Isometry games in Banach spaces (Q953988)

An old problem of Banach and Mazur asks whether every separable transitive Banach space necessarily is a Hilbert space. A Banach space \(X\) is said to be transitive if given two arbitrary norm-one elements \(a, b \in X\), there exists an isometric isomorphism of \(X\) sending \(a\) to \(b\). There exist non-separable transitive Banach spaces which are not Hilbert spaces; see \textit{P.\,Greim}, \textit{J.\,Jamison} and \textit{A.\,Kamińska} [Math.\ Proc.\ Camb.\ Philos.\ Soc.\ 116, No.\,3, 475--488 (1994); corrigendum ibid.\ 121, 191 (1997; Zbl 0835.46019)]. In the present paper, the authors introduce property \(WS\): a Banach space \(X\) has property \(WS\) if the second player has a winning strategy in a game consisting in constructing some isometric isomorphisms of \(X\). For separable Banach spaces, this property is (trivially) equivalent to transitivity. However, they give an example of a Banach space (a sum of \(L_p\) spaces, \(1 \leq p < \infty\), \(p \not= 2\)) which has \(WS\) but is not transitive. Propery \(WS\) is connected to Keisler's Back-and-Forth property, previously studied by the authors [Ann.\ Pure Appl.\ Logic 111, No.\,1--2, 115--143 (2001; Zbl 0980.03066), Math.\ Log.\ Q.\ 49, No.\,2, 150--162 (2003; Zbl 1020.03031)]. They show that if \(X\) is reflexive, then \(X\) has \(WS\) if and only if its dual has \(WS\). Other properties and examples are given. The paper ends with several open problems.