions that L be a simplex space and that OLK = oK cannot be dropped from the hypotheses of
Theorem 4.1. The next result is a generalization of the separable case of the Stone— Weierstrass Theorem, in the same spirit as Theorem 4.1. 4.2. THEOREM. Let 1 E L c M, where L and Mare closed linear subspaces which separate points of the compact metric space X, and such that of = OMX. Then L = M if and only if whenever 1, 0 a linear subspacc E of A(K) containing L space with d(g, E) < e and such that E is isometrically isomorphic to a space Proof.
Indeed if such a process can be carried out we can inductively take g to be a member of a countable dense set in the positive part of the unit ball of A(K) and a sequence of c's converging to 0, obtaining an increasing sequence of subspaces of A(K) with E1 = lin {1 } and d(g,,, < and such that E,1 is
isometrically isomorphic to a space Clearly we will then have cl U1 =A(K). Suppose then that L, g and c >0 are given, with ugh = 1. Since L is
and 1 is an extreme point of the unit ball S of
isometrically isomorphic to L with
S+1
IIfII2},
it is easy to see that there exists a basis for L with f1 0 for each .1 . . fi,. , and . +ffl =1. Define
byTx=(g(x),f1(x),...,f,,(x)), and let W be the range of T. If q denotes the projection of n coordinates, then
'onto the last
qW={tER': tjo,> Each t E a(q W) is of the form t = qs for some s E W and we can choose a i W such that qW = qW', where W' = finite collection s, Sm co {s1,. . . , Sm}, and such that for each (a, y) W there is a point (a', y) W'
with a — a'I < c. (In fact we can decompose qW into finitely many nonoverlapping polyhedra F1, . . , F', such that and /i2 have oscillations on each P, less than e/2, where h1, qW R are defined by .
= sup {a: (a, y)
W},
/22(y) = inf {a:
(a, y) W}.
Let E be the finite set of points in W of the form (h1(y), y), i = 1, 2, y . . , r. If E' is a finite subset of a W such that co E' can define W' = co B'.) Let A1,.. . , W' [0, 1] such that
SF1, / = 1,.
A(w)s1
co E then we
SIMPLEX SPACL3S
132
and Aj(w)=1
for each w E W', while A1(s1) = 1, 1 f < m. For each w E W\W' define A1(w) = A(w') for some w' E W' such that qw' = qw and p(w') —p(w)I where p denotes the projection of R"1 onto the first coordinate. For each
x€K we will have A.(Tx)=1, and 7"
g(x)--
Aj(Tx)s
<e.
1
We will show, in the lemma below, that these facts imply the existence of functions
lt,...,1mEA(K) such that rn
PU
4=1,
IIljII=1,
j1
and ljs,'O0, there exists a peaked partition of unity {gi,.. , gn+ i}for A(K)such that gulF =f,fori = 1,.. . , n + 1, and lie, —(g,+a,g,t. i)Ii<e . . ,
such that
.
fori=1,... ,n. Proof.
(1)
Let 5, E aF be peak points for e,, 1 j P:K-
F by Px
=
e,(x)s,,
n,
and define
x E K;
so P is an affine continuous projection from K into F which satisfies e, = e, ° P for I = 1,.. . , n. Since A(F) is separable there exists a dense sequence {u,,} in A(F)1 (the closed unit ball of A(F)), and hence 2 defines a metric in A(F)* which induces the
on Let E be the p-completion of A(F)* and define 'I' : K (E) (the non-empty closed convex subsets of E) by = P '(Px) F. It is evident that is affine, and also that P : K -* PK is an open map. Consequently, if U c E is open then {x
EK
is also open, so that 'I' is lower semi-continuous. By Lazar's Selection Theorem [145, Corollary 3.4] there exists a continuous affine map Q : K —' E such that Qx for each x in K, while Qy = y for each y in F. Therefore o is a projection from K onto F, and (P ° Q)(x) = Px for each x E K. Finally
we have e1oQ=(e1oP)oQ=e,oP=e1 for
1jn.
SIMPLEX
135
c aK such that e,(s1) = 1. Since F is a proper face of (ii) Choose s,. . K, 3K\F is dense in K and so there exists a point t,,+1 E 9K\F such that .
<e and if we putt1 while if s1€Fwe choose that e,(tj) — e,(s,)1 0, there exists an m >0 such that
d(Vk,Fm)0 there exists an n such that is an c-net for then, as in the proof of Theorem 7.1, it S = co follows that S is a compact simplex with of S.
=5, and clearly K is a closed face
A consequence of Theorem 7.5 is that is homogeneous. Moreover, every Polish space is, by Theorems 7.1 and 7.6, homeomorphic to a closed subset of aSp.
In particular [0, 1] is homeomorphic to an arc in aS,,. Using Theorem 7.5 it now follows easily that aS,, is arcwise-connected.
Notes
Theorem 7.2 was proved by Lazar and Lindenstrauss [1491. As we noted above, the spaces appearing in 7.2 can be chosen to be isometrically
SIMPLEX SPACES
138
isomorphic to
and the isometric embedding of +
=
and {u1,,,}, {Vk,,,÷1}, 1
are suitable bases of
and
j
into
where
a1,,,
0,
n, 1
k
n+1
takes the form a1,,,
1
respectively. The triangular matrix {a,,,,},
1 j n, n = 1, 2,..., is called a representing matrix for A(K). The
representing matrices for A(K) are not unique, hut in some cases they do determine certain properties of K. For example, Lazar and Lindenstrauss [149] show that K is a Bauer simplex with 0K totally disconnected if and only if A(K) can he represented by a matrix {a1.,,} where, for each n, a1,,,,, = a1,,, = 0 for j for some/n with 1 j,, n.
1,
Sternfeld [201] gives characterizations of Bauer simplexes, and of the Poulsen simplex, in terms of representing matrices. The Uniqueness Theorem 7.5 and the Universality Theorem 7.6, for the Poulsen simplexes, are due to Lindenstrauss, Olsen and Sternfeld [154]. They also prove that OS,, is homeomorphic to 12. The proof of 7.5 uses ideas similar to those of Lusky [155] in proving that the Gurarij space is unique. (A Gurarif space is a separable predual of an L1-space such that for any 8>0
and for any finite-dimensional Banach spaces E F and linear isometry T:E-*G, there exists a linear extension T:F-*G of T with (1—e)IIxII IITxJI (1 + e)IIxII for all x in F.) Some analogies between A(S,,) and G have
been given by Lusky [156].
8. NON-COMPACT SIMPLEXES AND CONVEX SETS
Let K be a convex set in a real vector space E such that K is linearly compact, that is every line in E meets K in a (possibly empty) compact subset of the line. Without loss of generality we can suppose that K is contained in a hyperplane of the form e'(l), and also that un K =E. Let C be the cone generated in E by K, with vertex 0, and partially order E so that
C={xE.E: x0}. In these circumstances we say that K is a simplex if E is a vector lattice. A result of Choquet and Kendall (see Kendall [139]) shows that a linearly compact set K is a simplex if and only if K has the property that, whenever
A 0 and x
E, there exists ,a 0 and y E E such that K (AK + x) = K be a linearly compact simplex embedded in the
partially ordered vector space (E, C).
NON-COMPACT SIMPLEXES AND CONVEX SETS
139
(i) The set = CO (K u —K) is the closed unit balifor a lattice norm in E, such that the norm is additive on C. (ii) If —f, g are bounded convex functions on K with f g, then there exists a bounded affine function h on K with f h g. (iii) If E is complete for the norm in (i) then every closed face of K is split. (i) If p is the gauge for then p is evidently a semi-norm, and since E C: e(x) 1} it follows that p and e coincide on C; in particular p is additive on C. If x E E then we have Proof.
')+p(x) If p(x) 0 if x 0, and that = p(x). The statement (i) now follows If we define f(ax) = af(x), g(ax) = ag(x) for a 0, x E K, then —f, g are subadditive positive homogeneous functionals on C. The methods of so
Theorem 2.4.2. can now be used to obtain the required h. (iii) If E is complete for the norm in (i) then E is an L-space (2.6.2) and hence is an order-complete vector lattice. If F is a closed face of K then the continuity of
the lattice operations in E implies that un F is an order-complete lattice ideal in E, and consequently there exists an ideal L in E such that E is the order-direct sum of un F and L (see Peressini [168, p. 39]). If F' = L K it is now easy to check that F and F' are complementary split faces of K. Statement (i) of Theorem 8.1 does not hold in general for non-simplexes, as the example (2.1.6) shows. Statement (ii) is a simple analogue of Edwards
Separation Theorem for compact simplexes (Theorem 2.7.6), but this property does not characterize linearly compact simplexes. In fact if E consists of the real polynomials on [0, 1] and if K = {fe f(t) dt = then E possesses the Riesz Decomposition Property and so statement (ii)
holds; however since E is not a vector lattice K is not a simplex. We now give an example to show that statement (iii) may fail if E is not a Banach space. 8.2. EXAMPLE.
Let E = CR[0, 1], K ={fe E':
F=jf€K: IJo f(t)dt=0
f(t) dt =
1)
and let
-
Then Kis a linearly compact simplex andFis a closed face ofKfor the induced norm in E. However, F is not a split face of K.
All the statements, except the final one, are evident. Suppose that F =0 for all is a split face of K with complementary face F'. Then, since Proof.
SPACES
140
f€ F there must exist a u E F' with u(4)> 0. But then we can easily decompose u = Ag + (1 — A )h with 0 : = co (K u —K) is linearly bounded, that is every line in E intersects>: in a bounded (possibly empty) subset of that line. Then the gauge of>: is a norm forE which is additive on C, and the dual space E* is
order and isometrically isomorphic to the Banach space Ab(K) of all bounded affine real-valued functions on K, with the supremum norm. If>: is the closed unit ball for E then the proof of Theorem 2.7.2 shows that K is a simplex if every closed face of K is split.
We now illustrate, in a very special case, some results concerning the structure of Banach spaces. Let K be a linearly compact convex set such that>: is the closed unit ball in E for a complete norm in E. A linear projection P E E such that o P I, for the pointwise ordering, will be called an L-projection, and the range of an L-projection will be called an L-ideal. 8.3. THEOREM.
(i) 1 is an L-ideal in E if and only if I = un Ffor some split face Fof K.
(ii) The L-projections on E form a commutative subset of B(E) the bounded linearoperators on F, and they are precisely the extreme points
of the set {T€B(E): 0 (iii) If {I,,} is a family of L-ideals in F then fla ía and Ua are also L-ideals in E. and (iv) If is a family of split faces of K then fl,, Fa are also split faces of K.
(i) Let F be a split face of K with complementary face F', so that F is the direct sum of un F and lin F'. If P is the projection of E onto un F then Proof.
it is clear that 0 P 1, so that un F is an L-ideal.
Conversely, let P be an L-projection and let F = K P(E) and F' = K (I — P)(E). Then since 0 P I and since un F nun F' = {0}; it is easy to check that F and F' are complementary split faces of K.
