Kadec–Pełczyński decomposition for Haagerup L p -spaces

Source 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 n k } ∞ k =1 of {φ n } ∞ n =1 , a decomposition φ n k = 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.