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GRUENHAGE COMPACTA AND STRICTLY CONVEX DUAL NORMS

arXiv:0710.5396v1 [math.FA] 29 Oct 2007

RICHARD J. SMITH Abstract. We prove that if K is a Gruenhage compact space then

C (K)? admits an equivalent, strictly convex dual norm. As a corollary, we show that if X is a Banach space and X ? = span|||·||| (K), where K is a Gruenhage compact in the w ? -topology and ||| · ||| is equivalent to a coarser, w ? -lower semicontinuous norm on X ? , then X ? admits an equivalent, strictly convex dual norm. We give a partial converse to the ?rst result by showing that if Υ is a tree, then C0 (Υ)? admits an equivalent, strictly convex dual norm if and only if Υ is a Gruenhage space. Finally, we present some stability properties satis?ed by Gruenhage spaces; in particular, Gruenhage spaces are stable under perfect images.

1. Introduction and preliminaries In renorming theory, we determine the extent to which the norm of a given Banach space can be modi?ed, in order to improve the geometry of the corresponding unit ball. Naturally, the structural theory of Banach spaces plays an important part in this ?eld but, in recent times, there has been a move toward a more non-linear, topological approach. This new outlook led to the solution of some long-standing problems, as well as producing some completely unexpected results. Recall that a norm || · || on a real Banach space X is called strictly convex, or rotund, if ||x|| = ||y|| = 1 ||x + y|| implies x = y. We say that || · || is locally 2 uniformly rotund, or LUR, if, given a point x and a sequence (xn ) in the unit sphere SX satisfying ||x + xn || → 2, we have xn → x in norm. If || · || is a dual norm on X ? then || · || is called w? -LUR if, given x and (xn ) as above, we have xn → x in the w? -topology. For a dual norm, evidently LUR ? w? -LUR ? strictly convex. It turns out that, in some contexts, these ostensibly convex, geometrical properties of the norm can be characterised relatively simply in purely non-linear, topological terms. Given a compact, Hausdor? space K, we denote the Banach space of continuous real-valued functions on K by C (K), and identify C (K)? with the space of regular, signed Borel measures on K. Raja proved that if K is a com? pact space then C (K) admits an equivalent, dual LUR norm if and only if K is σ-discrete [8]; that is, K is a countable union of sets, each of which is discrete in ? its subspace topology. Moreover, C (K) admits an equivalent w? -LUR norm if and

Date: October 2007. 2000 Mathematics Subject Classi?cation. Primary 46B03; Secondary 46B26. Key words and phrases. Gruenhage space, rotund, strictly convex, norm, tree, renorming theory, perfect image, continuous image. Some of this research was conducted during a visit to the University of Valencia, Spain. The author is grateful to A. Molt? for the invitation and subsequent discussions. He also wishes to o thank S. Troyanski and V. Montesinos for interesting discussions and remarks.

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RICHARD J. SMITH

only if K is descriptive [9]; the de?nition of a descriptive compact space is given below. Raja also proved that X ? admits an equivalent w? -LUR norm if and only if BX ? is descriptive in the w? -topology. Regarding strictly convex norms, the authors of [6] recently showed that X, which can be a dual space, admits an equivalent, strictly convex, σ(X, N )-lower 2 semicontinuous norm if and only if the square BX has a certain linear, topological decomposition with respect to a given norming subspace N ? X ? . In this paper, we examine what can be done without the linearity, and without explicit reference to the square. Using Gruenhage compacta, we obtain a su?cient condition for a dual space X ? to admit an equivalent, strictly convex dual norm. This condition covers all established classes of Banach space known to be so renormable, including the duals of all weakly countably determined, or Vaˇ?k, spaces. It also covers the sa more general class of ‘descriptively generated’ dual spaces, introduced recently in [7]. We de?ne descriptive compact spaces and related notions. All topological spaces are assumed to be Hausdor?. A family of subsets H of a topological space X is called isolated if, given H ∈ H , there exists an open set U that includes H and misses every other element of H ; i.e. H is discrete in the union H . The family H is called a network for K if, given t ∈ U , where U is open, there exists H ∈ H such that t ∈ H ? U . In other words, a network is a basis, but without the requirement that its elements be open subsets. Finally, we say that a compact space K is descriptive if it has a network H that is σ-isolated ; that is, H = n Hn , where each Hn is a isolated family. The class of descriptive compact spaces is rather large. It includes two classes of topological spaces that have featured prominently in non-separable Banach space theory, namely Eberlein and Gul’ko compacta; see, for example [2]. It also includes all σ-discrete compact spaces; in particular, all compact K such that the Cantor derivative K (ω1 ) is empty, where ω1 is the least uncountable ordinal. More information about descriptive compact spaces can be found in [7]. More generally, we say that a topological space X is fragmentable if there exists a metric d on K, with the property that given ε > 0 and non-empty E ? T , there is an open set U such that U ∩ E is non-empty and the d-diameter of E ∩ U does not exceed ε. General fragmentable compact spaces are not particularly wellbehaved from the point of view of renorming. Indeed, since every scattered space is fragmented by the discrete metric, the compact ω1 +1 is fragmentable, and it is well? known that C (ω1 + 1) does not admit an equivalent, strictly convex dual norm; see, for example [1, Theorem VII.5.2]. On the other hand, if X ? does admit an equivalent, strictly convex dual norm, then BX ? is fragmentable in the w? -topology [11]. The class of Gruenhage compact spaces ?ts between those of descriptive and fragmentable spaces. De?nition 1.1 (Gruenhage [4]). A topological space X is called a Gruenhage space if there exist families (Un )n∈N of open sets such that given distinct x, y ∈ X, there exists n ∈ N and U ∈ Un with two properties: (1) U ∩ {x, y} is a singleton; (2) either x lies in ?nitely many U ′ ∈ Un or y lies in ?nitely many U ′ ∈ Un . If we were to follow Gruenhage’s de?nition to the letter, the sequence (Un ) above would have to cover X as well, but this demand is not necessary as property (1)