(ii) Let P1 and P2 be L-projections on E, and let F, = K n P1(E) for j = 1,2. Now F1 nF2 is a split face of K (see the proof of 1.1) and so there is an associated L-projection Q. If x K then x can be uniquely decomposed as
x=
4
>: A,y,,
4
whereA10,
>: A,=1
j-=1
NON-COMPACT SIMPLEXES AND CONVEX SETS
141
and
But then Qx =A1y1 =P1P2x
P2P1x,
so that the linear operators Q, P1P2 and P2P1 agree on K, and hence are equal. Therefore, since clearly IIP1I 1 for any L-projection P 0, the L-projections form a commutative subset of B(E). Let A ={T E B(E): 0 T I} and suppose that P is an extreme point of A. Then P2 E A and also
2P—P2——P+P(I---P)EA.
The equation P =
P
P2
P=
P for some T1, T2 E A. Then we have
P=
+
and 0=
—
P)T1 +
—
P)T2,
and because PT, P and (l—P)T1 0, forj = 1, 2, it follows that T, = P7 = P for / = 1, 2. Consequently P E (iii) Let {Ia} be a family of L-ideals. To prove that fla Ia is an L-ideal we can assume that the family is decreasing, because the intersection of finitely many 4 is an L-ideal (using (i)). If {Pa} is the corresponding family of decreasing in C for each x C; since E is L-projections we have is norm-convergent to a limit complete and the norm is additive on C, converges strongly to an L-projection P such Px E C'. Therefore the net that P(E) = fla Ia. Moreover the complementary projections The final part of I P and (I — P)(E) = cl Ua (I— (iii) now follows. (iv) The required facts follow by direct verification using the results in U) and (iii).
By the above result the L-projections on E generate a commutative Banach subalgebra of B(E), with identity I. This Banach algebra is denoted by and is called the C'unningham algebra for E.
An operator T E B(E) is said to be order-bounded if —Al T Al for some constant A >0, and we denote infimum of such A by 1T111. It is easy to
see that
is a norm and that = sup
We
x K} 11Th1.
can identify A"(K) with A(K1) where K1 is the state space of the
complete order unit normed space A"(K). Suppose that T E B(E) satisfies
of T (1— a)I, where 0 0 and choose /h representing x such that
/h(f)—f(x)>0. If g: T -, R is given by g(t)
COMPLEX FUNCTION SPACES
152
then g is continuous with g 0 and g(1) >0. Hence
dt = L
0
d/h(y) dt —invf(x)
= /h(inv f) — mv f(x) (mv f)A(x) — mv f(x).
is a probability measure on X x T we can regard ji as a measure on the Since ext U c TX, any maximal measure on U is supported by TX. If
homeomorphic image TX c U =
1.4. PRoPosmoN. Let j.Z be a probability measure on TX and let = H1.Z. Then (i) (ii)
Proof.
ft
is maximal on U if and only
oir' is maximal on SA.
IffEA then d4 =
ff(tx) d12 (x, t) =
tf(x) d/Z (x, t) = ft d(Hj2)
= Li d/.L.
If f is an invariant Borel function on U then
ffd1z = ff(x) djZ(x, t) gives
f is continuous and convex on U then Proposition 1 .3(u) and (iii) 4(B(invf))=4 °
Hence by Proposition 1.3(v) 4 is maximal on U if and only if ji oir'
is
supported by X c SA, a face of U, so that 4 olr' is maximal on U if and only if it is maximal on SA. maximal with support in SA on U. But 4
is
1.5. THEOREM. LCtLEA* with 11L11 1. (i) There is an element/h E 0AM(X)1 such that /h represents L. (ii) The measure in (i) is unique in the boundary measures of norm one if and only if L is represented by a unique maximal probability measure 4 on Given L E A* with IIL1I = 1 we can choose, by the real Choquet-Bishop—dc Leeuw Theorem a maximal 4 on TX C U representing L. Let = H4. Then p = M(X)1 and represents L by Proposition 1.4(i). Since is maximal on SA by Proposition 1.4(u) it follows from Proposition 4° 1.1 that is maximal. Hence j.i. is a boundary measure. The =4 uniqueness equivalence of statement (ii) now follows from Proposition 1.2. Proof.
COMPLEX REPRESENTING MEASURES
153
We say the subspace A c C(X) has the uniqueness property if each L E A* represented by a unique boundary measure with We charac.. = terize this in terms of the geometry of in the following corollary. Further
applications of uniqueness to Lindenstrauss spaces will be discussed in Section 9. We call a (not necessarily closed) proper face F of a simplex if the cone spanned by F in A* is a lattice. This is of course consistent with the usual definition of simplex in case F is (Chapter 2, Section 7). Since F is proper,
Fc{x eAr: IIxIi= l}. 1.6. COROLLARY. The subspace A c C(X) has the uniqueness property if and only if each proper face of A1c is a simplex.
Proof. Let U = Ar and let F be a proper face of U. If A has the uniqueness property then Theorem 1.5 shows the resultant map is one-to-one from a face of the maximal measures in MT(U) onto F. But the maximal measures themselves form a face of the simplex MT(U) (see the remark preceding
Theorem 2.7.3) so that F is affinely equivalent to a simplex. For the converse let lix Ii = 1 in U and let F be the (algebraic) face of U generated by x. Then
Fc{yc U: IIyiI= l} is a simplex. We show that maximal measures on U representing x must agree on the cone Q(U) of continuous convex functions in Since the linear span of Q(U) is dense in CR(U) this will establish the uniqueness property, by Theorem 1.5(u). Thus let fE Q(U) and let be a maximal measure representing x. By Theorem 1.6. 1(viii) there is a —g E Q(U) with
fg and
f gd/h0. Since the cone spanned by F has the Riesz decomposition property it follows from Chapter 2, Section 4 (cf. Theorem 2.7.3) that flF, given by sup { fA1f(y1): y
is affine on F. Similarly,
F
is affine and
onF.
COMPLEX FUNCTION SPACES
I j4
Thus, using Proposition 1.6.2 and Theorem 1.6.5 we have
1(x)Jfdi2 Since >0 is arbitrary we have
f
=/(x).
Notes
The Representation Theorem (Theorem 1.5(i)) is referred to as the Hustad Theorem. Hustad [132] constructed the map H and used it to obtain the representing measure Hirsberg [127] showed that in general H11 is a boundary measure if Z is maximal on The construction of the inverse map in Proposition 1.2 is due to Fuhr and Phelps [112] who proved the uniqueness statement Theorem 1 .5(u). They also proved the geometric version of the uniqueness property in Corollary 1.6. Unit balls with that property are referred to as simplexoids. Many of the techniques herein employed, and in particular those of Proposition 1.3, are due to Effros [96]. For the analogous theory for subspaces A not containing constants we refer to Phelps [172].
2. INTERPOLATION SETS Let A be a closed subspace of Cc(X) and let E be a closed subset of X. If '1' is
the restriction map from C(X) to C(E) then the Tietze Extension Theorem shows 'I' is onto and in fact maps the unit ball U of C(X) onto the unit ball V of C(E). Then embeds (M(E), V°) onto the subspace (N, are (M(X), U°). If then In, = = complementing projections with N = range in. Let 9 = 'VIA and denote range 0 by AlE. We say E is an interpolation set for A if is closed in C(E). Since 'VU = V, we have a subset B of C(E) is closed if and only if \V '(B) is closed in C(X). In particular A is closed if and only if A +M is closed in C(X), where
M = ker 'V = {fE C(X): lIE If qx,
0}.
denote the quotient maps qx :M(X)-+M(X)/A1 =A* = :
then
=
0
INTERPOLATION SETS
Let U =qx(U°),
155
= qxN. Then
)Q
'IE(V) with
identifies I
We can now formulate Theorem 1.3.5 in this context.
2.1. THEOREM. Let E be a closed subset of X and let A be a closed separating subspace of C(X) with constants. The following are equivalent: (i) E is an interpolation set for A; (ii) (A + M) (U + M) c r(A U + M) for some r 1;
r 1;
(iii) (A + M) (mt U + M) c r(A (iv) Nis w*_closed in A*; (v) N is norm-closed in A*;
forsomer1.
(vi)
In particular, if E is an interpolation set then for each g E and a> 1 (using (iii)) there is an f E A such that flE = g and HflIx Another formulation of 2. 1(vi) is useful. By taking inverse images under qx we see that (vi) is equivalent to
(U°+A or
N Since 0*0
=
+A-
a
Nil =
the last statement is the same as + (AIE)111
+A111
for all
E M(E).
The following are equivalent: (i) E is an interpolation set for A;
2.2. COROLLARY. (ii)
r 1 and all
+
(iii) lIirim + (AIE)'ll Proof.
EN;
rllir2tnll for all m E A'.
As noted above, (ii) is a reformulation of 2.1(vi). If (ii) holds and
m€A1 then llirim + (AIE)11 rllirim +A'll
= rIl—ir2m +A111
rum — ir2m +A—II
rjI'n.2m11.
If (iii) holds and €N, m E A, then +mfl=fl'7r2m +(irim + (AIE)111 + (1/r)llirim + + irim) — (grim + (AlE) ')ll = Hence
+A111Il,z +(AIE)ii.
+ (AlE)' II.
COMPLEX FUNCTION SPACES
156
We say E is a full interpolation set for A if AlE in fact equals C(E). This is equivalent to E being an interpolation set for which (AlE)1 = {O} in
N=M(E). The following are equivalent: (i) E is a full interpolation set for A;
2.3. COROLLARy.
EN;
(ii) +A111 for some r 1 and (iii) Iliri mu rIIir2mII for all m E A1.
Proof.
The above remark shows (i) implies (ii). The proof of (ii) implies (iii)
is the same as in Corollary 2.2. If (iii) holds then, in particular, E is an interpolation set. If E (AlE)1 then E A1 N and IIILII
so
rlllr2Ihll = 0
=
that (AlE)1 = {0} and E is a full interpolation set.
We will say a subset B of M(X) is A-stable
if by &AB so that
qx(B
aAM(X)). We will denote B if and only if
= qx(B) B is A-stable
Hustad's Theorem shows U° is A-stable. If, in addition, N is A-stable then
we can modify the measure theoretic conditions of Corollary 2.2 and Corollary 2.3 by restricting attention to the boundary annihilating measures, aA1.
2.4. COROLLARY. Let N = M(E) be A-stable. Then the following are equivalent: (i) E is an interpolation set for A; r 1 and all m E (ii) llirim + (AIE)111 Proof.
If N is A-stable then for
E aM(X), =
÷aAil
since
/.L=sv+m;
v€U°,
mEA'
v'EW,
m'EA.
implies, by Hustad's Theorem, that
p.=sv'+m'; Since
ii' E
t9M(X), it follows that m' ÔA1. Thus, if E is an interpolation
INTERPOLATION SETS
157
set then Corollary 2.2 shows + (A IE)II
+
for all E Thus (ii) follows form E since ir1m E oA also. Conversely, if (ii) holds and x E N U, choose p, E oN with = x and let m be any element of M1. Then the computation of Corollary 2.2(iii) implies 2.2(u) shows + mu
+ (AIE)'91
and hence
IIgIlE.
We close this section by observing some instances where N =
M(E)
is
A-stable. 2.7. PROPOSITION. If E is a closed interpolation set forA such that either (i) E aAX, or (ii) E = F X, Pa closed face of SA, then M(E) is A-stable.
Also, (ii) implies E = k(E).
