Relatively weakly open subsets of the unit ball in functions spaces
The existence of a denting point in \(B_X\), the unit ball of a Banach space \(X\), implies the existence of relatively weakly open subsets of \(B_X\) of arbitrary small diameter. In particular, this is the situation in spaces with the RNP. The extreme opposite situation is when all relatively weakly open subsets of \(B_X\) have diameter 2. Let us call such spaces diameter 2 spaces. It is known from \textit{O. Nygaard} and \textit{D. Werner} [Arch. Math. 76, 441--444 (2001; Zbl 0994.46006)] that uniform algebras are diameter 2 spaces. Later on the authors, together with A. Rodriguez, proved that \(C^\ast\)-algebras are diameter 2 spaces as well; cf. \textit{J. Becerra Guerrero, G. Lopez Pérez} and \textit{A. Rodriguez} [J. Lond. Math. Soc. 68, 753--761 (2003; Zbl 1060.46007)]. Let \(K\) be infinite, compact Hausdorff, let \(\tau\) be some locally convex Hausdorff topology on the Banach space \(X\), and denote by \(C(K,(X,\tau))\) the vector space of continuous functions from \(K\) into \((X,\tau)\). By Mackey's theorem, we can give \(C(K,(X,\tau))\) the sup norm when \(\tau\) is compatible with the dual pair \((X,X^\ast)\) or, if \(X\) is a dual, with \((X_\ast,X)\), where \(X_\ast\) is some predual of \(X\). The first result is that in these situations, \(C(K,(X,\tau))\) are diameter 2 spaces. Earlier, Rao showed that there are no denting points in \(B_{C(K,(X,\tau))}\) when \(\tau\) is the weak topology on \(X\), cf. \textit{T. S. S. R. K. Rao} [Proc. Am. Math. Soc. 127, 2969--2973 (1999; Zbl 0932.46013)]. Many corollaries follow; we get the information that often operator ideals are diameter 2 spaces. In particular, is this so for \(L(L_1 (\mu))\), \(L(c_0)\) and \(L(\ell_\infty)\). The second part of the paper is devoted to the study of when operator ideals \(A(L_1(\mu),X)\subset L(L_1(\mu),X)\) are diameter 2 spaces. Here \(\mu\) is a positive measure. The result is that \(A(L_1(\mu),X)\) is not a diameter 2 space if and only if both \(L_1(\mu)\) is finite-dimensional and \(X\) is not a diameter 2 space. So \(A(L_1(\mu),X)\) has a clear tendency of being a diameter 2 space. At the end, the situation for \(L_1(\mu,X)\) and \(L_\infty (\mu,X)\) is investigated, \(\mu\) is now finite and positive. The answers are very satisfactory: \(L_\infty (\mu,X)\) fails to be a diameter 2 space exactly when \(L_\infty (\mu)\) is finite-dimensional and \(X\) is not a diameter 2 space. \(L_1 (\mu,X)\) fails to be a diameter 2 space exactly when \(\mu\) contains atoms and \(X\) is not a diameter 2 space.
scientific article; zbMATH DE number 5016147
scientific article