GRUENHAGE COMPACTA AND STRICTLY CONVEX DUAL NORMS

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forces the sequence to cover all points of X, with at most one exception. Gruenhage calls such sequences σ-distributively point-?nite T0 -separating covers of X. In the next section, we investigate the role of Gruenhage spaces in renorming theory. In the third section, we give a partial converse to Theorem 2.6, the principal result of the second section, and, by virtue of examples, get some measure of the gap between descriptive compact spaces and Gruenhage compact spaces. The last section is devoted to proving certain stability properties of the class of Gruenhage spaces and its subclass of compact spaces. 2. Gruenhage compacta and renorming We shall say that a family H of subsets of a topological space X separates points if, given distinct x, y ∈ X, there exists H ∈ H such that {x, y} ∩ H is a singleton. It should be noted that some authors demand more of point separation, namely that H can be chosen in such a way that {x, y} ∩ H = {x}. The next proposition brings together some equivalent formulations of Gruenhage’s de?nition that will be of use to us. Proposition 2.1. Let X be a topological space. The following are equivalent. (1) X is a Gruenhage space; (2) there exists a sequence (An ) of closed sets and a sequence (Hn ) of families such that n Hn separates points, and furthermore each element of Hn is an open subset of An and disjoint from every other element of Hn ; (3) there exists a sequence (Un ) of families of open subsets of X and sets Rn , such that n Un separates points and U ∩ V = Rn whenever U, V ∈ Un are distinct. Proof. (1) ? (2) follows directly from [15, Proposition 7.4]. Suppose that (2) holds. To obtain (3), simply de?ne Rn = K\An and set Un = {H ∪ Rn | H ∈ Hn }. Finally, if (3) holds, de?ne Vn = {Rn }. Given distinct x, y ∈ X, there exists n and U ∈ Un such that {x, y} ∩ U is a singleton. Let us assume that x ∈ U . There are two cases. If x ∈ Rn then y ∈ Rn because Rn ? U , thus {x, y} ∩ Rn is a singleton / and, since Vn is a singleton, x is in exactly one element of Vn . Alternatively, we assume that x ∈ Rn . Then x ∈ V ∈ Un forces V = U . Hence x is in exactly one / element of Un . This shows that X is Gruenhage. The second formulation presented in the proposition above prompts the following de?nition. De?nition 2.2. Let X be a topological space. We call (An , Hn ) a legitimate system if An and (Hn ) are as in Proposition 2.1, part (2). We say that H = n Hn is the union of the system. The next result follows easily. Corollary 2.3. A descriptive compact space is Gruenhage. Proof. In [9], Raja shows that if K is a descriptive compact space then there exists a legitimate system (An , Hn ) such that its union H is a network for K. We will spend a little time preparing our legitimate systems for battle. We can and do assume for the rest of this section that every legitimate system (An , Hn ), with union H , satis?es three properties:

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RICHARD J. SMITH

(1) H is closed under the taking of ?nite intersections; (2) K\An ∈ H for all n; (3) An \ Hn ∈ H for all n. Indeed, we ?rst extend the system (An , Hn ) by adding the pairs (K, {K\An }) and (An \ Hn , {An \ Hn }) for every n. We denote the extended system again by (An , Hn ) and then consider, for each non-empty, ?nite F ? N, the pairs (AF , HF ), where AF = n∈F An and HF = {

n∈F Hn

| Hn ∈ HF }.

A family H of pairwise disjoint subsets of K is called scattered if there exists a well-ordering (Hξ )ξ<λ of H such that ξ<α Hξ is open in H for all α < λ. Equivalently, H is scattered if, given non-empty M ? H , there exists H ∈ H such that M ∩ H is non-empty and open in M . Scattered families naturally generalise isolated ones. The following lemma is a simple extension of Rudin’s result that Radon measures on scattered compact spaces are atomic. We can state it in greater generality than required, without compromising the simplicity of the proof. We will say that H ? K is universally Radon measurable (uRm) if, given positive ? ∈ C (K)? , there exist Borel sets E, F such that E ? H ? F and ?(E) = ?(F ); equivalently, H can be measured by the completion of each such ?, which we again denote by ?. Lemma 2.4. If H is a scattered family of uRm subsets of a compact space K then ? H is uRm and ?( H ) = H∈H ?(H) for every positive ? ∈ C (K) . Proof. Take a well-ordering (Hξ )ξ<λ of H and open sets Uα , α < λ, such that Hα ? Uα and Uα ∩ Hβ is empty whenever α < β. We proceed by trans?nite induction on λ; note that by σ-additivity, we can assume that λ is a limit ordinal of uncountable co?nality. Set Dα = Uα \ ξ<α Uξ for α < λ. Given positive ? ? ∈ C (K) , by the uncountable co?nality, let α < λ such that ?(Dβ ) = 0 for α ≤ β < λ. The regularity of ? ensures that ?(D) = 0, where D = α≤β<λ Dβ . By inductive hypothesis, there exist Borel sets E, F such that E ? ξ<α Hξ ? F and ?(E) = ?(F ) = ξ<α ?(H), so the conclusion follows when we consider E and F ∪ D. It is evident that, given a legitimate system (An , Hn ), the family Hn′ = Hn ∪ {K\An , An \ Hn } is scattered and has union K. Moreover, the family Dn = {

i≤n

H1 ∩ . . . ∩ Hn | Hi ∈ Hi′ }

also enjoys these properties. Readers familiar with related literature will recognise that these families lead directly to fragmentability, via Ribarska’s characterisation of fragmentable spaces [10]. Elements of the proof of the following result appear in [15]. We denote both canonical norms on C (K) and C (K)?? by || · ||∞ , and that of ? C (K) by || · ||1 . We will be identifying certain subsets of K with their indicator ?? functions, either in C (K) or C (K) . Lemma 2.5. Let (An , Hn ) be a legitimate system that separates points, with union H and which satis?es properties (1) – (3) above. Then N = span||·||∞ (H ) is a subalgebra of C (K)?? that is 1-norming for C (K)? . Proof. Let Dn be the families introduced above, with union D. As H sepa? rates points, so does D. If ? ∈ C (K) has variation |?| then we have ||?||1 =

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D∈Dn |?|(D) by Lemma 2.4. Thus, given ε > 0, we can take ?nite subsets Fn ? Dn and compact subsets KD ? D, D ∈ Fn , such that n |?|(K\ D∈Fn KD ) < ε. Put M = n D∈Fn KD and MD = M ∩ KD = M ∩ D. If Mn = {MD | D ∈ Fn } then Mn is family of pairwise disjoint sets with union M , and Mn+1 re?nes Mn . Moreover, each MD is clopen in M and, as D separates points of K, so M = n Mn separates points of M . Therefore, by the Stone-Weierstrass Theorem, C (M ) = span||·||∞ (M ). It follows that we can take non-empty, disjoint MDi ∈ M and ai ∈ [1, ?1], i ≤ n, such that |?|(M ) ? i≤n ai ?(MDi ) < ε. Now MDi , MDj = ? and MDi ∩ MDj = ? implies Di ∩Dj = ?. Therefore, i≤n |?|(Di \MDi ) ≤ |?|(K\M ) < ε. We conclude that ||?||1 ? i≤n ai ?(Di ) < 2ε. Since D ? H , we are done.