If (i) holds then each element of MT(E) is maximal (cf. Theorems 1.6.3 and 1.6.5) so that M(E) C aM(X). If (ii) holds and is an clement of
Proof.
representing x F the M(E) (Proposition 2.10.6). Since (M(E)) is spanned by F we have M(E) is A-stable. Notes
As indicated in Chapter 2 the closed range conditions of Theorem 2.1 have been recognized in various forms by many researchers. The seminal work in this regard is contained in the 1962 paper by Glicksberg [118]. In particular he gives the equivalences of Corollary 2.2(i) and (ii) as well as Corollary 2.3. The equivalence of Theorem 2.1(i), (iv)—(vi) appears in Edwards [87] for the face of K. Alfsen [3] also realA(K) case where N is spanned by a
discusses this situation and exhibits the constant r in statement (vi) as a bound on the norm of extensions. General formulations of this principle for A(K) and for complex function spaces are given in Asimow [23,26].
The formulation of Corollary 2.2(iii) in the general case is given by Gamelin [113]. The use of boundary measures of course awaited the maturation of Choquet theory and first appears in Alfsen—Hirsberg [9] where stability is guaranteed by the hypothesis that E is a compact subset of aAX (cf. Proposition 2.7(i)). They prove (ii) implies (i) in Corollary 2.4 for the case r =0. This very important case will be discussed in detail in Section 4. Corollary 2.4 is proved in its present form by Briem [50] where the same stability condition (E c aAX) is assumed. The alternate stability condition (Proposition 2.7(u)) appears in Hirsberg [126]. Further abstract versions of the closed range conditions can be found in
Andô [19], Roth [182] and Asimow [28, 29]. Roth [182] also discusses conditions involving boundary measures in a quite general setting.
GAUGE DOMINATED EXTENSIONS AND COMPLEX STATE SPACE
159
The equivalence of the geometric definition of k(E) and the statement of Proposition 2.6 has been well known in the context of function algebras. It is shown in Alfsen and Hirsberg [9] for the case &i E a parallel face of SA and in general by Briem [50]. Examples of closed subsets that are not interpolation sets are plentiful.
We show, for example, in Section 6 that for A the disc algebra and E a proper closed arc of T we have AJE dense in, but not equal to, C(E).
3. GAUGE DOMINATED EXTENSIONS AND COMPLEX STATE SPACE
If E is a closed interpolation set in X for a subspace A c C(X) then Theorem 2.1 shows there is an r 1 such that each g E is the restriction of an fe A with 0 sufficiently small
g =1 ± e(f—f(x))2 (+ for j = I and — for j=2) satisfies
and
g(z)—8. Let g=e1. Then =e
i
and
onK.
(5)
Thus
But since o
is
multiplicative, o(g) =
=
= re cr(g) = o'(re g).
Hence
(f 1
do')
2(1
(6)
Now
0=j'gh dm =Jh dma _f(1 —g)hp do'-I-j'
gh
(7)
COMPLEX FUNCTiON SPACES
206
with
fgh
e
by (5), (2) and (3).
Likewise
i L (1—
g)hp doj
211h11
f
P1 d(T +
L 1(1 —
du
by (1), (6) and (4).
Therefore (7) gives
Since e
is
arbitrary, ma E A and thus
m
so is
—
m4.
7.18. COROLLARY. Let A be the disc algebra on T and a' normalized Lebesgue measure on T. If =0 for allf E A such that o(f) =0 then m is absolutely continuous with respect to a'.
M
Proof. Let ={fEA: a'(f) =0}. Then, since cr(zk)=0 for all k = 1,2,... we have A spanned by M and the constants. Thus if m €M and c = m (1), =0 for E by the theorem. Thus m — CuE A-so that (m — = If dm5. We have all k =0 for all k n then consider di' =
0.
for j
1
LzuI so
that v E
M'.
But the
above then yields
o=f dv =f(i)"
We conclude ifzç(n) =0
= ii A1. Hence
" dm.ç(z)= rhç(n +1).
for all n so that
=0.
7.19. CoRol LARY
THEOREM). LetEbe a subset of T with u(E) =0. Then E is an M-hullforthe disc algebra A on T. In particularE is a peak set and a full exact p-interpolation set for any dominator p. Proof.
Let m e A1. Then dm = h du for a u-integrable h on T, by the
preceding corollary. Thus mE =0. Then Theorem 7.1 shows E is an M-hull and the rest follows from Corollary 2.3 and Corollary 6.10.
207
M-HULLS AND FUNCTION ALGEBRAS
We can apply Corollary 7.18 to obtain another example of interpolation. Let B= so that B is the subalgebra of C(T) given by and
B=A' Let E={—1, 1}.
7.20. EXAMPLE. With B and E as above, E is an approximate, but not exact, full interpolation set forB in the uniform norm. Proof.
then
mEA1 and hence dm = h dcr with
h d(r =0. Thus
=hodoH-ae1; if N = M(E) then clearly B
N = {0} and = al.
Since Ct and h0 dcr are mutually singular
= llh0 doll j h0 doj = Ial = lkri,LIl. This shows E is a full approximate interpolation set. 1ff E C(E) is ZIE then
there is no extension g (even in C(T)) satisfying
1
=
dt7.
As noted after Theorem 7.1 the extension constant e(B, E) must be 0 or 1. Here it is one and exact interpolation fails. Notes
General reference books for function algebras are Browder [57], Gamelin [114], Leibowitz [150] and Stout [204].
The fact that, for function algebras, generalized peak sets are polation sets was shown by Bear [36] and by Hoffman and Singer [129]. in Theorem 7.1 the equivalence of (i) and (ii) is given by Glicksberg [1181 using
COMPLEX FUNCTION SPACES
208
arguments of Bishop [38],
as
indicated in the notes to Sections 2 and 4; the
equivalence of (iv) and (v) is due to Ellis [102]. For Dirichiet algebras Glicksberg [118] has shown that interpolation sets are generalized peak sets.
Proposition 7.2 is due to De Branges [76]. The antisymmetric decomposition for function algebras was introduced by Bishop [38], where he shows that maximal antisymmetric sets are generalized peak sets. The
notion of essential set was introduced and developed by Bear [35]. Theorem 7.6 and Corollaries 7.11—7.14 are due to Briem [52] and Ellis [102, 103]. Corollary 7.9 is a theorem of Hoffman and Wermer [130] (see also Browder [56]); the generalization of the Hoff man—Wermer Theorem given
in Theorem 7.15 is due to Sidney and Stout [194]. Theorem 7.17 and Corollary 7.18 are versions of the F. and M. Riesz Theoreni, and for the history of these results we refer to the book by Stout [204]. Example 7.20 is due to Glicksberg [118] and Gamelin [113]. 8.
FACIAL TOPOLOGIES AND DECOMPOSITIONS
We recall from Chapter 3 that the facial topology on ext K (K compact convex) has as its closed sets all subsets of ext K of the form (ext K) F, F a closed split face of K. This topology is Hausdorif if and only if K is a Bauer X0 = ext K. We shall interpret this in simplex, in which case A(K)
the context of A(ZA) where A is a (closed, separating with constants) subspace of
and ZA
Co (SA
—ZSA) is the complex state space.
We denote aZA ext ZA (=aAX —IaAX) with
= ext SA. We define the closed sets of the symmetric facial topology on aZA as those of the form aZA F, F a symmetric split face of ZA. Since the collection of symmetric split faces is closed under intersections
Now F is a and finite convex hulls this is indeed a topology on symmetric split face if and only if F X c SA is an M-hull for A. Thus a set E0
is (relatively) closed in aAX if and only if
EO=aAE for an M-hull E of A. For this reason we call the restriction of the symmetric facial topology to f3AX the M-topology. This is of course weaker than the restriction of the facial topology of 9ZA to 8.1. THEOREM. aAX is compact in the M-topology.
The M-topology is weaker than the facial topology on äAX in SA which is compact by Theorem 3.1.1. Proof.
We next develop the notion of central functions and the result of Theorem 3.1.4 to the present context. Let (IA = A(ZA) and let f be a real function on
FACIAL TOPOLOGIES AND DECOMPOSITIONS
209
aZA. We say f is symmetric if f(x) =f(—ix) for all x I9AX. If u is a real function on aAX then we denote the symmetric extension to aZA by 17. We see that f is facially continuous if and only if there is an h E A such that for X
äAX re
h(x)=f(x)
im h(x)=f(—ix)
and for each a E A there is a b E A such that
(re h)(re a)=re b
(imh)(ima)=imb pointwise Ofl aAX.
It is easy to see that if f is facially continuous and symmetric then f is continuous in the symmetric facial topology. Likewise, if u is continuous on aAX in the M-topology then 17 is symmetric facially continuous on As in Section 7 we denote A11
{u E A: u real-valued}.
Finally, observe that 1ff is real-valued on
then f is real-valued on X.
Let u0 be a real-valued function on aAX. The following are equivalent: (i) u0 is continuous in the M-topology, (ii) 170 = OhIÔZA; h = U + iu, u E A0 and Oh central in A(ZA), (iii) = ulax for u A11 such that for each a E A there is a b E A with ua b pointwise on 8.2. THEOREM.
If is symmetric facially is M-topology continuous then continuous, and hence facially continuous, on OZA. Thus there is an h E A with = and the symmetry of shows h = u + iv with u = v on aAX. But then h — ih e A and is real-valued on aAX, hence on X. This shows U = v E A0 and thus, (i) implies (ii). If (ii) holds the above formulas show
Proof.
u(re a) = re b
u(ima)=imb on so that ua = b on aX. This in turn shows that O(u + lu) is central in A(ZA) so that 170 is symmetric facially continuous and hence u0 is Mtopology continuous.
The following are equivalent: (1) The M-topology on aAX is Hausdorff; (ii) aAX is compact and A is isometric to C(aAX).
8.3. COROLLARY.
COMPLEX F(JNC1JON SPACES
210
Proof. If (i) holds w*-.topology on
then the M-topology, the facial topology and the
8.4. COROLLARY.
Let A be a function algebra.
coincide. Thus SA is a Bauer simplex with A(SA) isometric to Nowf E A implies re flax is continuous and = hence continuous in the M.-topology. Then Theorem 8.2 shows re f€ A so that A is self-adjoint, so that in particular re AIax is closed in But re is always dense in A(SA) = CR(aX) so that re AIax = and (ii) follows. The converse is clear.
(i) A real function u0 on ax is M-topology continuous if and only if u0 =
for some Li
A0.
(ii) The M-topology is Hausdorff if and only if A = C(X).
Theorem 8.2 shows (i). For (ii) we note that Corollary 8.3 gives A isometric to C(aX) and in particular, A is self-adjoint. Then the Stone.-. Weierstrass Theorem gives A C(X). Proof.
8.5. THEOREM. LetA be a function algebra. The following are equivalent: (i) Otis central in A(ZA); (ii) f=u+iv; U, v EA0 on the essential set K.
