Contemporary

Weak ∗-extreme points of injective tensor product spaces

Krzysztof Jarosz and T. S. S. R. K. Rao

Abstract. We investigate weak∗-extreme points of the injective tensor prod- uct spaces of the form A ! E,whereA is a closed subspace of C (X) and E is a . We show⊗ that if x X is a weak peak point of A then ∈ f (x) is a weak∗-extreme point for any weak∗-extreme point f in the unit ball of A ! E C (X,E).Consequently,whenA is a function algebra, f (x) is a ⊗ ⊂ weak∗-extreme point for all x in the Choquet boundary of A;theconclusion does not hold on the Silov boundary.

1. Introduction

For a Banach space E we denote by E1 the closed unit ball in E and by ∂eE1 the set of extreme points of E1. In 1961 Phelps [16] observed that for the space C(X) of all continuous functions on a compact Hausdorff space X every point f in

∂e (C (X))1 remains extreme when C (X) is canonically embedded into its second dual C (X)∗∗. The question whether the same is true for any Banach space was answered in the negative by Y. Katznelson who showed that the disc algebra fails that property. A point x ∂eE1 is called weak∗-extreme if it remains extreme in ∈ ∂eE1∗∗;wedenoteby∂e∗E1 the set of all such points in E1. The importance of this class for geometry of Banach spaces was enunciated by Rosenthal when he proved that E has the Radon-Nikodym property if and only if under any renorming the unit ball of E has a weak∗-extreme point [19]. While not all extreme points are weak∗-extreme the later category is among the largest considered in the literature. For example we have:

strongly exposed à denting à strongly extreme à weak∗-extreme.

We recall that x E1 is not a strongly extreme point if there is a sequence xn ∈ in E such that x xn 1 while xn 9 0 (see [3] for all the deÞnitions). We s k ± k → k k denote by ∂e E1 the set of strongly extreme points of E1.Itwasprovedin[14] s s that e ∂ E1 if and only if e ∂ E∗∗ (see [9], [13], or [17] for related results). ∈ e ∈ e 1 Examples of weak∗-extreme points that are no longer weak∗-extreme in the unit ball of the bidual were given only recently in [6]. In this paper we study the weak∗-extreme points of the unit ball of the injective tensor product space A E,whereA is a closed subspace of C(X).SinceC(X) E ⊗ ⊗ Both authors were supported in part by a grant #0096616 from DST/INT/US(NSF- RP041)/2000.

c 0000 (copyright holder) ° 1 2 KRZYSZTOF JAROSZ AND T. S. S. R. K. RAO can be identiÞed with the space C(X, E) of E-valued continuous functions on X, equipped with the supremum , elements of A E can be seen as functions on ⊗ X. We are interested in the relations between f ∂e∗ (A E) and f (x) ∂e∗E1, for all (some?) x X. Since any Banach space can∈ be embedded⊗ as a subspace∈ A of a C (X) space no∈ complete characterization should be expected in such very general setting. For example if A is a Þnite dimensional naturally embedded in C (X),withX = A∗,anddim E =1, then any norm one element f of A E is 1 ⊗ obviously weak∗-extreme however the set of points x where f (x) is extreme is very small consisting of scalar multiples of a single vector in A1∗. Hence in this note we will be primarily interested in the case when A is a sufficiently regular subspace of C (X) and/or x is a sufficiently regular point of X. It was proved in [5]that s s (1.1) f ∂ C(X, E)1 [f (x) ∂ E1, for all x X] . ∈ e ⇐⇒ ∈ e ∈ It follows from the arguments given during the proof of Proposition 2 in [6]that for a function f (A E) we have ∈ ⊗ 1 [f (x) ∂∗E1 for all x X with δx ∂eA∗]= f ∂∗ (A E) , ∈ e ∈ ∈ 1 ⇒ ∈ e ⊗ 1 wherewedenotebyδx the functional on A of evaluation at the point x.Inthis paper we obtain a partial converse of the above result (Theorem 1). Our proof also s s shows that if f ∂e (A E)1 then f (x) ∂eE1 for any weak peak point x (see Def. 1), extending∈ one⊗ of the implications∈ of (1.1). It follows that when A is a function algebra then any weak∗-extreme point of A1 is of absolute value one on the Choquet boundary ChA (and hence on its closure, the Shilov boundary) and consequently is a strongly extreme point [17]. Since we have concrete descriptions of the set of extreme points of several standard function algebras (see e.g. [12], page 139 for the Disc algebra) one can give easy examples of extreme points that are not weak∗-extreme. Recently several authors have studied the extremal structure of the unit ball of function algebras ([1], [15], [18]). It follows from their results that the unit ball has no strongly exposed or denting points. Our description that strongly extreme and weak∗-extreme points coincide for function algebras and are precisely the functions that are of absolute value one on the Shilov boundary completes that circle of ideas. We also give an example to show that the weak∗-extreme points of (A E)1 in general need not map the Shilov boundary into ∂eE1. Considering the more⊗ general case of the space of compact operators (E,F) (we recall that under K assumptions of on E or F ∗, (E,F) can be identiÞed with Kp E∗ F ) we exhibit weak∗-extreme points T ( ) for 1