We say that a norm || · || on X is pointwise uniformly rotund, or p-UR, if there exists a separating subspace F ? X ? such that, given sequences (xn ) and (yn ) satisfying ||xn || = ||yn || = 1 and ||xn + yn || → 2, then f (xn ? yn ) → 0 for all f ∈ F ; see, for example [12]. Evidently, p-UR norms are strictly convex. We can now present the main theorem. Theorem 2.6. If K is a Gruenhage compact then: (1) C (K)? admits an equivalent, strictly convex, dual lattice norm; ? (2) C (K) admits an equivalent, dual p-UR norm. Proof. The lattice norm is constructed ?rst. We take a legitimate system (An , Hn ) ? satisfying the conclusion of Lemma 2.5. For ? ∈ C (K) and m ≥ 1, de?ne the seminorm

?1 ||?||2 n,m = inf{m 2 H∈Hn |λ|(H)

+ ||? ? λ||2 | λ ∈ C (An ) }. 1

?

We observe that ||?||n,m ≤ ||?||1 and that || · ||n,m is w? -lower semicontinuous. We can verify the lower semicontinuity by applying a compactness argument. Alternatively, if we denote the open set ( Hn ) ∪ (K\An ) by U , we observe that ||?||n,m = sup {?(f ) | f ∈ B}, where B = {f ∈ C0 (U ) | m

2 H∈Hn ||f?H ||∞

+ ||f ||2 ≤ 1}. ∞

In this way, we see that || · ||n,m is also a lattice seminorm. ? We de?ne a dual lattice norm on C (K) by setting ||?||2 = ||?||2 + 1

n,m

2?n?m ||?||2 . n,m

Now suppose that ||?|| = ||ν|| = Fact II.2.3]) yields (1)

1 2 ||?

+ ν||. A standard convexity argument (cf. [1,

2||?||2 + 2||ν||2 ? ||? + ν||2 = 0 n,m n,m n,m

for all n and m. By appealing to compactness or the Hahn-Banach Theorem, there ? exist ?n,m , νn,m ∈ C (An ) such that

?1 ||?||2 n,m = m 2 H∈Hn |?n,m |(H)

+ ||? ? ?n,m ||2 1

and likewise for ν. Hence, by applying further standard convexity arguments to equation (1), we obtain (2) 2|?n,m |(H)2 + 2|νn,m |(H)2 ? |?n,m + νn,m |(H)2 = 0

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RICHARD J. SMITH

for all n, m and H ∈ Hn . Now we estimate ||??An ??n,m ||1 = ||? ? ?n,m ||1 ? ||??K\An ||1 ≤ ||?||n,m ? ||??K\An ||1 ≤ [m?1 ≤ m? 2

1

2 H∈Hn |?|(H)

+ ||??K\An ||2 ] 2 ? ||??K\An ||1 1

1

because || · ||1 ≤ || · ||. A similar result holds for ν. Therefore, we conclude from equation (2) that 2|?|(H)2 + 2|ν|(H)2 ? |? + ν|(H)2 = 0 for all H ∈ Hn and n ∈ N. As N from Lemma 2.5 is norming, we certainly 1 obtain |?| = |ν| = 2 |? + ν|. This gives ? = ν by the following lattice argument, included for completeness. If λ = ?+ ? ν? then |?| = |ν| implies λ = ν+ ? ?? , meaning ? + ν = 2λ. Hence ?+ + ?? = |?| = 1 |? + ν| = λ+ + λ? . We see that 2 λ+ = (?+ ? ν? )+ ≤ (?+ )+ = ?+ , hence ?+ = λ+ and ?? = λ? . We conclude that ? = ν as claimed. Now we construct the p-UR norm, using the norming subspace N . First, we claim that || · || above already satis?es the p-UR property if ?k and νk are positive. Suppose that ?k and νk are positive measures such that ||?k || = ||νk || = 1 and ? ||?k + νk || → 2. As above, we can ?nd ?k,n,m , νk,n,m ∈ C (An ) such that

?1 ||?||2 k,n,m = m 2 H∈Hn |?k,n,m |(H)

+ ||?k ? ?k,n,m ||2 1

and likewise for νk . By convexity arguments, we obtain (3) 2|?k,n,m |(H)2 + 2|νk,n,m |(H)2 ? |?k,n,m + νk,n,m |(H)2 → 0

as k → ∞. Moreover, if H ∈ Hn , we estimate |?k ? ?k,n,m |(H) ≤ ||?k?An ??k,n,m ||1 ≤ m? 2 and likewise for νk . Therefore, by ?xing m large enough and appealing to equation (3), we get 2?k (H)2 + 2νk (H)2 ? (?k + νk )(H)2 → 0 whence (?k ? νk )(H) → 0. It follows that ξ(?k ? νk ) → 0 for all ξ ∈ N , thus completing the claim. Now we set |||?|||2 = ||?+ ||2 + ||?? ||2 . To see that this de?nes a dual norm, observe that as || · || is a lattice norm, we have ||?+ || = sup{?(f ) | f ∈ C (K), f ≥ 0 and ||f || ≤ 1} where || · || also denotes the predual norm. Thus ? → ||?+ || is w? -lower semicontinuous, and likewise for ? → ||?? ||. Now, given general ?k and νk satisfying 2|||?k |||2 + 2|||νk |||2 ? |||?k + νk |||2 → 0 we get 2||(?k )+ ||2 + 2||(νk )+ ||2 ? ||(?k )+ + (νk )+ ||2 → 0 and similarly for (?k )? and (νk )? . Therefore, we can apply the claim twice to get (?k ? νk )(H) → 0 for all H ∈ H .