If (1) holds we can apply the multiplicative formulas for central functions with a = 1 to obtain b1 A such that on ax Proof.
u = re b1 v =im b1. Thus b1 is real and so b1 A0. If a = o
—i
we obtain b2 e A with
re b2
= im b2
on ox. Then b2 is imaginary so that ib2 A0 and equals v on ax. Therefore b1 — b2 =fon x and it follows that b1 = u on x, ib2 = v on x, giving u, v A0. Let w = u — v. We show Ow is central in A(ZA). Given a E A choose h A such that
u(re a)=re b
v(ima)=imb on ax. Then b — va = w(re a) on OX. But b —va A and hence b — va E A0. Thus (Ow)(Oa) = O(b — va)
onax on —iax.
FACIAL TOPOLOGIES AND DECOMPOSITIONS
211
Thus Ow is central. Now E aZA:
for a closed split face F of Z. But —iaX c F implies
F=co (Flu—iSA): F1
X. But then WIK.
If (ii) holds and a E A is given let b = u(re a) + iv(im a)
on X.
Then b C(X) and b ua on K. But ua A and so Theorem 7.5(v) gives b A. It follows that Of is central. The function algebra A is antisymmetric if and only if the only centralfunctions in A(ZA) are constants. 8.6. COROLLARY.
In general we say the centre of A(K) is trivial if the only central functions are constants. We next consider abstract versions in A(K) of the Silov decomposition for function algebras. We first define the Bishop decomposition for function algebras and then give A(K) analogues of both decompositions.
Thus, let A be a function algebra and let
denote the collection of
maximal (with respect to inclusion) antisymmetric subsets of X. Since each {x} is antisymmetric and the closure of the union of all antisym.metric sets containing x is again antisymmetric, we see that is a partition of X into closed subsets. is called the Bishop decomposition of A. 8.7. THEOREM. (1)
(ii) each E E
is a generalized peak set;
(iii) a ={f€ C(X): fIEEAIE for allE E (i) is the conclusion of Theorem 7.3(u). For (ii), let E1 denote the intersection of all peak sets containing E. Then E1 is a generalized peak set and hence an M-hull. If E is a proper subset of E1 then the maximality of E shows there is an f€ A, fjE1 real and not constant. But fIB is constant, say fIB = c, so that for e >0 small, Proof.
g= 1—B(f—c)2 is
a real peak function on E1 with
Ec{xeEj:
COMPLEX FUNCTION SPACES
212
But then E2 is an M-hull for AlE1 and hence an M-hull for A. It follows that E2 is a generalized peak set such that
E E2 c E1. Since this contradicts the minimality of E1, it must be the case that E = E1.
The proof of (iii), using Proposition 7.2, is identical to that of Theorem 7.3(iii).
We see that in both .9' and
the elements are M-hulls and therefore
correspond to (symmetric) split faces of ZA. If K is a compact convex set we let be the collection of non-empty closed split faces of K. A sub-collection of is called a decomposition of K if it consists of pairwise disjoint split faces covering 3K (= ext K). Let be the sub-collection of sets FE 3' such that fIF is constant wheneverf is a centralfunction in A(K). We denote by the collection of maximal (with respect to inclusion) elements in 8.8. THEOREM. The collection isa decomposition of K with restriction to aK consisting of the maximal sets of constancy in 9K of central functions in
A(K). 0 and 1€ A(K) is constant on F1 If F, (I = 1, 2) E and F1 and F2 then f is constant on F1 u F2, hence Ofl Co (F1 u F2) which is also a are pairwise disjoint. If x E split face. It follows that the elements of let E = {y E aK: f(x) = f(y) for all f central in A(K)}. Proof.
Since each such f is facially continuous, E is facially closed in Thus E aK F, Fe Since F = aF (= ext F), if f is central then flE constant implies fIF is constant so that F e 3". If F c F1 €3" then the definition of E c Hence F is maximal and so, 3's covers aK. shows that The collection is termed the abstract S1ilov decomposition of K If A is a function algebra then the abstract Silov decomposition of ZA determines the Silov decomposition of A in the following sense. 8.9. THEOREM. Let A be a function algebra with complex state spa-e ZA. Then E e .5°, the Silov decomposition of A if and only if
E=Fr'mXCSA
for Fe Proof.
the
abstract
decomposition for ZA.
Given Fe .9' then F is an M-hull so that, with 8E F
c SA,
213
FACIAL TOPOLOGIES AND DECOMPOSITIONS
Let x E 1E c aZA and let x E FE
If Of is central in A(ZA) then Theorem
8.5 shows ref = u E A0. Hence u is constant on E so that Of aF = UIaE is cF X and hence E c constant. It follows from Theorem 8.8 that Xc Conversely, u E A0 implies (by Theorem 8.5) that O(u ÷ iu) is central and hence constant on F. Thus u is constant on F X so that This shows E
We shall see that the abstract
decomposition determines A(K) in a
fashion analogous to Theorem 7.3(iii) for function algebras. We develop this pcoperty in conjunction with the abstract Bishop decomposition which is described as follows. We let, for the closed non-empty split faces of K,
= {F
the centre of A (F) is trivial}.
The elements of 2?' are called abstractly antisymmetric. Let maximal elements of 2?'. 8.10. THEOREM.
The collection
is
denote the
a decomposition of K such that for
with F1 F2 0. Let F = Co (F1 u First, let (1= 1, 2) E E Now, central functions in A(F) restrict to central functions in A(F,) and since F is as well. This shows elements of are pairwise disjoint. If x E t9K let be the smallest element of containing x. Then must be antisymmetric, for if f is central in there such that is a split face G of Proof.
G = {y Hence x E 0 =
aFt: f(x) = f(y)}.
so that f is constant and
€2?'.
if
={G: XEGE2?'} then
U
is non-empty. Thus, let F be the smallest element in containing We show F is antisymmetric. Let f be central in A(F) and let
E = {y äF: f(y) =f(x)}.
If G all G
then G c E. But E = 9F n G0 for
f is constant on G and
a split face of F which contains IIG.for Thus G0 = F and so f is constant on F. Clearly F is maximal in 2?'
and FcF0€ As in the decomposition for function algebras, we can associate the abstract Bishop decomposition with the Bishop decomposition of A.
COMPLEX FUNCTION SPACES
214
8.11. THEOREM. LetA be a function algebra with Bishop decomposition Then E 6 if and only if E = F X SA for some F€ the abstract Bishop decomposition for ZA. Proof. Let E E (F0 u 6
so that E is an
with F0 =
(E) and G =
the split faces of ZA. Now G is the complex state space of
AlE so that if h is central in A(G) then h = Oflo, fE A, and Theorem 8.5 shows f = u + iv with tilE, VIE 6 (AIE)() and u v on the essential set of AlE. But E antisymmetric for A implies that uJn, VIE are constant functions and that AlE is essential. Thus u = v on E and Of is constant on G. Thus G E so that
with F1 c SA. Let E1 = F1 F Co (F1 u—iF2) Theorem 7.6 shows E1 is an M-hull and hence so is E = E1 u E2. Let 0 be the symmetric split face of ZA such that G X = E. We show F is antisymmetric for A. Let satisfies Theorem 8.5(u) so that fEA such that fJE e(AIE)o. Then
O(f+ if)Io is central in A(G). Thus its restriction to F is constant, say, c. Then O(f+ if)IE, = re tIE, = a = re fIE2 = O(f+ so
that fJE = re fIE = a. Hence we have, since E is antisymmetric,
We now take up the question of to what degree these abstract positions determine A(K). Let X = (ext K) and consider A = A(K)Ix as a subspace of CR(X) with state space K. As usual, for subsets C of MR(X), aAC or äC denotes the intersection of C with the boundary measures in for (ext K) E for subsets E c K. MR(X). We continue to use We first establish the abstract analogue of Theorem 7.6. 8.12. THEOREM. Proof.
then supp1a
If IJ
Let F0 be the smallest element in
containing supp p. We show F0
is antisymmetric. Let f be centraFin A(F0) and assume Of 1. Then for each a€A(F0) there is a b€A(F0) with af—b on and hence on supp
This gives
j so that f f dp + (1—f)
af
=0
and (1—f) EAt. Since is extreme in At and we have f = A A a constant. Then f = A a.e.
FACIAL TOPOLOGIES AND DECOMPOSITIONS
A on supp
SO that
213
If G is the split face of F0 such that aF()
G = {x aF0: f(x)
then fA on G so that suppjicG, implying
On F,,. Thus F0 is antisymmetric.
f
Similarly
We need to show next that if f€ C(X) is annihilated by a(ext At) then f is To accomplish this it is necessary to introduce a annihilated by all variant of the Choquet ordering on a set. If W is a compact convex set then we say a subset T is hereditary-up if partially ordered
T and
ii
>
implies
e
We say T is extremal if
x=Ay+(1—A)zET; y,ZEW, impliesy,zeT. For example, T may be a union of faces. A function f on W is isotonic if
v> '1 implies f(v) 8.13. LEMMA. Let (W, >-.) be as above with the continuous convex isotonic functions separating the points of W. If T is a closed extremal hereditary-up subset of W then there is an element of T which is maximal with respect to and belongs to ext W. Proof.
Let
= {E c T: 0 E is closed, extremal and hereditary-up}.
Since the properties of elements of are preserved under intersections every sub-collection with the finite intersection property has non-empty intersection. Thus there is a minimal (with respect to inclusion) element 1ff is continuous, convex and isotonic and if E00 is the subset of E0 E() E is non-empty, extremal and, where f attains its maximum on E0, then since E0 is hereditary-up, E00 is as well. Thus E00 E Therefore E00 = E0 and must be a singleton by the point separation hypothesis. It follows that and extremal, hence extreme in W. this point must be maximal
For f€ C(X) let / and J be the upper and lower envelopes of f on K as in Chapter 1, Section 6.
8.14. PRoPosrrroN. Let X = (ext K) and f€ C(X). If Proof.
Define >. on Af by p >..
if it(p1)+
=0 for all
COMPLEX FUNCI1ON SPACES
216
for all p1' P2 continuous convex functions on K. If p is continuous, convex and non-negative then the maps (p)
are continuous and convex on A By considering the pairs p, 0 and 0, p they are seen to he isotonic as well. Since —P2 :p1 continuous, convex on K} is
dense in CR(K) we see that the elements of At are separated by the continuous convex isotonic functions. Now, givenfE C(X), let E with 0. We show v(f)> 0 for some ii E a(ext A Since is a boundary measure, are maximal so that /z. and
=
(--I) =
P2 continuous and convex on K with
Therefore there exist
and —
(f) +
/L (Pi) + /L (p2)
(—f)) =
I
0.
Let
a = sup {p '(pi)+ ir(p2):
ii E
At}
and
T is closed and hereditary-up in At. Furthermore, T is extremal since a >0 implies lvii = 1 for all v T. But then if ii E T and
vEAj
(i1,2)
then Ikil = 1. Since these measures annihilate the function 1, liv
I
1 I
and
ii,
12 I
1.—
If B1, B2 is the Jordan decomposition of X for ii then v
1
(B1) 1
so that p
+i4)
and
+i.',).
Thus
and it follows that E T. T and 11 maximal Therefore Lemma 8.13 yields i' (ext are maximal measures so that latter clearly implies that p and
vca(ext Ar).
The
FACIAL
T gives ii
If X = (ext K), let A = A(K)Jx
and denote
1=1=1
L ={f€
(p2)=a >0.
on Xc
K}
where / and fare the upper and lower envelopes off relative to K. 8.15. PROPOSITION.
A ={fEL:
Ea(extAt)}.