2. The result

As noticed earlier, for A C (X) and a point f ∂∗ (A E) we may not ⊂ ∈ e ⊗ 1 have f (x) ∂e∗E1 for all x X even in a Þnite dimensional case. Hence we need to deÞne a∈ sufficiently regular∈ subset of X in relation to A. WEAK∗-EXTREME POINTS 3

Definition 1. A point x0 X is called a weak peak point of A C (X) if for ∈ ⊂ each neighborhood U of x0 and ε>0 there is a A with 1=a (x0)= a and ∈ k k a (x0) <ε for x X U;wedenoteby∂pA the set of all such points in X. | | ∈ \ There are a number of alternative ways to describe the set ∂pA.Ifx0 X is a weak peak point of A C (X), µ is a regular Borel measure on X annihilating∈ ⊂ A, and aγ is a net in A convergent almost uniformly to 0 on X x0 and such \{ } that aγ (x0)=1then µ ( x0 )=limγ aγ =0. Hence if a∗ A∗ and ν1,ν2 { } X ∈ are measures on X representing a∗ we have ν1 ( x0 )=ν2 ( x0 ),consequently χ x R { } { } { 0} ν ν ( x0 ) is a well deÞned functional on A∗. 7−→ { } On the other hand if χ x0 A∗∗ then µ ( x0 )=0, for any annihilating mea- { } ∈ { } sure µ, and x0 is a weak peak point. To justify the last claim notice that A1 is weak∗- dense in A1∗∗ so χ x0 is in the weak∗-closure of the set K = f A1 : f (x0)=1 . { } { ∈ } Let U be an open neighborhood of x0 and AX U be the space of all restrictions of \ the functions from A to X U.WedeÞne the norm on AX U as sup on X U.Let \ \ \ KX U be the set of restrictions of the functions from K and cl KX U be the norm \ \ closure of KX U AX U .If0 / clKX U then there is G AX U ∗ , represented \ ⊂ \ ∈ \ ∈ ¡ \ ¢ by a measure η on X U and separating KX U from 0: \ \ ¡ ¢

Re G (h) >a>0=χ x0 (µ) , for all h clKX U . { } ∈ \ The measure η extends G toafunctionalonA so K is functionally separated from 0 in A contrary to our previous observation. Hence 0 clKX U so there is a function ∈ \ in K that is smaller then ε outside U which means that x0 is a weak peak point. The concept of weak peak points is well known in the context of function alge- bras where ∂pA coincides with the Choquet boundary ([8], p. 58). For more general df spaces of the form A0 = f0a C (X):a A , where A C (X) is a function al- { ∈ ∈ } ⊂ gebra and f0 a nonvanishing continuous function on X we have ChA ∂pA0. Spaces of these type appear naturally in the study of singly generated modules⊆ and Morita equivalence bimodules in the operator theory [2].