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We apply the theorem above to obtain renorming results for more general Banach spaces. First, we give a modest generalisation of the classic transfer method for LUR renormings, applied to strictly convex renormings; cf. [1, Theorem II.2.1]. A proof is provided for completeness. Proposition 2.7. Let (X, || · ||) , (Y, || · ||) be dual Banach spaces, with (Y, || · ||) strictly convex. Further, let ||| · ||| be a coarser, w? -lower semicontinuous seminorm on X ? , T : X ? ?→ Y ? a bounded, linear operator and set Z = T ? Y ? Then there exists an equivalent dual norm | · | on X, such that whenever f ∈ Z, f ′ ∈ X we have |||f ? f ′ ||| = 0. Proof. De?ne seminorms | · |n on X ? by |f |2 = inf{|||f ? T ? g||| + n?1 ||g|| | g ∈ Y ? } n and set |f |2 = ||f ||2 + n≥1 2?n |f |2 . Since |||·||| is coarser than ||·||, our new norm n | · | is equivalent to || · ||. As in Theorem 2.6, by a w? -compactness argument or the Hahn-Banach Theorem, | · |n is a w? -lower semicontinuous seminorm, and the in?mum in the de?nition is attained. Now let f and f ′ satisfy the above hypothesis. ′ By convexity arguments and in?mum attainment, we can take gn , gn ∈ Y ? such that (4) (5) and

′ 1 ′ ||gn || = ||gn || = 2 ||gn + gn ||. ′ The last equation tells us that gn = gn for all n, meaning that we have ′ |||f ? f ′ ||| ≤ |||f ? T ? gn ||| + |||f ′ ? T ? gn ||| |||·||| ? ? ?

? X ?.

and

1 |f | = |f ′ | = 2 |f + f ′ |

|f |2 = |||f ? T ? gn |||2 + n?1 ||gn ||2 , n

′ |||f ? T ? gn ||| = |||f ′ ? T ? gn |||

Since f ∈ Z, we have |f |n → 0, so by equations (4) and (5), this leads to ′ |||f ′ ? T ? gn ||| = |||f ? T ? gn ||| → 0, giving |||f ? f ′ ||| = 0 as required. Using this, we can obtain our general renorming result. Proposition 2.8. Let (X, || · ||) be a Banach space, F ? X ? a subspace and ||| · ||| ? a coarser norm on X, such that F ∩ (X, ||| · |||) separates points of X. Further, let K ? X be a Gruenhage compact in the σ(X, F )-topology and suppose X = span|||·|||(K). Then: (1) there is a coarser, σ(X, F )-lower semicontinuous, strictly convex norm | · | on X; (2) X admits an equivalent, strictly convex norm. Moreover, if F is a norming subspace then | · | is equivalent to || · ||. Proof. Since F is separating, we can identify ((X, || · ||), σ(X, F )) as a topological ? subspace of ((F, || · ||) , w? ) by standard evaluation and consider K as a w? -compact subset of F ? . Now elements of F act as continuous functions on (K, w? ) and the ? map S : C (K) ?→ F ? , given by (S?)(f ) = K f d?, is a dual operator. Let ? ||| · ||| also denote the canonical norm on G = (X, ||| · |||) , and de?ne the w? -lower semicontinuous seminorm |||ξ|||1 = sup{ξ(f ) | f ∈ F and |||f ||| ≤ 1}

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RICHARD J. SMITH

on F ? . By Proposition 2.7, there exists an equivalent, dual norm | · |1 on F ? , such that if ξ ∈ SC (K)

? |||·|||1

, ξ′ ∈ F ?

and |ξ|1 = |ξ ′ |1 =

1 2 |ξ

+ ξ ′ |1

then |||ξ ? ξ ′ |||1 = 0. Let | · | be the restriction of | · |1 to X and note that | · | is both σ(X, F )-lower semicontinuous and coarser than ||·||. Moreover, X = span|||·|||(K) ? SC (K) . 1 Therefore, whenever |x| = |x′ | = 2 |x + x′ |, we have |||x ? x′ |||1 = 0. Since F ∩ G separates points of X, it follows that x′ = x. This gives (1). For (2), observe that the sum || · || + | · | is an equivalent, strictly convex norm on X. Finally, if F is norming then | · | is equivalent to || · ||. Let us assume that the coarser norm ||| · ||| of Proposition 2.8 is σ(X, F )-lower semicontinuous. By a standard polar argument |||x||| = sup{f (x) | f ∈ F, |||f ||| ≤ 1} and, in particular, F ∩ (X, ||| · |||) separates points of X. Corollary 2.9. Let X be a Banach space and X ? = span|||·|||(K), where K is a Gruenhage compact in the w? -topology and ||| · ||| is equivalent to a coarser, w? lower semicontinuous norm on X ? . Then X ? admits an equivalent, strictly convex dual norm. The result above applies to all established classes of Banach spaces known to admit equivalent strictly convex dual norms on their dual spaces; for example, Vaˇ?k spaces. We move on to discuss a property of Banach spaces, introduced in sa [3] and shown there to be a su?cient condition for the existence of an equivalent, strictly convex dual norm. De?nition 2.10 ([3]). We say that the Banach space X has property G if there exists a bounded set Γ = n∈N Γn ? X, with the property that whenever f, g ∈ BX ? are distinct, there exist n ∈ N and γ ∈ Γn such that (f ? g)(γ) = 0 and, either |f (γ ′ )| > 1 |(f ? g)(γ)| for ?nitely many γ ′ ∈ Γn , or |g(γ ′ )| > 1 |(f ? g)(γ)| 4 4 for ?nitely many γ ′ ∈ Γn . As well as showing that all Vaˇ?k spaces possess property G, the authors of [3] sa remark that the property is closely related to Gruenhage compacta. Proposition 2.11. If X has property G then the dual unit ball BX ? is a Gruenhage compact in the w? -topology. Proof. We can and do assume that Γ is a subset of the unit ball BX . Given γ ∈ Γ and q ∈ (0, 1)∩Q, we let U (γ, q) = {f ∈ BX ? | f (γ) > q}. We prove that, together, (Un,q ) and (Vn,q ), n ∈ N and q ∈ (0, 1) ∩ Q, satisfy (1) and (2) of De?nition 1.1, where Un,q = {U (γ, q) | γ ∈ Γn } and Vn,q = {?U (γ, q) | γ ∈ Γn }. Given distinct 1 f, g ∈ BX ? , take γ ∈ Γn with the property that α = 4 |(f ? g)(γ)| > 0. It follows that either |f (γ)| > α or |g(γ)| > α; without loss of generality, we assume that the former inequality holds. Now suppose that f (γ) > 0. We choose rational q to satisfy f (γ) > q > max{g(γ), α} if f (γ) > g(γ), or g(γ) > q > f (γ) otherwise. Either way, U (γ, q) ∩ {f, g} is a singleton, giving (1). Since q > α, (2) follows. If f (γ) < 0, we repeat the above argument with ?f and ?g.

