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

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”.

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 |