ASYMPTOTIC SMOOTHNESS IN BANACH SPACES, THREE SPACE PROPERTIES AND APPLICATIONS

We study four asymptotic smoothness properties of Banach spaces, denoted Tp, Ap, Np and Pp. We complete their description by proving the missing renorming theorem for Ap. We prove that asymptotic uniform flattenability (property T∞) and summable Szlenk index (property A∞) are three space properties. Combined with the positive results of the first named author, Draga, and Kochanek, and with the counterexamples we provide, this completely solves the three space problem for this family of properties. We also derive from our characterizations of Ap and Np in terms of equivalent renormings, new coarse Lipschitz rigidity results for these classes