Theorem 1. Let E be a Banach space, X a compact Hausdorff space, and A a closed subspace of C (X).Iff A E isaweak∗ (strongly) extreme point of ∈ ⊗ the unit ball then f (x) isaweak∗(strongly) extreme point of the unit ball of E for any x ∂pA. ∈ In particular if f (C (X, E)) is a weak∗(strongly) extreme point then f (x) ∈ 1 is a weak∗(strongly) extreme point of E1 for all x X. ∈ We Þrst need to show that for a weak peak point x0 X there exists a function ∈ in A not only peaking at x0 butthatisalsoalmostrealandalmostpositive. Lemma 1. Assume X is a compact Hausdorff space, A is a closed subspace of C (X),andx0 is a weak peak point of A. Then for each neighborhood U of x0 and ε>0 there is g A such that ∈ g =1=g (x0) , k k (2.1) g (x) <ε, for all x X U, and | | ∈ \ Re+ g g <ε, − where Re+ z =max°0, Re z . ° {° } ° 4 KRZYSZTOF JAROSZ AND T. S. S. R. K. RAO

Proof. Put U1 = U and let g1 A be such that ∈ g1 =1=g1 (x0) and g1 (x) <εfor x/U1. k k | | ∈ Put U2 = x U1 : g1 (x) 1 <ε and let g2 A be such that { ∈ | − | } ∈ g2 =1=g2 (x0) and g2 (x) <εfor x/U2. k k | | ∈ Put U3 = x U2 : g2 (x) 1 <ε . Proceeding this way we choose a sequence { ∈ | − | } 1 gn n 1 in A. Fix a natural number k such that k>ε and put { } ≥ 1 k g = g . k j j=1 X We clearly have g =1=g (x0) and g (x) <εfor x/U.Letx U, then either k k | | ∈ ∈ x belongs to all of the sets Uj,j k,inwhichcase g (x) 1 <ε,or there is a ≤ | − | natural number p

gn =1=gn (x0) , k k 1 1 (2.2) gn (x) < ,if f (x) f (x0) , and | | n k − k ≥ n + 1 Re gn gn < , − n Hence ° ° ° ° 1 sup f(x) f(x0) f (x) + gn (x) en , f (x) gn (x) en max k − k≥ n {k k | |k k} 1 k ± k ≤ sup f(x) f(x0) < f (x) gn (x) en ½ k − k n {k ± k} ¾ 1 1 + 1 max 1+ , + f (x0) Re gn (x) en + ≤ n n ± n ½ ¾ 3 ° ° 1+ . ° ° ≤ n Therefore f gnen 1 but (δ (x0) e∗)(gnen)=gn (x0) e∗ (en) 9 0.This k ± k → ⊗ contradiction shows that f (x0) is a weak∗-extreme point. Thesamelineofargumentsshowsthatf (x) is strongly extreme for any strongly extreme f (A E) . ∈ ⊗ 1 ¤ Since for a function algebra A the Choquet boundary ChA coincides with ∂pA ([8], p. 58) and the Shilov boundary ∂A is equal to the closure of ChA we have: WEAK∗-EXTREME POINTS 5

