Geometric and topological properties of certain \(w^ *\) compact convex subsets of double duals of Banach spaces, which arise from the study of invariant means (Q1084619)

The paper deals with \(w^*\)-compact subsets of the dual of a Banach space X, which are big. A \(w^*\)-compact subset A is called big if there is a surjection from X to \(\ell^{\infty}\) such that the dual map transforms \({\mathcal F}\) into A, when \({\mathcal F}\) consists of the states on \(\ell^{\infty}\), which annihilate \(c_ 0.\) The main theorem describes, when a \(w^*\)-compact subset of a dual space is big. ''It states loosely that if X is a Banach space, k a bounded subset of \(X\subseteq X^{**}\) and S is a countable subset of operators \(s_ n:\) \(X\to X\) and \(A=\{y\in w^*-c\ell (k)|\) \(s_ n^{**}(y)=0\), all \(n\}\) : If some non-void \(G_{\delta}\)-section of \(A_ 0\) of A can be pushed outside the \(w^*\)-sequential closure of k then \(A_ 0\) is big.'' The proof uses H. P. Rosenthal's characterization of Banach spaces containing \(\ell^ 1.\) A big set does not have the WRNP (Weak Radon-Nikodym Property). The main theorem is applied to the study of invariant means.