On weak sequential convergence in JB$^*$-triple duals
Tom 160 / 2004
Studia Mathematica 160 (2004), 117-127
MSC: Primary 17C65, 46L05, 46L70.
DOI: 10.4064/sm160-2-2
Streszczenie
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.
