Geometric characterization of Generalized Property (II)

Abstract

Let $\mathcal{A}$ be a compatible collection of bounded subsets of a Banach space $X$. In this paper, we introduce the notion of $\mathcal{A}$-Property (II) and prove that $X$ has $\mathcal{A}$-Property (II) if and only if every $f$ in $S_{X^{\ast }}$ is $\mathcal{A}$-semi PC of $B_{X^{\ast }}$.