? ? |||·|||1

GRUENHAGE COMPACTA AND STRICTLY CONVEX DUAL NORMS

9

Corollary 2.12 ([3]). If X has property G then X ? admits an equivalent, strictly convex dual norm. Proof. Combine Proposition 2.11 and Corollary 2.9. We ?nish this section with an open problem. Problem 2.13. If C (K)? admits a strictly convex dual norm then is K Gruenhage? More ambitiously, if X ? is a dual Banach space with strictly convex dual norm, is BX ? Gruenhage? 3. A topological characterisation of Y -embeddable trees In this section, we present a partial converse to Theorem 2.6. We call a partially ordered set (Υ, ) a tree if, for each t ∈ Υ, the set (0, t] = {s ∈ Υ | s t} of predecessors of t is well-ordered. Given t ∈ Υ, we denote by t+ the set of immediate successors of t in Υ; that is, u ∈ t+ if and only if t ? u and t ? ξ ? u for no ξ. The locally compact, scattered order topology on Υ takes as a basis the sets (s, t], s ? t, where (s, t] = (0, t]\(0, s]. To ensure that this topology is also Hausdor?, we demand that every non-empty, totally ordered subset of Υ has at most one minimal upper bound; trees satisfying this property are themselves called Hausdor?. We study the space C0 (Υ) of continuous, real-valued functions on Υ that vanish at in?nity, and the dual space of measures. To date, most of the results about renorming C0 (Υ) and its dual have been order-theoretic in character: [5], [13] and [14]. Such order-theoretic results, while well-suited in this context, are deeply bound to the tree-structure and, as such, do not o?er obvious generalisations. Here, we are able to give a purely topological characterisation of trees Υ, such that C0 (Υ)? admits an equivalent, strictly convex dual norm. The following de?nition ?rst appears in [13]. De?nition 3.1. Let Y be the set of all strictly increasing, continuous, trans?nite sequences x = (xα )α≤β of real numbers, where 0 ≤ β < ω1 . We order Y by declaring that x < y if and only if either y strictly extends x, or if there is some ordinal α such that xξ = yξ for ξ < α and yα < xα . We say that a map ρ : Υ ?→ Σ from a tree to a linear order is increasing if ρ(s) ≤ ρ(t) whenever s ? t, and strictly so if the former inequality is always strict. The next theorem is the key result of this section. Theorem 3.2. If Υ is a tree and ρ : Υ ?→ Y is a strictly increasing function then Υ is a Gruenhage space. Theorem 2.6 and Theorem 3.2, together with [13, Proposition 7] and a result from [14], allows us to present the following series of equivalent conditions and, in particular, provides our partial converse to Theorem 2.6. Observe that a locally compact space is Gruenhage if and only if its 1-point compacti?cation is. Corollary 3.3. If Υ is a tree then the following are equivalent: ? (1) C0 (Υ) admits an equivalent, dual p-UR norm; ? (2) C0 (Υ) admits an equivalent, strictly convex dual lattice norm; (3) C0 (Υ) admits an equivalent, G?teaux smooth lattice norm; a ? (4) C0 (Υ) admits an equivalent, strictly convex dual norm; (5) there is a strictly increasing function ρ : Υ ?→ Y ;

10

RICHARD J. SMITH

(6) Υ is a Gruenhage space. It is proved in [13] that the 1-point compacti?cation of a tree Υ is descriptive, equivalently σ-discrete, if and only if there is a strictly increasing function ρ : Υ ?→ Q. As trees go, those that admit such Q-valued functions are relatively simple. The order Y is considerably larger than Q in order-theoretic terms; indeed, given any ordinal β < ω1 , the lexicographic product Rβ embeds into Y . Accordingly, there is an abundance of trees that admit strictly increasing Y -valued maps, but not strictly increasing Q-valued maps [13]. Therefore, the class of Gruenhage compact spaces encompasses appreciably more structure than the class of descriptive compact spaces. A little preparatory work must be presented before giving the proof of Theorem 3.2. We recall some material from [13]. De?nition 3.4 ([13]). A subset V ? Υ is called a plateau if V has a least element 0V and V = t∈V [0V , t]. A partition P of Υ consisting solely of plateaux is called a plateau partition. If V is a plateau then V \{0V } is open, so given a plateau partition P of Υ, the set H = {0V | V ∈ P} of least elements of V is closed in Υ. De?nition 3.5 ([13]). Given a tree Υ, let (Pβ )β<ω1 be a sequence of plateau partitions with the following properties: (1) if α < β and V ∈ Pα , W ∈ Pβ , then either W ? V or V ∩ W is empty; (2) if β is a limit ordinal and W ∈ Pβ , then W = {V | V ∈ Pα , α < β, W ? V };

(3) if t ∈ Υ, there exists β < ω1 , depending on t, such that {t} ∈ Pβ . We call such a sequence of plateau partitions admissible. De?nition 3.6 ([13]). Let (Pβ )β<ω1 be admissible and let T be the tree {(α, V ) | V ∈ Pα , α < ω1 } with order (α, V ) ? (β, W ) if and only if α ≤ β and W ? V . Then the subtree Υ(P) = {(β, V ) ∈ T | U is not a singleton whenever (α, U ) ? (β, V )} of T is called the partition tree of Υ with respect to (Pβ )β<ω1 . It is evident that if V is a plateau then so is V , with 0V = 0V . A subset of a tree Υ is called an antichain if it consists solely of pairwise incomparable elements. With respect to the interval topology, antichains are discrete subsets. We make the following elementary, yet important, observation. Lemma 3.7. Let E be an antichain in a partition tree Υ(P). If (α, V ) and (β, W ) are distinct elements of E then both intersections V ∩ W and V \{0V } ∩ W \{0W } are empty. Proof. We can assume that α ≤ β. That the ?rst intersection is empty follows directly from the de?nition of the partition tree order. To see that the same is true for the second, note that if (α, U ) (β, W ) then W \{0W } ? U \{0U }, so all we need to do is prove that if t ∈ V \{0V } ∩ U \{0U } then V and U intersect nontrivially and are thus equal. Given such t, we have that 0V and 0U are comparable. If 0V 0U then since there exists s ∈ (0U , t] ∩ V , we have 0U ∈ V as V is a plateau. Likewise, if 0U 0V then 0V ∈ U .

