Geometrical implications of upper semi-continuity of the duality mapping on a Banach space

Big points in C*-algebras and JB*-triples

“…Most results in this section are inspired by the Giles-Gregory-Sims paper [34]. In particular, the proof of Proposition 2.1 immediately below follows the lines of that of [34,Theorem 3.3]. There it is shown that, if X is a Banach space, and if there is 0 < k , 1 such that s k ðXÞ ¼ S X ; then X is nicely smooth and Asplund.…”

“…For instance, a convex transitive Banach space X lies in J if (and only if) s 0 (X) has non-empty interior relative to S X , and this is the case if (and only if) s 0 ðXÞ ¼ S X (that is, DðX; uÞ ¼ DðX ** ; uÞ for every norm-one element u of X). In fact, the last characterization follows straightforwardly from [34,6]. Note that non-reflexive Banach spaces X satisfying s 0 ðXÞ ¼ S X do exist.…”

“…Concerning nicely smooth Banach spaces, the reader is referred to [35,3]. Most results in this section are inspired by the Giles-Gregory-Sims paper [34]. In particular, the proof of Proposition 2.1 immediately below follows the lines of that of [34,Theorem 3.3].…”

“…Following [34], we say that the duality mapping of X is upper semicontinuous ðn -tÞ at u if for every t-neighbourhood of zero (say B) in X * there exists a norm-neighbourhood of u (say C) in S X such that DðX; xÞ # DðX; uÞ þ B whenever x belongs to C. Denote by w and n the weak and norm topologies on X * , respectively. Corollary 4.4 of [36], applied in the proof of Lemma 2.4, asserts that the strong subdifferentiability of the norm of X at u is equivalent to the upper semicontinuity ðn -nÞ of the duality mapping of X at u, and hence implies the upper semicontinuity ðn -wÞ of the duality mapping of X at u.…”

“…Y * stands for the Dixmier projection. The constant s ðX; uÞ is implicitly considered in [34], where it is shown that if X is a Banach space, and if there is 0 < k , 1 such that s ðX; uÞ < k for every u [ S X ; then X is Asplund and no proper closed subspace of X * is norming for X. In Theorem 2.5 we shall prove that, if X is a Banach space, and if there is a big point u of X with s ðX; uÞ , 1 and such that the norm of X is strongly subdifferentiable at u, then X is Asplund and no proper closed subspace of X * is norming for X.…”