Clearly A is contained in the set on the right. Suppose then, f€ L Ai). Then Proposition 8.14 shows = Ofor all E =0 for all p e A , implying that We prove that fEA A. Thus, given p A with p Owe have p = p — p and we can 1. Now choose v1 and maximal on assume Ip 'jJ = p(l) = p. (1) = K with Proof.
and f is annihilated by all
and v.>p. If a EA(K) then
'(a)—(a)=p.(a)=() so that 7.' =
Also
— P2 E
p.
=
= p.
(f) =
0 we can, by the Bishop—Phelps Theorem (Chapter 1, Section 7), choose a support pair (a, x) with flail = and fix — xoji < s. Then using the above iixii
=
= ((Fl) — =).
Thus iixoll
lix — xoli + llxil < 5e.
)+
)+
= 411x —xofl0, then we will prove by induction on n that there exist h, .1, j=0, 1,. . . , ii, such that h = h, and —e 0. if +1,, then is an M-ideal and . . +1,, = A(K), so that there exist
1+8, that
)'- = F,,_1
h,,-1 €A(co
€J,_1 for some 6E(0,e). Let F,, F,,=J: so = 0 on F,,_1 F,,. Clearly we can define nF,, and
LF,,)) so that h,,. = 1
on F,, and h,,1 = 0 on F,,_1; in
226
CONVEXITY THEORY FOR
u F,,). Since co (F,, F,,) is particular we have 0 h i øt CO a closed split face of K we can, by Theorem 2.10.5, extend h,,_1 to K so that A(K) and 0 h71_1 h, — 8). Now if we put h,, = — J,,, 0 h,,_1 < 1 + e, —e Owe can assume that the carrier g of th is
Since
disjoint from e". This contradicts the inaximality of {e1j, and therefore = C,,. Suppose
that we have 0(e') =
0 is a normal functional on A** 0 0 6(e,j, and so 0(e') = 6(e1) for a countable subfamily {ej} of the {ea}. Because 0(e' -- e,) 0 and the carrier h of 0 is we see that h c1. If th1 F are such that e1 is the carrier of then 2 'th1 belongs to F with carrier 2 'c1 h. The first part of the proof of (i) now gives 0 E F. Consequently we obtain F {Oe: 0 E K, 0(e') 0}, the reverse inclusion following by the corresponding argument for given above. The
preceding result gives M={a**e'Oe?:OEK, a**EA**}. Now if '1' a**eIOe'EK we have and hence M n K F, and the proof of (i) is complete. (ii) Let denote the closed unit ball of We need to show that
Al r is
is a Suppose that net in M with limit /. The polar decomposition (cf. Sakai [188, 1.14.4], Dixmier [81, with w,, E A**, Ija 0, 12.2.4]) enables us to write = =
=
kind
In
particular we have ii,, M (r)
co (F u {0}). We can
therefore, by choosing a subnet if neccssary, assume that is convergent to some ij e co (Pu {0}). We have, for each a E A, 1&(a)i2 so that Ø(a)12r,(a*a) and consequently for all b in A**. Now let 4, be the polar decomfor all
position of th. Then
subnet such that we see that If a E A,, we obtain, for all = IthII ku = a
= urn
= urn
A
A2
and this implies that is
+A)j2
ij((a +A)2)
Consequently 4, =
r,(a2)
M
r
THE IDEAL STRUCTURE OF A UNITAL C*.AI.GEP3RA
229
2.2. CoRolLARY. Every closed face of the state space K of the C A is semi-exposed.
We prove directly that F is a semi-exposed face of K if and only if and the result then follows from 2.1. Let F be a semi-exposed face of K, so that F = (F1)' = ((F1)1)- by 2.8. If aE then a2 Ilalla, so that qS(a2) =0 for all q5 in F, and hence a E Proof.
F=
Therefore we have (F ) c
F'0, and it follows immediately that
F=(LF)'. and let c/, E K\F. There exists an a E L11 with Conversely, let F = 0 by the Cauchy—Schwarz inequality. Since 0, and hence and hence F is semi-exposed. Lj; is a left ideal a*a belongs to Our next investigation in this section is the connection between closed two-sided ideals in A and closed split faces of K. A subset E of K is invariant belongs if for each th E F and each a E A with gó(a*a) >0 the functional = for each b E A. to F, where
2.3. THEOREM. Let F be a closed face of the state space K of a unital A. Then the following statements are equivalent.
(i) Fis invariant; (ii) F0 is a two-sided ideal in A; (iii) F,. is positively generated; (iv) Fis a split face of K. (iv) implies (iii). This follows from (2.10.5). (ii) implies (iv). We have F = (Fr) by Corollary 2.2. Let ç& 6 K and let ir denote the representation of A induced by 1, with cyclic vector in the sense of the Gelfand—Neumark--Segal construction. If H denotes the HilProof.
bert space associated with IT let L be the closed linear subspace of H spanned by {rr(a)x: a EF0, XE H}, and let M be the orthogonal complement of L. Since F0 is a two-sided ideal L is invariant under ir(A), and hence also is M; write IT1, 1T2 for the restrictions of IT to L and M respectively, and write for the projections of onto L and M respectively. Forj = 1, 2 where a 6 A. Then and are clearly positive define th1(a) = linear functionals on A and, using orthogonality, we have for a 6 A
+ b2(a) = =
=
+
+
1(a).
Similarly we obtain 4.'2(a) =0 whenever a 6 F0, and also that lkb1Il
Suppose
now that b =
112
=
+ '/4 where
= 0 on F0 and
=
Foil.
Then if {u,j is an approximate identity for the ideal F0, and if IT' denotes the
CONVEXITY THEORY FOR
230
representation associated with
and with cyclic vector con= converges to (x) for each x A, and similarly converges to = vanish on F0, and F0 for each a, it follows that b1(x). Since i52 and and Finally the set N' K: II'YII = II'YIF0II} is convex because I'YIF0I! = urn (cf. Dixmier [80, 2.1.5]), and it is therefore evident that F is a split face of K. (iii) implies (ii). As in the proof of 2.2 we have c Fo. Since is self-ad joint it follows that F0 = line; (Fj' c Lp, so that F0 = LF. Hence F0 is a seif-adjoint left ideal and therefore a two-sided ideal. (i) implies (ii). We know that F = Let u E A be unitary, so that EF it follows that u*au whenever b E F. Hence if a and therefore au = u(u*au) e is a left ideal. Since A is the complex linear span since of its unitary elements we deduce that LF is a right ideal also. In particular
verges to
Consequently
is a strongly Archimedean ideal because Ah/(A,, n L1.) is isometrically order-isomorphic to (A/LF),,. Therefore Ah = F1 (cf. Chapter 2 Section 8), and also LF = F0. (ii) implies (i.). Let F = I', where 1 is a closed two-sided ideal in A,.and let F and a E A with *a) > (1. Then if h E I we have a */,a E I, so that
&(h) =0. Hence & F and F is invariant.
If A is a unital
with state space K then is an algebra homomorphism. Moreover, the following statements are equivalent: (i) A,, is a vector lattice; (ii) K is a simplex; (iii) A is commutative. 2.4. COROLLARY.
is a split face of K if and only if
Proof.
If tb is an algebra homomorphism then
'(0) is a closed two-sided
'(0)) is a split face of K by Theorem 2.3. ideal, and hence b = 1(0) is a two-sided ideal, Conversely, if is a split face of K then by Theorem 2.3. It is elementary now to verify that q5 is an algebra
homomorphism. (i) implies (ii) is trivial. If A is commutative then the Gelfand—Neumark Theorem shows that A = C(X), and hence that A,, = cR(X), for some compact Hausdorif space X; consequently we obtain (iii) implies (i). Finally,
if K is a simplex then each b aK is a split face of K, and hence b is an algebra homomorphism. Therefore = for all a, h E = A and &K, so that ab = ba by the Krein--Milman Theorem; consequently (ii) implies (iii).
State spaces of unital concerning split faces.
provide some interesting examples
TILE IDEAl. STRUCTURE OF A UNITAL C*.ALGEBRA
231
2.5. EXAMPLE. Let A denote the (the bounded linear and let K be the state space of A and F = operators on where L is the
two-sided ideal of compact operators jn A. Then F is a closed split face of K
and there existf, g A(K)4, h
with h f, g on F, and such that no extension h'€A(K) of h exists satisfying gon K. Proof.
for n
1
e21,
be the coordinate basis By 2.3, F is a closed split face of K. Let define and for n =0, = —1 -2 -'1/2 e2,,, --1 —n (n — n )e2,, 1—n
1 +(n
1
1
C2n.
It is easily checked that f and g =f+k are projections withL kwhile gonF. Puth = L.Wehaveof, g onK f, that there exists an h' A(K)' extending h with h' g. If q denotes the range projection for h' we have q f, g, so that the intersection A'! of the ranges of f and g contains the range of q. However, inspection shows that A'! = {0}, giving a contradiction.
The above example should be compared with 2.10.5. In Chapter 2 we saw that every closed split face is semi-exposed; the following example answers a natural question in this connection. Let Hhe a non-separable Hilbert space, let A = B(H) and let K be the state space of A. Then if L denotes the compact operators on H, F = L1 is a closed split face of K which is not an intersection of exposed split faces of K. 2.6. EXAMPLE.
Proof. Since L is a minimal closed two-sided ideal in A, F is a maximal proper closed split face of K. It will therefore be sufficient to show that F is f(x)> 0 for not exposed. Suppose that there exists an f€ A(K) with fE all x K\F. Then f identifies with a compact Hermitian operator T on H, H with and since H is non-separable there exists a 1 and =0. But then if j5 is defined on A by K and = (at, we have for H, then f(qS) = (TE, =0. However, if a E L satisfies a-ri = çb(a) = I so that j5 K\F, giving a contradiction.
We now give a generalization of Corollary 4.9.5. 2.7. ThEoREM. If a complex unital Banach algebra A is a complex Lindenstrauss space then A = C(X) for some compact Hausdorff space X. Proof. Let e denote the algebraic identity of A and let S denote the state space of A. The second Banach dual A** of A is also a complex Banach algebra with respect to the Arens product and has unit ê, the canonical embedding of e in A**. Let S** denote the state space of A** relative to ê.
CONVEXITY THEORY FOR
232
Since A is a complex Lindenstrauss space A** is linearly isometric to the space C( Y) for some compact Hausdorif space Y. Let T: A** -÷ C( Y) The Bohnenblust—Karlin Theorem implement this isometry, and let u = [43] shows that ê is an extreme point of (A**)i, and so U is an extreme point of (C(Y))1. Hence ui = 1 on Y. Let S' denote the state space of C(Y) and define p: (C( Y))* by r(d4(f) = q5 (fü) for q5
(C(Y))* and f€ C(Y). Now r is a linear isometry onto (C(Y))* and
= 1 = b(1) so that hence p is a linear isometry onto A***. If b ES' then = = = çb(uu) = 1, giving Jp(b)Ii = I, and also =I= we have = Conversely, if E and (p p(b) E = (T* = S'. = ili(ê) 1, so that We have shown that is an affine isomorphism between the simplex S' and S**, and therefore S** is a simplex. Similarly pco (S' u —iS') is an affine isomorphism between the simplex co (S' —iS') and co u _j5**) so that S** is a split face of Z**. Therefore A** is a (see 4.0.2), and Corollary 2.4 shows that A** is commutative. The subalgebra A of A** is consequently a function algebra, and so the result follows from Corollary 4.9.5.