GRUENHAGE COMPACTA AND STRICTLY CONVEX DUAL NORMS

11

The next result shows that if there is a strictly increasing function ρ : Υ ?→ Y then Υ admits a partition tree Υ(P), on which may be de?ned a strictly increasing, real-valued function. It is important to note that the order of the partition tree is related to the order of Υ through the second, albeit technical, property below. If t ∈ Υ then the wedge [t, ∞) is the set {u ∈ Υ | u t}. Proposition 3.8 ([13]). Let Υ be a tree. If ρ : Υ ?→ Y is strictly increasing then there exists an admissible sequence of partitions (Pβ )β<ω1 that yields a partition tree Υ(P), and a strictly increasing function π : Υ(P) ?→ [0, 1]. Moreover: (1) P0 = {[r, ∞) | r ∈ Υ is minimal}; (2) for any non-maximal (β, V ) ∈ Υ(P), the map 0W ?→ π(β + 1, W ) is strictly decreasing on the subtree of least elements H(β,V ) = {0W | (β + 1, W ) ∈ (β, V )+ }. In the proof below, we will assume the partition tree Υ(P) and function π from Proposition 3.8. Proof of Theorem 3.2. We construct a legitimate system on Υ. As Υ(P) admits a strictly increasing, real-valued function π, its isolated elements may be decomposed into a countable union of antichains (Fn ). Indeed, if (β, W ) ∈ Υ(P) is isolated and non-minimal, then it has an immediate predecessor (α, V ), and we can pick τ (β, W ) ∈ Q∩(π(α, V ), π(β, W )). Then consider the antichain of minimal elements, together with the ?bres (τ ?1 (q))q∈Q . If V is a plateau then V \V is an antichain and hence discrete. Note that here, closure is taken with respect to Υ. From Lemma 3.7, the family {V \{0V } | (β, V ) ∈ Fn } is a pairwise disjoint collection of open sets in Υ. Hence Dn = {V \V | (β, V ) ∈ Fn } is discrete. Given q ∈ Q, consider the set Eq of successor elements (β +1, W ) ∈ (β, V )+ , with (β, V ) ∈ Υ(P) arbitrary, such that π(β, V ) < q < π(β + 1, W ). Observe that Eq is an antichain in Υ(P). Indeed, if (α + 1, U ) ? (β + 1, W ) and (β + 1, W ) ∈ Eq then (α + 1, U ) (β, V ) ? (β + 1, W ), thus π(α + 1, U ) ≤ π(β, V ) < q. It follows that (α+1, U ) ∈ Eq . Given non-maximal (β, V ) ∈ Υ(P), property (2) of Proposition 3.8 / tells us that, in particular, the set of relatively isolated points in the least elements H(β,V ) can be decomposed into a countable union of antichains (F(β,V ),m ) in Υ. Given (β + 1, W ) ∈ (β, V )+ such that 0W ∈ F(β,V ),m , set Eq,(β,V ),W = {(β + 1, W ′ ) ∈ Eq ∩ (β, V )+ | 0W and Eq,m = {Eq,(β,V ),W | (β + 1, W ) ∈ (β, V )+ and 0W ∈ F(β,V ),m }. We observe that each Eq,m is a family of disjoint subsets of Eq . Indeed, let Eq,(β,V ),W , Eq,(β ′ ,V ′ ),W ′ ∈ Eq,m . If (β, V ) = (β ′ , V ′ ) then (β, V )+ ∩ (β ′ , V ′ )+ is empty and we are done, so we assume that this is not the case. If W = W ′ then 0W and 0W ′ are incomparable in Υ, so Eq,(β,V ),W and Eq,(β ′ ,V ′ ),W ′ must be disjoint. By Lemma 3.7, it follows that the sets Jq,(β,V ),W = {W ′ | (β + 1, W ′ ) ∈ Eq,(β,V ),W }, 0W ′ }

Eq,(β,V ),W ∈ Eq,m , are also pairwise disjoint. We prove that J = Jq,(β,V ),W is a plateau. Evidently 0W is the least element of J. Now suppose t ∈ J and 0W s t. We have to show that s ∈ J. As

12

RICHARD J. SMITH

0W , t ∈ V and V is a plateau, s ∈ V and so there exists (β + 1, W ′ ) ∈ (β, V )+ such that s ∈ W ′ . We know that t ∈ W ′′ , where (β + 1, W ′′ ) ∈ Eq ∩ (β, V )+ and 0W ′′ and, by condition (2) of Proposition 0W ′ 0W 0′′ . Thus we have 0W W 3.8, π(β + 1, W ′ ) ≥ π(β + 1, W ′′ ) > q. It follows that (β + 1, W ′ ) ∈ Eq and s ∈ J. At last, we have enough information to de?ne our legitimate system. Begin by setting A = Υ and H = {{t} | t ∈ Υ is isolated}. Then de?ne An = Dn and Hn = {{t} | t ∈ Dn }. Again using Lemma 3.7, we are permitted to de?ne A′ = Υ n and Hn′ = {V \{0V } | (β, V ) ∈ Fn }. From the above discussion, given q ∈ Q and m ∈ N, we can de?ne Aq,m = Υ and Hq,m = {Jq,(β,V ),W \{0W } | Eq,(β,V ),W ∈ Eq,m }. We claim that, together, the families H , Hn , Hn′ and Hq,m separate points of Υ in the manner of Proposition 2.1, part (2). Let s, t be distinct elements of Υ. If s or t is an isolated point of Υ, we can separate using H . Henceforth, we will s assume that both s and t are limit elements of Υ. Let Vβ be the unique element of β containing s, and likewise for t. Let γ < ω1 be minimal, subject to the condition s that Vγ = Vγt . Such γ exists by property (3) of De?nition 3.5. By property (2) s of De?nition 3.5, γ cannot be a limit ordinal. If γ = 0 then V = Vγ = [r, ∞) by property (1) of Proposition 3.8. Being minimal in Υ, r is isolated, so s ∈ V \{0V }. As (0, V ) is minimal in Υ(P), it is an element of Fn for some n. Consequently, we can separate s from t using Hn′ . s We ?nish by tackling the case where γ = β +1 for some ordinal β. Let W = Vβ+1 ′ t and W = Vβ+1 . If s ∈ W \{0W } then as (β + 1, W ) is isolated in Υ(P), we can separate using some Hn′ as above. We can argue similarly if t ∈ W ′ \{0W ′ } so, from now on, we assume that s = 0W and t = 0W ′ , i.e. s, t ∈ H(β,V ) . If 0W is an immediate successor with respect to H(β,V ) , i.e. if there exists 0U ∈ H(β,V ) such that 0U ? 0W and no element of H(β,V ) lies strictly between the two, then 0W ∈ U \U . Indeed, if r ? 0W then as 0W is a limit in Υ, there exists ξ ∈ (max{r, 0U }, 0W ]\{0W }. Now ξ must lie in U because 0U is the immediate predecessor of 0W in H(β,V ) . It follows that 0W ∈ U as required. Now (β + 1, U ) is in Fn for some n, so {0W } ∈ Hn , thus separating 0W from 0W ′ . As above, we can argue similarly if 0W ′ is an immediate successor with respect to H(β,V ) , so now we assume that neither 0W nor 0W ′ are such elements. As H(β,V ) has a least element and is a Hausdor? tree in its own right, the greatest element less than both 0W and 0W ′ is some 0U ∈ H(β,V ) and, without loss of generality, we can assume that 0U ? 0W . If 0U ′ is the immediate successor of 0U in H(β,V ) then 0U ′ ? 0W , because 0W is not such an element. Consequently, 0U ′ ∈ F(β,V ),m for some m so, given rational q strictly between π(β, V ) and π(β + 1, W ), we have 0W ∈ J\{0U ′ }, where J = Jq,(β,V ),U ′ . Since 0U ′ 0W ′ by maximality of 0U , it follows that J\{0U ′ } separates 0W from 0W ′ . 4. Stability properties of Gruenhage spaces Our ?rst stability property is purely topological. Theorem 4.1. If X is a Gruenhage space and f : X ?→ Y is a perfect, surjective mapping, then Y is also Gruenhage. Proof. Let X be a Gruenhage space and assume that we have families (Un ) and sets Rn satisfying Proposition 2.1 (3). By adding new families { Un } if necessary,

