Convex combinations of weak*-convergent sequences and the Mackey topology

TL;DR

This paper investigates property (K) in Banach spaces, which concerns the convergence of convex combinations of weak*-convergent sequences under the Mackey topology, and explores its stability under various constructions.

Contribution

It extends known results by proving property (K) is preserved under certain sums and explores its relation to strongly weakly compactly generated spaces.

Findings

Property (K) is preserved by ll^1-sums of fewer than \u2118 summands.

Property (K) is stable under ll^p-sums for 1<p<.

Connections between property (K) and strongly weakly compactly generated spaces are established.

Abstract

A Banach space $X$ is said to have property (K) if every $w^*$-convergent sequence in $X^*$ admits a convex block subsequence which converges with respect to the Mackey topology. We study the connection of this property with strongly weakly compactly generated Banach spaces and its stability under subspaces, quotients and $\ell^p$-sums. We extend a result of Frankiewicz and Plebanek by proving that property (K) is preserved by $\ell^1$-sums of less than $\mathfrak{p}$ summands. Without any cardinality restriction, we show that property (K) is stable under $\ell^p$-sums for $1<p<\infty$.