Kadec-Pelczynski decomposition for Haagerup L_p-spaces

TL;DR

This paper extends the Kadec-Pelczynski subsequence decomposition to Haagerup L^p-spaces associated with general von Neumann algebras, revealing structural properties of bounded sequences in these spaces.

Contribution

It generalizes classical decomposition results to non-semi-finite von Neumann algebras' Haagerup L^p-spaces, including applications to subspace structures in duals of C*-algebras.

Findings

Decomposition of bounded sequences into weakly compact and disjoint support parts

Existence of asymptotically isometric copies of l_1 in non-reflexive subspaces

Failure of fixed point property in certain non-reflexive subspaces

Abstract

Let M be a von Neumann algebra (not necessarily semi-finite). We provide a generalization of the classical Kadec-Pelczynski subsequence decomposition of bounded sequences in L^p[0,1] to the case of the Haagerup L^p-spaces (1\le p<\infty). In particular, we prove that if (\phi_n)_n is a bounded sequence in the predual M_* of M, then there exist a subsequence (\phi_{n_k})_k of (\phi_n)_n, a decomposition \phi_{n_k}= y_k+ z_k such that {y_k, k\ge 1} is relatively weaklycompact and the support projections s(z_k)\downarrow_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 l_1 and therefore fails the fixed point property for nonexpansive mappings. These generalize earlier results for the case of preduals of semi-finite von Neumann algebras.