GRUENHAGE COMPACTA AND STRICTLY CONVEX DUAL NORMS

13

we assume that given n, there exists m such that Rm = Un . If G ? N is ?nite, de?ne VG = { i∈G Ui | (Ui )i∈G ∈ i∈G Ui }. Given a perfect, surjective map f : X ?→ Y , we set VF,G,k = {Y \f (X\(

i∈F

Ri ∪

F )) | F ? VG and card F = k}

for ?nite F, G ? N and k ∈ N. Since f is perfect, every element of VF,G,k is open in Y . Let y, z ∈ Y be distinct. We show that there exists ?nite F, G ? N, k ∈ N and F ? VG with cardinality k, such that {y, z} ∩ Y \f (X\(

i∈F

Ri ∪

F ))

is a singleton. Moreover, if G ? VG has cardinality k and y ∈ Y \f (X\(

i∈F

Ri ∪

F ))

is non-empty, then G = F . From this, it follows immediately that Y is Gruenhage. To prove this claim, we ?rst construct a pair of decreasing sequences of compact sets. Set A0 = f ?1 (y) and B0 = f ?1 (z). Given r ≥ 0, if, for all n, it is true that (Ar ∪Br )∩Rn = ? or Ar ∩Br ? Rn , then we stop. If not then let nr+1 be minimal, subject to the requirement that (Ar ∪ Br ) ∩ Rnr+1 = ? and (Ar ∪ Br )\Rnr+1 = ?. Put Ar+1 = Ar \Rnr+1 and Br \Rnr+1 . Continuing in this way, either we stop at a ?nite stage or continue inde?nitely. If the process stops at a ?nite stage r ≥ 0, set A = Ar and B = Br . Evidently (A∪B)∩Rn = ? or A∪B ? Rn for all n. If the process above continues inde?nitely, then we obtain a sequence n1 < n2 < . . . and decreasing sequences (Ai ), (Bi ) of ∞ ∞ non-empty, compact sets. Put A = i=0 Ai and B = i=0 Bi . Then, given any n, again we have (A ∪ B) ∩ Rn = ? or A ∪ B ? Rn , lest we violate the minimality of the ?rst ni > n. If A = ? then by surjectivity, and compactness if necessary, there is some r ≥ 1 such that Ar = ?. Since (Ar?1 ∪ Br?1 )\Rnr = ? by construction, it is not the r r case that Br is empty, thus f ?1 (y) ? i=1 Rni and f ?1 (z) ? i=1 Rni . Put F = {n1 , . . . , nr } and let G be arbitrary. Then Y \f (X\ i∈F Ri ) is the only element of VF,G,0 and {y, z} ∩ Y \f (X\

i∈F

Ri ) = {y}.

If B = ? then we proceed similarly. Now suppose that A = ? and B = ?. De?ne K = A ∪ B and let I = {n ∈ N | K ∩ Rn = ? and K ? Un }.

We have K = {K ∩ U | U ∈ Un } whenever n ∈ I. Moreover, the sets in each {K ∩ U | U ∈ Un }, n ∈ I, are pairwise disjoint. In fact slightly more can be said; if x ∈ K ∩ U ∩ V for U, V ∈ Un and n ∈ I then U = V . Indeed, if x ∈ K ∩ U ∩ V , U, V ∈ Un and U = V then x ∈ Rn , so n ∈ I. Given distinct a, b ∈ K, there exists n and U ∈ Un such that {a, b} ∩ U is a singleton. Firstly, this means n ∈ I. Indeed, if K ∩ Rn = ? then K ? Rn , meaning a, b ∈ U , which is not the case. Now suppose K ? Un . We have Un = Rm for some m, so K ∩ Un = K ∩ Rm is empty, which again is not the case. Thus n ∈ I. In particular, this means we can assume that {a, b} ∩ U = {a} because

14

RICHARD J. SMITH

{K ∩ V | V ∈ Un } partitions K. By compactness, it follows that there is ?nite G ? I and ?nite E ? i∈G Ui such that A ? {K ∩ U | U ∈ E } and B ? {K ∩ U | U ∈ E }.

Let x ∈ K ∩ U , where U ∈ E . For every i ∈ G, we know from above that there is a unique Ui ∈ Ui such that x ∈ Ui . By de?nition i∈G Ui ∈ VG , and since U ∈ Uj for some j ∈ G, we have U = Uj and x ∈ i∈G Ui ? U . This allows us to take a ?nite subset F ? VG , such that A ? {K ∩ V | V ∈ F } and B ? {K ∩ V | V ∈ F }.

