On weak sequential convergence in JB$^*$-triple duals

Leslie J. Bunce¹ ; Antonio M. Peralta²

¹ University of Reading Reading RG6 2AX, Great Britain
² Departamento de Análisis Matemático Facultad de Ciencias Universidad de Granada 18071 Granada, Spain

Studia Mathematica, Tome 160 (2004) no. 2, pp. 117-127

Résumé

We study various Banach space properties of the dual space $E^*$ of a homogeneous Banach space (alias, a JB$^*$-triple) $E$. For example, if all primitive $M$-ideals of $E$ are maximal, we show that $E^*$ has the Alternative Dunford–Pettis property (respectively, the Kadec–Klee property) if and only if all biholomorphic automorphisms of the open unit ball of $E$ are sequentially weakly continuous (respectively, weakly continuous). Those $E$ for which $E^*$ has the weak$^*$ Kadec–Klee property are characterised by a compactness condition on $E$. Whenever it exists, the predual of $E$ is shown to have the Kadec–Klee property if and only if $E$ is atomic with no infinite spin part.

Détail

DOI : 10.4064/sm160-2-2

Keywords: study various banach space properties dual space * homogeneous banach space alias * triple example primitive m ideals maximal * has alternative dunford pettis property respectively kadec klee property only biholomorphic automorphisms unit ball sequentially weakly continuous respectively weakly continuous those which * has weak * kadec klee property characterised compactness condition whenever exists predual shown have kadec klee property only atomic infinite spin part