Non-commutative subsequence principles

We prove that if is a hyperfinite semi-finite von Neumann algebra equipped with a normal semi-finite trace and (x _n ) _{n = 1} is a bounded sequence in L ¹ ( ) then there exists a subsequence (y _n ) _{n = 1} of (x _n ) _{n = 1} such that for any further subsequence (z _n ) _{n = 1} of (y _n ) _{n = 1} and > 0, there exists a projection p with (1–p)< and thus providing a non-commutative analogue of the classical Komlóss subsequence theorem. We also extend a classical result of Menchoff and Marcienkiewicz on convergence almost everywhere of subseries of orthogonal functions in L ² 0,1] to orthogonal operators in the non-commutative L ² -space associated to the type II ₁ hyperfinite factor. Mathematics Subject Classification (2000):46L51, 46L52, 40A05, 60F15.Supported in part by NSF grant DMS-0096696 and by a Miami University Summer Research Appointment.