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Norm Attaining Functionals on C(T) PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 1, January 1998, Pages 153{157 S 0002-9939(98)04008-8 NORM ATTAINING FUNCTIONALS ON C(T ) P. S. KENDEROV, W. B. MOORS, AND SCOTT SCIFFER (Communicated by Dale Alspach) Abstract. It is shown that for any infinite compact Hausdorff space T ,the Bishop-Phelps set in C(T )∗ is of the first Baire category when C(T )hasthe supremum norm. For a Banach space X with closed unit ball B(X), the Bishop-Phelps set is the set of all functionals in the dual X∗ which attain their supremum on B(X); that is, the set µ X∗:thereexistsx B(X) with µ(x)= µ .Itiswellknownthat { ∈ ∈ k k} the Bishop-Phelps set is always dense in the dual X∗. It is more difficult, however, to decide whether the Bishop-Phelps set is residual as this would require a concrete characterisation of those linear functionals in X∗ which attain their supremum on B(X). In general it is difficult to characterise such functionals, however it is possible when X = C(T ), for some compact T , with the supremum norm. The aim of this note is to show that in this case the Bishop-Phelps set is a first Baire category subset of C(T )∗. A problem of considerable interest in differentiability theory is the following. If the Bishop-Phelps set is a residual subset of the dual X ∗, is the dual norm necessarily densely Fr´echet differentiable? It is known, for example, that this is the case if the dual X∗ is weak Asplund, [1, Corollary 1.6(i)], or if X has an equivalent weakly mid-point locally uniformly rotund norm and the weak topology on X is σ-fragmented by the norm, [2, Theo- rems 3.3 and 4.4]. It is also known that each dual norm on X∗ is densely Fr´echet differentiable on X∗ provided that the Bishop-Phelps set of each equivalent norm on X is residual in X∗, [6]. No counterexamples are known, and a consequence of our result is that no space of the form X = C(T ) can be a counterexample to the problem above. Let T be an infinite compact Hausdorff space and C(T )bethespaceofall continuous real-valued functions on T with the supremum norm x sup x(t) : t T . We will denote by B the unit ball of C(T ); that is, B kxk≡C(T):{| x | 1∈. } ≡{ ∈ k k≤ }For every pair of real-valued functions on T (not necessarily continuous) we write x y if and only if x(t) y(t) for every t T .Wedenoteby1U the characteristic function≤ of a subset U of≤T ;thatis, ∈ 1ift U; 1U(t)= ∈ 0ift=U: ( ∈ Received by the editors December 18, 1995 and, in revised form, June 24, 1996. 1991 Mathematics Subject Classification. Primary 46E15. c 1998 American Mathematical Society 153 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 154 P. S. KENDEROV, W. B. MOORS, AND SCOTT SCIFFER Given λ C(T )∗ andanopenU T we define ∈ ⊂ λ∗ (U) sup λ(h):h C(T);0 h 1 : + ≡ { ∈ ≤ ≤ U} Clearly λ∗ (U) 0. Similarly we define + ≥ λ∗ (U) sup λ( h):h C(T);0 h 1U − ≡ { − ∈ ≤ ≤ } =sup ( λ)(h):h C(T);0 h 1 {− ∈ ≤ ≤ U} =( λ)∗(U): − + For our first result we will need to consider the following closed sets, λ S t T :λ∗(U)>0 for every open neighbourhood U of t ; + ≡{ ∈ + } λ S t T :λ∗(U)>0 for every open neighbourhood U of t : − ≡{ ∈ − } Theorem 1. A linear functional λ C(T )∗ attains its supremum on B at x0 if and only if ∈ 1 for every t Sλ ; ( )= + x0 t ∈ λ ( 1 for every t S : − ∈ − λ λ That is, λ is in the Bishop-Phelps set if and only if S+ S = ?. ∩ − Proof. Suppose λ attains its supremum over B at x but there exists a t Sλ 0 0 ∈ + such that x0(t0) <r<1. Then there is an open neighbourhood U0 of t0 such that x (t) <r<1 for every t U , by the continuity of x .Sincet Sλwe have 0 ∈ 0 0 0∈ + λ+∗ (U0) > 0. Then there exists some h C(T ), 0 h 1U0 , with λ(h) > 0. Take some 0 <<1 rand consider the function∈ x +≤h.Then≤ − 0 x0(t)+h(t) <r+<1fort U0; (x0+h)(t)= ∈ x(t)fort=U: ( 0 ∈ 0 Remembering h 0 we conclude that x + h B. On the other hand ≥ 0 ∈ λ(x0 + h)=λ(x0)+λ(h) >λ(x0) contradicting the assumption that λ attained its supremum on B at x0. Hence x (t ) = 1 for every t Sλ . 