Relatively weakly open convex combinations of slices and scattered C$^*$-Algebras

Abstract

We prove that given a locally compact Hausdorff space, $K$, and a compact C$^*$-algebra, $\mathcal{A}$, the C$^*$-algebra $C(K, \mathcal{A})$ satisfies that every convex combination of slices of the closed unit ball is relatively weakly open subset of the closed unit ball if and only if $K$ is scattered and $\mathcal{A}$ is the $c_0$-sum of finite-dimensional C$^*$-algebras. We introduce and study Banach spaces which have property $(\overline{\hbox{P1}})$, i. e. For every convex combination of slices $C$ of the unit ball of a Banach space $X$ and $x\in C$ there exists $W$ relatively weakly open set containing $x$, such that $W\subseteq \overline{C}$. In the setting of general C$^*$-algebras we obtain a characterization of this property. Indeed, a C$^*$-algebra has property $(\overline{\hbox{P1}})$ if and only if is scattered with finite dimensional irreducible representations. Some stability results for Banach spaces satisfying property $(\overline{\hbox{P1}})$ are also given. As a consequence of these results we prove that a real $L_1$-predual Banach space contains no isomorphic copy of $\ell_1$ if and only if it has property $(\overline{\hbox{P1}})$.