Corollary 1. Let A be a function algebra, E a Banach space and f (A ∈ ⊗ E)1 aweak∗-extreme point. Then

f(x) ∂∗E1, for x ChA,and f (x) =1, for x ∂A. ∈ e ∈ k k ∈ Remark 1. Theorem 1 is not valid for the spaces WC(X, E) of E-valued con- tinuous functions with E quipped with the . Even a strongly extreme point of WC(X, E)1 need not assume extremal values at all points of X [13]. We next give an example of a function algebra A and a 3-dimensional space E showing that a weak∗-extreme point f (A E)1 need not take extremal values on the entire Shilov boundary. Since E∈ is Þnite⊗ dimensional this function f maps the Choquet boundary into the set of strongly extreme points but f is not a strongly extreme point. Example 1. Put 3 2 2 3 Q = (z,w,0) C : z + w 1 (0,w,u) C :max w , u 1 , ∈ | | | | ≤ ∪ ∈ {| | | |} ≤ n o and B = convQ.Let be the norm on C©3 such that B is its unit ball. Noteª that k·k 2 2 (z,w,0) is an extreme point of B iff w =1and z + w =1.PutE = C3, , | | 6 | | | | k·k df ¡ ¢ X = 0 1 D (sin t, cos t, 0) : 0 t π , { }×{ }× ∪ { ≤ ≤ } df f0 : X E1,f0 (x) = x,and → A = h C (X):h (0, 1, ) A (D) , { ∈ · ∈ } where A (D) is the disc algebra. We have ChA = 0 1 ∂D (sin t, cos t, 0) : 0

T ∂∗ (E,C (X)) = T ∗ ∂∗C (X)∗ ∂∗E∗. ∈ e K 1 ⇒ e 1 ⊂ e 1 Thus more generally one can ask whether T ∗ (∂¡e∗F1∗) ∂e∗¢E1∗ for any T ∂e∗ (E,F)1. We give a class of counter examples with the help of⊂ the following proposition.∈ K Proposition 1. Let E be an inÞnite dimensional Banach space such that (E) K is an M-ideal in (E).IfT (E)1 then T ∗(∂eE∗) S(E∗). L ∈ K 1 6⊂ We recall that a closed subspace M of a Banach space E is an M-ideal if there is a projection P (E∗) such that ker P = M ⊥ and P (e∗) + e∗ P (e∗) = e∗ , ∈ L k k k − k k k for all e∗ E∗ (see [11] for an excellent introduction to M-ideals). ∈ Proof. Since (E) is an M-ideal it follows from Corollary VI.4.5 in [11]that K E∗ has the Radon-Nikodym property and hence the IP (see [10]). Also since (E) is a proper M-ideal it fails the IP. It therefore follows from Theorem 4.1 inK [10] that there exists a net x∗ ∂eE∗ such that x∗ x∗ in the weak∗-topology with { α} ⊂ 1 α → 0 x∗ < 1. Suppose T ∗(∂eE∗) S(E∗).SinceT ∗ is a by going k 0k 1 ⊂ through a subnet if necessary we may assume that T ∗(x∗ ) T ∗(x∗) in the norm. α → 0 Thus 1= T ∗(x∗) < 1 and the contradiction gives the desired conclusion. k 0 k ¤ Example 2. Banach spaces E for which (E) is an M ideal in (E) have been well extensively studied. Chapter VI of [11K] provides several examplesL including E = p, 1

Proof. Since the space A E∗∗ can be canonically embedded in (A E)∗∗ [7]wehave,foranynaturalnumber⊗ n ⊗ (2n) (2n 2) (2n) A E (A E − )∗∗ (A E) . ⊗ ⊂ ⊗ ⊂ ⊗ (2n+2) If f A E is an extreme point of (A E) then it is a weak∗-extreme ∈ ⊗ (2n) ⊗ (2n) point of (A E) ,asitalsobelongstoA E it is a weak∗-extreme point (2n⊗) ⊗ (2n) of A E . Hence by our theorem f (x0) is an extreme point of E . ⊗ 1 ¤ The next proposition characterizes strongly extreme points in terms of ultra- powers. WEAK∗-EXTREME POINTS 7