Notes
Theorem 2.1 is due, independently, to Ethos [92, 5.1] and Prosser [176, 5.11]; our treatment follows the proof of Prosser. Corollary 2.2 was noted by Chu [66], and the particular case for extreme points follows from a result of Glimm [120, Theorem 9]. The connection between closed two-sided ideals in a and the split faces of the state space as presented in 2.3 was
developed in the papers of Effros [92], StØrmer [206] and Alfsen and Andersen [6]. The implication (ii)
(iv) in Theorem 2.3 can also be proved
by a method of Smith and Ward [197, 5.3] by which they show that the M-ideals and the closed two-sided ideals coincide for any The equivalence of (i) and (iii) in Corollary 2.4 was first proved by Sherman [192]. Example 2.5 is due to Stefánson [200], Example 2.6 is due to Andersen and Theorem 2.7 to Ellis [105].
3. COMMUTATIVITY AND ORDER IN C*..ALGEBRAS
The relationship between commutativity and order properties in Cd'algebras has been considered by many authors; in this section we discuss some of their results. subspace of the bounded seif-adjoint operators Let V be a on a Hubert space H. Then if a E V we denote by a j the positive square root
COMMUTATIVITY AND ORDER IN
233
of a2. Even if a belongs to V it is not in general the lattice modulus of a in V unless the operators in V commute. 3.1. THEOREM.
Let V be a linear space of bounded linear seif-adjoint
operators on a Hubert space H. Then if a E V whenever a E V, and if V has the Riesz decomposition property, V is commutative. Proof.
Let L be the closure of V in B(I-f) for the strong (or weak) operator
topology. Then al e L whenever a EL (cf. Kadison [137]). If P {a c L: 0 a, flail 1} then its strong closure 0 is compact for the weak operator topology. If
Q
the Krein—Milman
Theorem, and hence so is V because V is positively generated since I
aj a, —a for each a E V.
is commutative we will prove that V is a vector Before showing that lattice with a equal to the lattice modulus of a, for each a E V. First suppose and that ab =0. If x E V and ..v a, b then, by the Riesz that a, h E
interpolation property, there exists y E V with 0, x y a, b. If z"2 denotes the positive square root of any z E V we have 0 a' "2ya"2 = 0 since a'12 is the limit of certain polynomials in a, and ab = 0. /2)? I/2)* so that a112y 1/2 = 0 and Therefore 0 = a '12ya 1/2 = (a "2y hence ay 0. Moreover, we have 0 y2 y"2ay112 =0 so that y2 = 0 = y. Hence every lower bound x for a and b belongs to — V , and so inf (a, h) 0. Now for any a V we have Ia I + a, a — a 0, and (lal a )(a I — a) =0, + a, a j — a) =0. Thereand therefore the preceding argument gives inf I
= laI, and V is a vector lattice with the required property. The strong continuity of the modulus operation now shows that L is an M-space with al the lattice modulus of a for each a EL. Since 0 is compact for the weak operator topology, and is also a directed set, Q contains a greatest element SO that L is a GR(X) space, for some compact Hausdorif space X. The extreme points of Q correspond to characteristic functions of open and closed sets in X. Hence if e,, e2 E &Q then e1 A e2, e1—e, A e2 and and are disjoint. Suppose that a, h E 0 and e2 — e, A e2 all belong to a A b =0. Then we have fore sup (a, —a)
Ia I — inf (Ia I + a, Ia I —a)
(a+h)2=(avb+aAb)2=(avb—aAh)2=Ia--b12=(a—b)2, so that ah ± ba 0. But then a2b2 —ahab = h2a2 and so the positive square root a of a2 also commutes with b2. Applying this argument again gives ab = ha, so that ab 0= ha. It follows that e, A e2, e, — e1 A e2 and A e2 commute, and hence e, and e2 commute. C2 — The above result enables us to extend Corollary 2.4 to the non-unital case immediately.
CONVEXITY THEORY FOR
234
Let A be a only if A,, is a simplex space. 3.2. COROLLARY.
Then A is commutative if and
Then A is commutative if and Let A be a only if it has the property that 0 a b in A implies that 2 b2. 3.3. COROLLARY.
Of course a belongs to A,, whenever a E A,1. The result will follow if we can show that A,, is a vector lattice; for this it is sufficient, as in the proof of 3.1, to show that a, h E with ab =0 implies that inf (a, h) =0. Suppose that x E A,, with x a, b. Then we have 0 b — x 0, (Al + S + T)(AI + T — S) = (A i(iTS — iST) = X + iY, where X and Y arc Hermitian. Moreover X is Proof.
S I
positive and invertible and since X "2yx""2 is self-adjoint the spectrum of I —iX'12YX 1/2 lies in {z C :rez = 1} and consequently (X + iY) =
X'12(1+iX
1X1"2 exists. Therefore it follows that the spectrum of (T — S) lies in the positive half-line, giving S T.] Similarly we + x) a, b. obtain xI 2b — x, and the two inequalities together give 0 I
As in the proof of Theorem 3.1 the fact that ab =0 now implies that x 0. Consequently we obtain 0 = inf (a, b) as
required. Notes
An extensive investigation of the relationship between order properties and commutativity in was carried out by Topping [208], where, in particular, he proved Theorem 3.1. Corollary 3.3 was originally proved by Ogasawara [166].
4. THE CENTRE OF A
WEAKLY CENTRAL ALGEBRAS, PRIME ALGEBRAS
In Section 3.1 we introduced the abstract notion of centre for A(K) spaces. We now show that this notion coincides for A,, with the algebraic centre, that is those elements in A,, which commute with all elements of the algebra. In version of carrying out this process we investigate the concrete the Dauns—Hofmann Theorem that we studied in Section 5.1. If A is a unital with state space K, and if bc&K, then the usual GNS construction gives an irreducible of A, and the kernel of is called a primitive ideal in A. The space of all primitive ideals in A will be denoted by Prim A, and will be endowed with the Jacobson (or hull-kernel) topology (see Dixmier [81]).
WEAKLY CENTRAL ALGEBRAS, PRIME ALGEBRAS
235
4.1. THEOREM. If K is the slate space of a unital then the following statements hold. (i) us a primitive ideal in A if and oniy if!' = {x E K; a(x) =0, Va E I} is a primitive face of K.
(ii) K satisfies Strmer's axiom. (iii) 1/f: Prim A R is continuous for the Jacobson topology then there exists an a E A,,, with a in the algebraic centre of A, such that for all çb €aK. f(ker (iv) The algebraic centre of A,, coincides with the centre of A(K).
so that I is a primitive ideal in A, and let I = ker *xy) e iiF, 'I' is the w*..lirnit of states of the form (see Dixmier [81, Corollary 3.4.3]), and since the primitive face containing j5 is invariant we see that Therefore we have F = E,,, a primitive then I = ker face of K. Conversely, if E is a primitive ideal, and again we obtain Fqs (ii) Let {Fa} be a family of closed split faces of K and let the family of closed two.sided ideals in A with F,. = (see 2.3). Then if I = fla and if E = aFa we clearly have Proof.
Let
E
F = I'. Then, If
a
a
a
4,e.E
Therefore 1 contains the set
the
=fl
(F,,,)o.
U, Fa. Now, if
of states of the form 3.4.2]). Therefore we obtain 'V E U, It follows that U. Fa face of K.
(ker ira)'), 'I' is )E, E) where / E (see Dixmier [81, (ker and consequently 'V and hence is a closed split
(iii) By the result of (i) the Jacobson topologies on Prim A and on Prim A(K) are equivalent. Therefore, by Theorem 1.1, there exists an aE belonging to the centre of A(K), such that f(ker = 4(a) for all We need to show that a belongs to the algebraic centre of A. Let I be a primitive ideal in A and let p E sf1. Then ker is a primitive ideal containing I, so that ker irs, belongs to the closure of I in Prim A(K) be
and thus f(I) =f(ker
In p(a —f(ker ira)) =0 for all p E
particular we see that p(a —f(I)) = so that a —f(I) belongs to I by the
Krein—Milman Theorem. Now for any b E A we have ab —f(I)h E I and ba —f(I)h E 1, so that ab — ha E I for any primitive ideal 1. Because A is semi-simple it now follows that ab = ha, that is a belongs to the algebraic centre of A. (iv) Let a belong to the centre of A(K). Then, by Theorem 1.1, the function f: Prim A R defined by f(ker = 4(a) is continuous. Hence the proof of (iii) shows that a belongs to the algebraic centre of A.
CONVEXITY THEORY FOR C*.ALOEBRAS
236
Conversely, let a belong to the algebraic centre of A, a Ah. We may assume that If 4) E .3K then 8 = — 4)(a) E (0, 1), and we can define A. If b 0 then we have 0 'P(b) =8 b — a)) for all b Therefore we see that 'V E K and also (1—8) '(4) — 8'I') that 0 'I' S K. Since 4) is extreme in K the equation 4) = 8'P+(4) —6'!') implies that 4) = '1', and hence 6q5(b) 4)(bd— a)) = (i— 4)(a))di(b). Consequently we obtain 4)(ba) = 4)(ab) = 4)(a)4)(b), and therefore a belongs to the centre of A(K). and Bishop decompositions for a compact In Chapter 4 we defined the convex set K, and we investigated their relationship with function algebras. A then these decompositions If K is the state space of a unital
give rise to natural decompositions for A. Using some of the results denotes the developed in this chapter it is not difficult to see that if is the family bf closed two-sided ideals in decomposition for K then A which arc generated by the maximal ideals of the centre of A. Similarly, if {F0} denotes the Bishop decomposition for K then {(F0)0} forms the family of closed two-sided ideals I which are minimal subject to the property that
A/I has trivial centre. If K is the state space of a
then the Bishop decomposition situation, and it is not known in general need not cover 0Km the if the Bishop decomposition determines A(K) amongst Banach subspaces of However, if some subspace L of has the form L = B,, for some unital B containing A as a *..subalgebra, and if L contains {fE
I
A(K)I(F0
then L = A,, by Glimm's Stone—Weierstrass Theorem [119]. In general the best result we have is the following, which shows that the Bishop decomposition always determines the centre of A (K). 4.2. THEOREM. Let K be a compact convex set satisfying Størmer's axiom and let {F0} he the Bishop decomposition for K. Then we have (centre A (K))
{f€
:fI(F0
is constant, V0}.
Proof. 1ff belongs to the centre of A(K) then certainlyf is constant on each F,3 because of antisymmetry. Conversely, letf E C14(.3K) be constant on each 3K. Since each primitive face of K is contained in some F0 we can i1 by g((FX).I) =f(x) for all x E 3K. If we define a function g: Prim can show that g is structurally continuous it will follow that f€ (centre A(K))IOK by Theorem 1.1.