We choose F so that it has minimal cardinality k. If necessary, we appeal to compactness to ?nd r ≥ 0 satisfying

r

f

r

?1

(y) ?

i=1

Rni ∪

r

F,

A ? f ?1 (y)\ i=1 Rni and B ? f ?1 (z)\ i=1 Rni . Let F = {n1 , . . . , nr }. Observe that if G ? VG and f ?1 (y) ? i∈F Ri ∪ G then A ? G , and likewise for f ?1 (z) and B. Thus f ?1 (z) ? i∈F Ri ∪ F and consequently {y, z} ∩ Y \f (X\(

i∈F

Ri ∪

F )) = {y}.

Now let y ∈ Y \f (X\( i∈F Ri ∪ G )), where G ? VG has cardinality k. It follows that A ? G . We show that G = F . Take W ∈ F . By minimality of k A ? {K ∩ V | V ∈ F \{W }}

thus there is x ∈ A ∩ W . Take V ∈ G such that x ∈ V . We claim that W = V . Indeed, W = i∈G Wi and V = i∈G Vi for some Wi , Vi ∈ Ui , i ∈ G. Since G ? I and x ∈ K ∩ Wi ∩ Vi , we have Wi = Vi for all i ∈ G, hence W = V ∈ G . Therefore F ? G and, by cardinality, we have equality as required. Next, something of a more functional analytic nature. Proposition 4.2. If K is a Gruenhage compact then so is BC (K)? . Proof. Let (An , Hn ) be a legitimate system satisfying properties (1) – (3), presented after Corollary 2.3. We can and do assume that ? ∈ Hn for all n. Given H ∈ Hn and q ∈ (0, 1) ∩ Q, de?ne the w? -open set U+

(n,q) (H,n,q)

= {? ∈ BC (K)? | ?+ (H ∪ (K\An )) > q}

(H,n,q) (n,q)

and U? in the corre| H ∈ Hn }. De?ne U? and let U+ = {U+ (n,q) (n,q) sponding manner. We claim that, with respect to U+ and U? , n ∈ N and q ∈ (0, 1) ∩ Q, BC (K)? is a Gruenhage compact in the sense of De?nition 1.1. Let ?, ν ∈ BC (K)? be distinct. Either ?+ = ν+ or ?? = ν? . We suppose that the former holds; if the latter holds then we repeat the argument below using the sets (H,n,q) (n,q) U? and U? . By Lemma 2.5, there exists n ∈ N and H0 ∈ Hn such that ?+ (H0 ) = ν+ (H0 ). If ?+ (K\An ) = ν+ (K\An ) then set H = ?. Otherwise, set H = H0 . Either way, we have ?+ (H ∪ (K\An )) = ν+ (H ∪ (K\An )) and, without loss of generality, we suppose that ?+ (H ∪ (K\An )) < q < ν+ (H ∪ (K\An )) for (H,n,q) (H ′ ,n,q) some rational q. Then {?, ν} ∩ U+ = {ν}. Moreover, if ? ∈ U+ for some H ′ ∈ Hn then ?+ (H ′ ) = ?+ (H ′ ∪(K\An ))??+ (K\An ) > q??+ (H ∪(K\An )) > 0.

(H,n,q)

GRUENHAGE COMPACTA AND STRICTLY CONVEX DUAL NORMS

15

Hence, as each Hn is a family of pairwise disjoint sets, ? can only be in ?nitely (n,q) many elements of U+ . We ?nish by using these two results to glean a further crop of stability properties. Proposition 4.3. (1) If K is a Gruenhage compact and π : K ?→ M is continuous and surjective then M is also Gruenhage; (2) if Xn , n ∈ N are Gruenhage spaces then so is n Xn ; (3) if X is a Banach space, F ? X ? is a separating subspace and K ? X is a Gruenhage compact in the σ(X, F )-topology then so is its symmetric, σ(X, F )-closed convex hull. Proof. (1) follows immediately from Theorem 4.1. To prove (2), we let Xn have a sequence (Un,m )m∈N of families of open sets satisfying De?nition 1.1. It is straightforward to verify that the families (Vn,m ), de?ned by Vn,m = {

i<n

Xi × U ×

i>n

Xi | U ∈ Un,m }

are witness to the fact that n∈N Xn is Gruenhage. To see that (3) holds, consider, as in Proposition 2.8, K as a subset of F ? and the map S restricted to BC (K)? , which is Gruenhage by Proposition 4.2. By (1), SBC (K)? ? F ? is a Gruenhage compact in the w? -topology, giving (3). References

1. R. Deville, G. Godefroy and V. Zizler, Smoothness and Renormings in Banach Spaces. Longman, Harlow, 1993 2. M. Fabian, G?teaux Di?erentiability of Convex Functions and Topology. John Wiley and a Sons, Inc., New York, 1997. 3. M. Fabian, V. Montesinos and V. Zizler, The Day norm and Gruenhage compacta. Bull. Austral. Math. Soc. 69 (2004), 451–456. 4. G. Gruenhage, A note on Gul’ko compact spaces. Proc. Amer. Math. Soc. 100 (1987), 371– 376. 5. R. G. Haydon, Trees in renorming theory. Proc. London Math. Soc. 78 (1999), 541–584. 6. A. Molt?, J. Orihuela, S. Troyanski and V. Zizler Strictly convex renormings. J. London Math. o Soc. 75 (2007), 647–658. 7. L. Oncina, M. Raja Descriptive compact spaces and renorming. Studia Math. 165 (2004), 39–52. 8. M. Raja, On dual locally uniformly rotund norms. Israel J. Math. 129 (2002), 77–91. 9. M. Raja, Weak? locally uniformly rotund norms and descriptive compact spaces. J. Funct. Anal. 197 (2003), 1–13. 10. N. K. Ribarska, Internal characterization of fragmentable spaces. Mathematika 34 (1987), 243–257. 11. N. K. Ribarska The dual of a G?teaux smooth space is weak star fragmentable. Proc. Amer. a Math. Soc. 114 (1992), 1003–1008. 12. J. Rycht?ˇ, Pointwise uniformly rotund norms. Proc. Amer. Math. Soc. 133 (2005) 2259– ar 2266. 13. R. J. Smith, On trees and dual rotund norms. J. Funct. Anal. 231 (2006), 177–194. 14. R. J. Smith, Trees, linear orders and G?teaux smooth norms. To appear in J. London Math. a Soc. 15. C. Stegall, The topology of certain spaces of measures Topology Appl. 41 (1991), 73–112. Queens’ College, Cambridge, CB3 9ET, United Kingdom E-mail address: rjs209@cam.ac.uk

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