Proposition 3. An element e of a Banach space E is a strongly extreme point of the unit ball E1 if and only if (e) is an extreme point of (E )1. F F s Proof. If e/∂e E1 then there is a sequence en n 1 E1 with e en 1 ∈ { } ≥ ⊂ k ± k → and infn N en > 0.Thus (e) (en) =1and (en) =0so (e) is not an extreme∈ point.k k k F ± F k k F k 6 F If (e) / ∂e(E )1 then there is 0 =(en) (E )1 with 1= (e) (en) = F ∈ F 6 F ∈ F k F ± F k lim e en . Thus for every ε>0 the set n N: e en 1+ε is none empty F k ± k { ∈ k ± k ≤ } as an element of . Hence there exists a sequence kn such that e ek 1 F { } k ± n k → but ekn 9 0 so e is not a strongly extreme point. k k ¤ References [1] P. Beneker and J. Wiegerinck, Strongly exposed points in uniform algebras, Proc. Amer. Math. Soc. 127 (1999) 1567-1570. [2] D. Blecher and K. Jarosz, Isomorphisms of function modules, and generalized approximation in modulus, Trans. Amer. Math. Soc. 354 (2002), 3663-3701 [3]R.D.Bourgin,Geometric aspects of convex sets with the Radon-Nikodym property, LNM 993, Springer, Berlin 1983. [4] A. Browder, Introduction to Function Algebras, W. A. Benjamin, New York 1969. [5] P. N. Dowling, Z. Hu and M. A. Smith, Extremal structure of the unit ball of C(K, X), Contemp. Math., 144 (1993) 81-85. [6]S.DuttaandT.S.S.R.K.Rao,On weak∗-extreme points in Banach spaces, preprint 2001. [7] G. Emmanuele, Remarks on weak compactness of operators deÞned on injective tensor prod- ucts, Proc. Amer. Math. Soc., 116 (1992) 473-476. [8] T. Gamelin, Uniform Algebras, Chelsea Pub. Comp., 1984. [9] B. V. Godun, Bor-Luh Lin and S. L. Troyanski, On the strongly extreme points of convex bodies in separable Banach spaces, Proc. Amer. Math. Soc., 114 (1992) 673-675. [10] P. Harmand and T. S. S. R. K. Rao, An intersection property of balls and relations with M-ideals, Math. Z. 197 (1988) 277-290. [11] P. Harmand, D. Werner and W. Werner, M-ideals in Banach spaces and Banach algebras, Springer LNM No 1547, Berlin 1993. [12] K. Hoffman, Banach spaces of analytic functions, Dover 1988. [13] Z. Hu and M. A. Smith, On the extremal structure of the unit balls of Banach spaces of weakly continuous functions and their duals, Trans. Amer. Math. Soc. 349 (1997) 1901-1918. [14] K. Kunen and H. P. Rosenthal, Martingale proofs of some geometric results in Banach space theory, PaciÞc J. Math. 100 (1982) 153-175. [15] O. Nygaard and D. Werner, Slices in the unit ball of a uniform algebra, Arch. Math. (Basel) 76 (2001) 441-444. [16] R. R. Phelps, Extreme points of polar convex sets, Proc. Amer. Math. Soc. 12 (1961) 291-296. [17] T. S. S. R. K. Rao, Denting and strongly extreme points in the unit ball of spaces of operators, Proc. Indian Acad. Sci. (Math. Sci.) 109 (1999) 75-85. [18] T. S. S. R. K. Rao, Points of weak-norm continuity in the unit ball of Banach spaces, J. Math. Anal. Appl., 265 (2002) 128-134. [19] H. Rosenthal, On the non-norm attaining functionals and the equivalence of the weak∗-KMP with the RNP, Longhorn Notes, 1985-86.

Department of Mathematics and Statistics, Southern Illinois University, Edwardsville, IL 62026-1653, USA E-mail address: [email protected] URL: http://www.siue.edu/~kjarosz/

Indian Statistical Institute, R. V. College Post, Bangalore 560059, India E-mail address: [email protected]