Kadec–Pełczyński decomposition for Haagerup L^p-spaces

Abstract

Let [Mscr ] be a von Neumann algebra (not necessarily semi-finite). We provide a generalization of the classical Kadec–Pełczyński subsequence decomposition of bounded sequences in L^p[0, 1] to the case of the Haagerup L^p-spaces (1 [les ] p < 1 ). In particular, we prove that if { φ_n}^∞_n=1 is a bounded sequence in the predual [Mscr ]∗ of [Mscr ], then there exist a subsequence {φ_nk}^∞_k=1 of {φ_n}^∞_n=1, a decomposition φ_nk = y_k+z_k such that {y_k, k [ges ] 1} is relatively weakly compact and the support projections supp(z_k) ↓_k 0 (or similarly mutually disjoint). As an application, we prove that every non-reflexive subspace of the dual of any given C*-algebra (or Jordan triples) contains asymptotically isometric copies of [lscr ]¹ and therefore fails the fixed point property for non-expansive mappings. These generalize earlier results for the case of preduals of semi-finite von Neumann algebras.