F0
It will be sufficient to show that g is upper semi-continuous for the :f(x) a} is non-empty. structure topology. Let a R such that E =
WEAKLY CENTRAL ALGEBRAS, PRIME ALGEBRAS
237
Since K satisfies StØrmer's axiom the set F = {UFX :f(x) a } is a closed Certainly we have (Fr).. J for each (Fr) E E. split face of K; put J =
J then is contained in F. Now the set On the other hand, if f(y) a} is compact and contains all with f(u) a, and G = {y E consequently G 2 by Milman's Theorem. Hence we havef(x) a so that belongs to E. We have therefore shown that E = h(J), and thus E is structurally closed and g is upper semi-continuous as required.
A is said to he weakly central if whenever I, J are maximal two-sided ideals in A, and Z is the centre of A, then I Z = .1 Z implies that I = J. A compact convex set K is weakly central if whenever F and G are minimal closed split faces of K, and is the centre of A(K), implies that F = G. We show next that the Bishop then F1 = A
decomposition always determines A(K) when K is weakly central. 4.3. THEoREM. If K is weakly central then the Bishop and &ilov decompositions for K coincide, consequently if A is a -algebra, or any weakly central unital the Bishop decomposition for the state space of A determines A.
Let F be a face in the decomposition for K, and let be the centre of A(K). Then if 0 and H are distinct minimal closed split faces of F we have = since each is constant on F. Therefore Fl = G because K is weakly central and this contradiction shows F has no non-trivial, mutually disjoint, closed split faces. Hence A(F) must have trivial centre, and consequently the Bishop and decompositions for K coincide. The preceding general theory now shows that the Bishop decomposition determines A(K) and hence A, whenever K is the state space of a weakly central unital Since every is weakly central (cf. now follows. Misonou [160]) the result for Proof.
4.4. A compact convex set K is weakly central if and only for every closed split face Fof K the centre of A(F) equals the restriction toFof the centre of A(K). consequently a unital A is weakly central if and only if for every closed two-sided ideal Tin A the centre of A/f is the quotient of the centre of A.
Suppose first that centre A(F) = (centre A(K))IF for every closed split face F of K. Let 0 and H be minimal closed split faces of K, so that, in particular G and I-f are disjoint. Let F = co (0 u I-I) and let f€ A(F) such that f(x) =0 for all x G and f(y) = I for all y I-I. Clearly f is central in
Proof.
238
CONVEXITY
FOR
A(F) and so there exists a g central in A(K)with g belongs which shows that K is weakly central. to G but not to Conversely, suppose that K is weakly central. Let F be a closed split face of K, and let q A(K) A (F) be the restriction (quotient) map. If Max denote the map centre A(K) and = centre A(F) let c7 : Max where N belongs to the maximal ideal space of q(N) f(x) = O} for some x aF so that belongs to Max because N = : f(xi)=O}he distinct from N, with E oF. Then there exist disjoint closed split faces G and G1 of F containing x and x1 respectively. But then x and x1 cannot belong to the same member of the Silov decomposition for K, arguing as in Theorem 4.3. Hence there exists a g Ej with g(x) g(x1), and consequently 4(N) 4(N1), which shows that 4 is an injection. It is easily seen that c7 is continuous. In fact let Na = {f E
:
f(Xa) = O} be a
f(x) = O}, that is for all net in Max j' converging to N = {f€ -* g(x), which means that so that f€ But if g we have 4(N). This argument shows that, under 4, Max is homeomorphic to a closed subset of Max 4_I to Max so Let f and let g be a continuous extension of fo we have g(x) = g({h h(x) = O}) = that g Max1. If x g(4({h€j': h(x)=O}))=f(x), so that g is an extension of f.
counterpart of Theorem 5.1. A We now consider the algebra A is called a prime algebra if whenever 1, J are closed two-sided ideals in A then IJ = {xy: x E I, y J} = {O} implies that either I = {O} or J={O}. 4.5. THEOREM.
Let A he a unital and oniy if Ah is an antilattice.
Then A isa prime algebra if
By Theorem 3.5.1, Ah will be an antilattice if and only if the state space K of A is a prime compact convex set. Let A be a prime algebra and let F, 0 be semi-exposed faces of K with Co (F G) = K. Then, by Theorem 2.1, there exists a closed left ideal L and and G = R . Now we have a closed right ideal R in A with F = RL R L (co (R u L' = {O}, and therefore AR and LA arc closed two-sided ideals with ARLA = {O}. Hence we obtain either AR = {O} or LA = {O}, that is either R = {O} or L = {O}. Therefore either F = K or G = K so that K is prime, and A,, is an antilattice. Conversely, suppose that K is prime and that I, J are closed two-sided = {O}. In fact I nJ is a ideals in A with JJ = {O}. We must also have I closed two-sided ideal containing If and, since (I J) A,, is positively Proof.
generated, we can deduce that I n I = If if we prove that every b €
UNiT TRACES FOR
239
belongs to If. Since b"2 is a uniform limit of polynomials in b, with no constant terms, we have b112 €1 n J, and hence h b 112b"2 E If as required. But now it follows that co (I = K, and hence = (I either I = K or f = K. Therefore either I = {O} or J = {O}, so that A is a prime algebra.
AZ
4.6. COROLLARY.
If A is a
with state
space K then the
following statements are equivalent.
(i) A isa factor; (ii) K is a prime compact convex set; (iii) A1, is an antilattice.
Prool The equivalence of (ii) and (iii) has already been established. If K is prime then A(K) is an antilattice, and so clearly the centre of A(K) must be the constants. Consequently A is a factor, by Theorem 4. 1(iv). Conversely, suppose that A is a factor. If A is of type H or type III then aK= K (see 011mm [119, p. 239]) and hence K is prime by Corollary 3.5.2. If A is of type I then there exists a f E uK associated with a faithful irreducible representation of A. If F is the smallest closed split face of K aK is the of states of A of the form containing then every (see Dixmier [81, Corollary 3.4.3]), and the invariance of F now gives F = K. If I, J are closed two-sided ideals in A with If = {O} then we K = Co (J1 u f'), as in the proof of 4.5, and since either €1' or conclude that either P = K or J = K. Therefore either I {O} or f {O} so that A is a prime algebra. Consequently A,, is an antilattice by Theorem 4.5. Notes The results contained in Theorem 4.1 arc due to Størmer [206] and to Alfsen and Andersen [6, 7, 16]. Theorems 4.2 and 4.3 were proved by Ellis [106]. was studied by Vesterstrøm in The theory of weakly central
[211], and Theorem 4.4 is an extension by Chu [64, 67] of some of Vesterstrøm's results to compact convex sets. Theorem 4.5 is due to Chu [66]; it was also proved by Archbold [20] using quite different techniques. Corollary 4.6 was first proved by Kadison [138]. Sonic further results of this kind are proved by Dixmier [82].
5. UNIT TRACES FOR We saw in Section 2 that the state space K of a unital
A is never
a simplex unless A is commutative. Nevertheless K does possess an algebraically important subset which is a simplex. Let T(A) = Va, b E A) denote the unit traces on A. We will e K: =
240
CONVEXITY THEORY FOR
require the fact, proved in the next section, that every Hermitian E A * has with 'V', a unique decomposition 'P = 'it — positive and II'Vll = II"Pil÷II'i'lI; this result is due to Grothendieck [123]. We will prove that link T(A) is an L-space with = sup ('iF, 0), 'P = sup (—'I', 0) for each 'PE "AR T(A).
5.1. THEOREM. The u/lit traces T(A) on a unital
A form a
compact simplex. Proof. Let V = T(A) and let 4) V. If u E A is unitary define and i41 belonging toA* by 'I'(a) =4) *au), 'P(a) =4) (u *aii)for each a E A, where 4) + and 4)- are as mentioned immediately prior to the theorem. We have, foreach a EA, and also ll'Pi1 + Il'Vil. The uniqueness of the 1I'i'÷II + I1'r + 114) 11=114)11
= decomposition 4) = and 'V = 4), that is — 4) now implies that = 4)(u*au) and 4)(a) = V(u*au) for all a E A and all unitaries u €A. Therefore cj'(ua)= cb+(u*uau)= a a linear combination of unitaries it follows that 4) and the theorem will follow if we V show that inf (p, q) exists in V for each pair p. q in V, or equivalently that [0, p1 is lattice-ordered for all p in Firstly, let p £ (A*) and let be the Hilbert space associated with p in
the usual Gelfand—Neumark-Segal construction, and let denote the representation of A on H,, induced by the left regular representation of A. the completion of A/Ne for the norm induced by the inner product = {x E A: p(x*x) = O} and I denotes the image of (a, b) = p(h*a), where x A in A/Na under the quotient map.] if q A* with 0 q then the Cauchy—Schwarz inequality for q gives q(bi*ai) = q(ba) whenever h1 =b and = ci, and hence there exists a Dq such that q(b*a) = (a, Dqb) for all a, b A. Since 0 q p it follows easily that 0 Dq where I is the = q(b*a*c) = identity operator on Moreover DqLj,(a)b) = (a*c,Dqb)=(Lp(a)*ê,Dqb)=(ê,Lp(a)Dqb), and hence Dq belongs to the commutant A}. Conversely, if D = of and we can define q(a)=(a,Di), and it is easy to check that 0 q p and that D = Dq. From this reasoning it follows that {q £ 0 D I}. A*: 0< q p} is order isomorphic to {D Now take p V with 0 q p. In this case the right regular and the representation of A induces a *..representation R,, of A on such that .Jâ = involution in A induces an isometry J on for a e A and (x, y) = (Jy, Jx) for x, y in 1-I,,. For all, a, b, c in A we have (a, = (ab*, Dq5) = = q(b*c*a) = (a, Dqcb) = (a, so that Dq where q' E A* and 0 q' p, Conversely if then
UNIT TRACES FOR C*.ALGEBRAS
241
q'(ab), that is q' V. Therefore we have shown that {q E V: 0 q p} is 0< D I}. order isomorphic to {D E
r If we can show that is a commutative then and T€ the result will follow. Let be 1-lermitian. Then in Ah with -* Si and 37, JS1 = Si, JT1 = Ti and so there exist Ti. = urn For each a E A we have Sâ = and Ta = similarly. Since each pair urn and commute it follows YPZ
that (STd, b) = (Td, Sb) = urn (TSd,
lim
=
h)so that ST= TS.
The first part of the above proof shows that, for p K, is an irreducible *representation of A if and only if p aK, whereas the second part of the proof shows that, for p T(A), L,, is a factor representation (i.e. is trivial) if and only if p aT(A). If p 9K T(A) then is a two-sided ideal and also, byTheorem 2.l,{p}= Nt so thatp is a split face of T(A) coincides with the characters of A. In K; consequently the set T(A) is not equal to 3T(A); in fact if A is the general of complex 2 x2 matrices then (cf. Størmer {201}) the state space of A is affinely isomorphic to a ball in R3, and the unique trace maps into the centre of the ball. In the case when A is separable a result of Dixmier [80] shows that is a primitive ideal whenever p E in this case is a primitive face of K containing p. Another subset of K which is a simplex, and which is of algebraic interest,
is the commutator face Cm(A) of K. Here we define Cm(A) = (comm where the commutator ideal comm 4 of A is the closed two-sided ideal generated by the commutators ab — ba, a, b A. Since T(A) = {ab —ba: a, b €A} it is clear that T(A)Cm(A). In the case when A = B(C"), T(A) is a singleton whereas Cm(A) is empty, because A is a simple algebra. 5.2. THEOREM. The commutator face Cm(A) of a unital A is the largest closed split face of the state space K which is a simplex. Moreover,
Crn(A) is the closed convex hull of the characters of A. Proof.
