(I)-envelopes of unit balls and James’ characterization of reflexivity. (English) Zbl 1139.46018

Kalenda, Ondřej F. K.

Stud. Math. 182, No. 1, 29-40 (2007).

This article is a continuation of the author’s article [Isr.J.Math.162, 157–181 (2007; Zbl 1149.46018)] where, motivated by the notion of “(I)-generation” due to V.P.Fonf and J.Lindenstrauss [Isr.J.Math.136, 157–172 (2003; Zbl 1046.46014)], the (I)-envelopes of sets in Banach spaces were introduced.
Let \(X\) be a Banach space and \(B\subset X^\ast\). The (I)-envelope of \(B\) is the following intersection
\[ \text{(I)-env}(B)= \bigcap\left\{\overline{\text{conv}\bigcup^\infty_{n=1} \overline{\text{conv}C_n}^{w^\ast}} ^{\|\cdot\|}\;\Biggl|\;B=\bigcup^\infty_{n=1}C_n\right\}. \]
The paper studies (I)-envelopes of \(X\) and its closed unit ball \(B_X\) considered as (canonically embedded) sets in the bidual space \(X^{\ast\ast}\). It is shown that (I)-env\((X)=X^{\ast\ast}\) if and only if \(X\) is a Grothendieck space, and that the only spaces \(X\) for which (I)-env\((B_X)=B_{X^{\ast\ast}}\), for any equivalent norm, are the reflexive ones. For an arbitrary \(\ell_1(\Gamma)\) space, it is proven that (I)-env\((\ell_1(\Gamma))=\ell_1(\Gamma)\) if and only if (I)-env\((B_{\ell_1(\Gamma)})=B_{\ell_1(\Gamma)}\), and it is stated as an open problem whether the same holds true for an arbitrary Banach space.
Using results from [Fonf and Lindenstrauss, loc. cit.] and [M. Morillon, Extr. Math. 20, No. 3, 261–271 (2005; Zbl 1121.46013)], the author also provides a quick proof of James’s famous characterization of reflexivity: \(X\) is reflexive whenever every continuous linear functional on \(X\) attains its norm at some point of \(B_X\). In particular, the proof deals with asymptotically isometric copies of \(\ell_1\) as the recent new proof of James’s theorem in [Morillon, loc. cit.].
The reviewer agrees with the author that his proof is “quite short but uses a large number of nontrivial results” and that “this proof makes the James theorem more accessible”.

Reviewer: Eve Oja (Tartu)

MSC:

46B20	Geometry and structure of normed linear spaces
46A55	Convex sets in topological linear spaces; Choquet theory
46B10	Duality and reflexivity in normed linear and Banach spaces
46B25	Classical Banach spaces in the general theory
46B26	Nonseparable Banach spaces

Keywords:

(I)-envelope; Grothendieck space; James’s characterization of reflexivity; \(\ell_1(\Gamma)\)-spaces; real-valued measurable cardinal

Citations:

Zbl 1046.46014; Zbl 1121.46013; Zbl 1149.46018