Banach spaces with property \((M)\) and their Szlenk indices (Q839820)

The authors investigate Banach spaces with property \((M)\) or \((M^*)\) in the sense of \textit{N.\,J.\thinspace Kalton} [Ill.\ J.\ Math.\ 37, No.\,1, 147--169 (1993; Zbl 0824.46029)] from the point of view of their Szlenk indices and their moduli of asymptotic uniform convexity and smoothness. This allows them to formulate a necessary and sufficient condition on a separable reflexive space with property \((M)\) to embed isomorphically into an \(\ell_p\)-sum of finite-dimensional spaces. As an application, the authors obtain for Lipschitz-isomorphic Orlicz sequence spaces \(h_F\) and \(h_G\) that their Boyd indices coincide; hence \(h_F\) and \(h_G\) contain the same \(\ell_p\) spaces.