By Theorem 2.3, Cm(A) is a closed split face of K; also Cm(A) can
be identified with the state space of the commutative A/comm A, and hence Cm(A) is a Bauer simplex. It is evident that Cm(A) contains all the characters of A. If x A\comm A then 0 I A/comm A,
and since A/comm A is commutative there exists a character 'I' of = 0; hence defines a character of A with q5(x) 0. It follows from this reasoning that comm A is precisely the intersection of the kernels of all the characters of A. Since each character of
A/comm A such that 'I'(I)
A is a split face of K, and since K satisfies Størmer's axiom, the closed
CONVEXITY THEORY FOR C*.ALOEBRAS
242
convex hull of the characters is also a closed split face F of K and hence F= (F1) L= (comm A) l. = Cm(A).
Suppose that 6 is a closed split face of K which is a simplex. Then, if aG, q5 is a split face of G and hence b is a split face of K. But then, by Corollary 2.4, is a character of A so that 4) Cm(A). Consequently Cm(A) contains G. q5
Notes
Theorem 5.1 is a result of Thoma [207], but the proof given here is due to Effros and Hahn [97]. A consequence of 5.1 is that each 4) T(A) is the barycentre of a unique maximal measure on T(A). This measure is the central measure corresponding to 4, (see Sakai [188, Theorem 3.1.18]). For a complete discussion of central measures for compact convex sets and for we refer to Alfsen [5, § 2.8], Sakai [188, Chapter 3] and Wils [214, 215,216]. The use of Choquet boundary theory in the decomposition of lower semi-continuous semi-infinite traces on separable is discussed by Davies [74]. is given by Alfsen Another approach to the trace simplex for and Shultz [10, § 12], and some further results concerning simplexes of states have been proved by Batty [33]. The theory of commutator ideals in C*_algebras is discussed by Arveson [21, § 3.3].
6. THE STATE SPACES OF JORDAN OPERATOR ALGEBRAS AND OF We firstly consider some recent work of E. M. Alfsen, F. W. Shultz and E. Størmer concerning characterizations and representations of certain Jordan algebras.
Let A be a real Banach space which is a Jordan algebra over R with identity I and such that (i) Ia . bil Ilaillibli, (ii) 11a211 = Ia and (iii) 11a211 112
a, b in A; A is called a J13 -algebra. The connection between A(K) spaces and JB-algebras is demonstrated in the following result. fla2
1)211 for all
6.1. THEOREM.
If A is a JB-algehra then A can be identified with an A(K)
space such that A(K) = A2 = {a2: a A}, 1(x) = a for all x E K, and 0 a2 1 whenever —1 a 1. Conversely, if A(K) is a Jordan algebra over R
with identity 1, and if 0 a2 1 whenever haIl
1, then A(K) is a JB-
algebra.
Let A be a JB-algebra. If a A then (cf. Jacobson [133, p. 36]) the polynomials in a form an associative subalgebra, and hence their closure Proof.
THE STATE SPACES OF JORDAN OPERATOR ALGEBRAS
243
C(a) is a commutative Banach algebra. Let a, b €A2\{O} and put a =llall, = llbll. Since la/a 11= 1 we can find d C(a) such that 1—a/a = d2, by the Binomial Theorem. Hence if a = c2 and f= 'lad we have a — a = ad2 so that la —all 111211 11c2+f211 = a; similarly we obtain —bli -continuous, and hence are the duals of projections F, 0: V V. Clearly we have IIPII, 1Q11 1. Suppose that Then, for p E K, the Cauchy—Schwarz inequality p*a =0 for some a = 0. Hence pa =0 and also ap = (pa)* =0, so gives p(pa)12 c (Q*A)+. Similarly we that Q*a = a — ap —pa +pap = a, that is (ker (p*A) obtain (ker Q*)+ Now if a E (p*Ar we have pap = a, so that Q*a = (1 —p)a(1 —p) = (1 —p)pap(1 —p) =0. Hence (P*A)+ = (ker and similarly (Q*A)+ = (ker p*)f• In order to show that F'> is smooth we need to shiw that when p 0 and p*)0, then p E (ker p*)0 Since PV is closed we need to show that pE we see that b p PV. Now if b = 1 so that b(p) = 0, and hence = 1 without loss of = p(l) = p(P*l) = IPpil, and we can assume generality. Note that p(l — p) = p(l — 1) =0, and so for each a E A we have, using the Cauchy—Schwarz inequality, p(a) = p(pap)+p(pa(1 —p)) ± p((l —p)ap) +p((l —p)a(1 —p)) = p(pap) = p(P'>a)). Therefore we obtain p = Pp and thus P'>, and similarly Q*, is smooth. Consequently is a p-projection. Conversely, let F'> : A-> A be a P-projection with quasi-complement Q* and let P'> 1 = p. Let 0< A , and also a = P'>a b =P*b But then Aa =p —(1 —A)b Ap, and we obtain a =p = b. This shows that p is an extreme point of [0, 1] and is therefore a projection in B (see Prosser [176,2.2]). Now if R* : A -> A is defined by R *a = pap we know that R * is also a P-projection with the same range as F'>, namely the order Proof.
CONVEXITY ThEORY FOR
246
ideal generated by p. The quasi-complement S* for R* will be given by S*a = (1 —p)a(1 —p) and since Q*1 = 1 —p, we can deduce that S*A = Q*A It follows that ker R* = ker and the first equivalence of the theorem is established. Let : A -+ A be any P-projection and let x e K PV, SO that (P*1)(x) = 1(Px) = 1. Conversely, let x EK such that (P*1)(x) = 1. Then, if Q* is
the quasi-complementary P-projection for P*, we have (Q* 1 )(x) = 1)(x) =0 so that x (ker = (PV). Hence we have shown that the projective face F,,, associated with P, has the form F,, = {x K: (P*1)(x) = 1}. (In particular, F,, is norm-exposed.) Now if F is any norm-closed face of K there exists a projection pp E B,, (the carrier of F) such that F = {x K: pp(x) = 1} (see proof of Theorem 2.1). The first part of the proof, together with the preceeding remarks, now shows that F is the projective face associated with the P-projection Ppa = (1
a€A. Suppose that F,, = for some P-projection S: V V. Then P and S have the same range and so arguing as we did above for the equality of is a bijection as required. P = S. Therefore the map F
If F is a split face of K then it is easily seen that the associated = A, so that PF = is central in A, and in B,, by Theorem 4.1(iv) (see also Section 3.8). Conversely, suppose that is central in A. Then if Q* is the quasi-complement to
P-projection P* is order-bounded in
I. It follows that K = co (F,, u F0) and hence, by Theorem 3.17, that F,, and FQ are complementary split faces of K. A is precisely the normal state Since the state space K of a unital A** the preceding result shows that every normspace of the closed face of K is projective. The following result proves, and generalizes, the result concerning unique decompositions which was used in the proof of Theorem 5.1. 6.4. THEOREM. Let K he a compact convex set such that every normexposed face is projective. Then each p E A(K)* has a unique decomposition p = o — r, where o', 0 and = HI + HI. Proof. Let p = — r where cr, r 0 and = flcrll + lrll = 1,0W, r 0. Choose f€A"(K) such that I =II/1I=f(p), and let g +f), h = 1 —g so that it 0g, h 1. Since 1 = (g—h)(p) =g(o)—g(r)—h(cr)±h(r) =
that g(o-) = licrIl, h(-r) IIrII and g(r) = hfrr) 0. Put F = K: g(x) = 0} so that F is a norm-exposed face of K containing r/IIrII, and let P: V - V (V = A(K)*) be the P-projection such that F = F,,. Let k A"(K) be defined by k = (PI)*l where P' is the quasi-complement of P. Now follows {x
247
TIlE STATE SPACES OF JORDAN OPERATOR ALGEBRAS
g =0 on F,, = = (ker gE(kerPs)O=(P?)*Ab(K). Therefore, since (P?)*1 = k
P' is smooth we have we have
1.
It is evident that Pr = r and hence P'r 0. Moreover, since g(u) 114 we have = ho-Il so that (P*1)(cr) = (1 ._(pl)* 1)(o-) 0. Hence o- belongs to (ker = (im P')', and therefore P'p = Since P' did not depend on the
choice of decomposition p =
—r
it follows that this decomposition is
unique. Continuing with our notation in which V = un K is a base normed space with dual space A = A"(K), we define (V, A) to be in spectral duality if for each a E A and A R there exists a P-projection P, with quasi-complement 0, such that a (x) A for x E F,, and a (x) > A for x E F0, and such that P satisfies the following conditions: (1) P*a ± Q*a = a; (ii) whenever S and T are
quasi-complementary P-projections on V with S*A ± T*a =
a,
then P
commutes with S and T. In the particular case where A and V satisfy the hypotheses of Theorem 6.3, (V, A) is in spectral duality. In fact let a €A, A €R and let eA be the
spectral projection for a corresponding to the value A. Then eA A and commutes with a, so that eAaeA + (1 — eA)a(l
—
eA) = a,
and thus condition (i)
holds for the P-projection P corresponding to eA. Moreover if p E A is a projection such that pap + (1— p)a(1 —p) = a then, multiplying on the left by p and also on the right by p, we see that p commutes with a, and therefore p commutes with eA. Evidently now (ii) is also satisfied. Finally, if x F1 then 4(x) = I so that a(x) = J A, while if x EFQ we have (x) =0 so
that a(x) =
> A.
In the general case where (V, A) are in spectral duality Alfsen and Shultz
([10, Theorem 6.8]) have proved that for each a A there exists a unique family A E R, of P-projections in V the spectral resolution for a, such = that the family satisfies for A
(i)
(ii)
= inf
= 1, and such that a = JA Riemann—Stieltjcs Integral.
(iii) inf
in the sense of a norm-convergent
When K is a compact convex set such that (V, A b(K)) are in spectral duality, K is said to be spectral. If, in addition, for each a A(K) the family consists of upper semi-continuous functions in A"(K) then K is said to be strongly spectral. 6.5. THEOREM. (1) Every compact simplex is spectral.
(ii) A compact simplex is strongly spectral if and only if it is a Bauer simplex.
(iii) The state space of a anita!
is strongly spectral.
CONVEXITY THEORY FOR Ca-ALGEBRAS
248
(i) Since every norm-closed face of a compact simplex is split and hence projective, the P-projections in V clearly commute. Hence, given a Ab(K) and A E R we need to find complementary faces F and F' such Proof.
that a (x) A on F and a (x)> A on F'. Without loss of generality we can take A =0. Let a = a — a - be the lattice decomposition of a and let F = {x E K: a 4(x) = O} and G = {x E K: a(x) = 0}. Then F and G are split faces of K with a (x) 0 for x E F. If we can show that K = co (F u G) it will follow that
forx€F'. F'cG, and hence that Fix an x in K, and note that a (x) = sup {a (y): 0 y x}. Thus we can choose {y,,} with 0 y,, x, such that a — a(y,,) < 2". We have, for each
ii,
a(y,,)—a(y,, A y,,ii)=a(yn Vy,,