Kadec-pe Lczyński Decomposition for Haagerup L-spaces

Let M be a von Neumann algebra (not necessarily semi-finite). We provide a generalization of the classical Kadec-Pe lczynski subsequence decomposition of bounded sequences in L[0, 1] to the case of the Haagerup L-spaces (1 ≤ p < ∞). In particular, we prove that if (φn)n is a bounded sequence in the predual M∗ of M, then there exist a subsequence (φnk)k of (φn)n, a decomposition φnk = yk + zk such that {yk, k ≥ 1} is relatively weakly compact and the support projections s(zk) ↓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 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.