0 0 0 ∈ + Similarly, or by noting that λ∗ =( λ)+∗, we conclude that x0(t0)= 1 for every λ − − − t0 S . ∈Conversely,− suppose 1 for every t Sλ ; ( )= + x0 t ∈ λ ( 1 for every t S ; − ∈ − and y B.Set 0 ∈ (y x)+(t)=max (y x )(t); 0 0− 0 { 0− 0 } and (y x )−(t)=min (y x )(t); 0 ; 0 − 0 { 0− 0 } License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NORM ATTAINING FUNCTIONALS ON C(T ) 155 + + so (y0 x0)=(y0 x0) +(y0 x0)−.ConsiderT= t T:(y0 x0) (t) , a compact− subset− of T disjoint− from Sλ .Foreach{t∈Tthere− is an open≥ } + ∈ neighbourhood U of t for which λ+∗ (U) = 0. These form an open cover of T,andso there is a finite subcover U of T for i =1;2;:::;n, which together with U (T )c i 0 ≡ form an open cover of T .Letpibe a partition of unity subordinate to this cover, [5, p. 171]. Thus n λ((y x )+)=λ p (y x )+ 0 − 0 i 0 − 0 i=0 ! X λ(p (y x )+) ≤ 0 0 − 0 λ p ((y x )+) ≤k kk 0 0 − 0 k λ . ≤k k Similarly λ((y0 x0)−) λ , and so we conclude that λ(y0 x0) 2 λ .Since this is true for− any >≤k0wehavek λ(y) λ(x) for all y −B,so≤ λkattainsk its 0 ≤ 0 0 ∈ supremum over B at x0. We need one more definition to formulate our next proposition. Given λ C(T )∗ and t T , define ∈ ∈ λ r (t) = inf λ∗ (U):t U; U open and + { + ∈ } λ r (t) = inf λ∗ (U):t U; U open : − { − ∈ } λ Theorem 2. For every λ C(T )∗ the sets t T : r+(t) > 0 and t T : λ ∈ λ { ∈ λ} { ∈ r (t) > 0 are countable. Moreover, t T r+(t) < and t T r (t) < . − } ∈ ∞ ∈ − ∞ Proof. Let ti T for i =1;2;:::;n,P be distinct points in TPand let >0. There ∈ λ λ i exist disjoint open sets Ui such that ti Ui.Sinceλ+∗(Ui) r+(ti)>r+(ti) /2 , ∈ ≥ − i there exists some hi C(T ), with 0 hi 1Ui , such that λ(hi) >r+λ(ti) /2 . Then h n h is∈ in B since the open≤ sets≤ U are a disjoint family. Thus− ≡ i=1 i i P n n n 1 λ λ(h)= λ(h ) > rλ (t ) : k k≥ i + i − 2i i=1 i=1 i=1 X X X n λ λ Consequently λ + > i=1 r+(ti), which shows that t T r+(t) < ,andso k k λ∈ ∞ the set t T : rλ(t) > 0 must be countable. That t T r (t) < ,andtheset { ∈ }P P − ∞ t T:rλ(t)>0 is countable, follow similarly. ∈ { ∈ − } P λ λ We note that the points t T where r+(t) > 0(r (t)>0) are points with ∈ − positive (negative) λ-measure. They are related to the \l1-part" of the functional λ,anl1-weighted sum of point evaluations. λ λ Lemma 3. Given λ C(T )∗ and t T where r+(t) > 0(r (t)>0) then there ∈ ∈ µ µ − exists an open neighbourhood A of λ such that r+(t) > 0(r (t)>0) for each µ A. − ∈ λ λ Proof. We consider the case r+(t) > 0. Choose >0 such that r+(t) > 3,and consider any µ A µ C(T)∗ : λ µ < .LetUbe any open neighbourhood ∈ ≡{ ∈ k − k } t,thenλ+∗(U)>3. Then there exists an h B; 0 h 1U, such that λ(h) > 2. Since λ µ <we conclude that µ(h) >∈. Hence≤rµ ≤(t) >0. k − k + ≥ The case rλ (t) > 0 follows similarly. − License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 156 P. S. KENDEROV, W. B. MOORS, AND SCOTT SCIFFER All is in readiness for our main theorem. Theorem 4. For any infinite compact Hausdorff space T , the Bishop-Phelps set is of the first Baire category in C(T )∗. Proof. The proof splits into two cases, T is scattered or not. Case 1. T scattered.InthiscaseTcontains a convergent sequence ti of distinct points with limit point t T , [3, Lemma 5.3]. Consider the sets { } ∞ ∈ µ Gi µ C(T)∗ :forsomej>i;r+(tj)>0 ; ≡{ ∈ µ } Hi µ C(T)∗ :forsomej>i;r (tj)>0 : ≡{ ∈ − } Now these are open by Lemma 3. They are also dense since for any λ C(T )∗ λ λ ∈ and any >0 there exists a j>isuch that r (tj) <(r+(tj)<)andsofor − µ µ µ(x) λ(x)+x(tj )(µ(x) λ(x) x(tj )) we have r+(tj ) > 0(r (tj)>0), and so ≡ ≡ − − λ λ µ Gi (µ Hi).
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