ACTA issues

Two-parameter SLLN in noncommutative $L_2$-spaces in terms of bundle convergence

Barthélemy Le Gac, Ferenc Móricz

Acta Sci. Math. (Szeged) 67:1-2(2001), 273-298

Abstract. The notion of bundle convergence for single (ordinary) sequences in von Neumann algebras and their $L_2$-spaces was introduced by Hensz, Jajte, and Paszkiewicz in 1996. We adopt this notion for double sequences. Bundle convergence is stronger than almost sure convergence, and it enjoys the property of additivity. We prove an extension of the classical two-parameter strong law of large numbers to an orthogonal double sequence of vectors in a noncommutative $L_2$-space. Our main tool is a two-parameter extension of the classical Rademacher--Menshov inequality to the noncommutative case. We also prove the extension of another strong law, even the one-parameter version of which seems to be new.

AMS Subject Classification (1991): 46L10, 46L53, 60B12

Keyword(s): \phi, L_2({\eufm A}, von Neumann algebra {\eufm A}, faithful and normal state, completion, ; Gelfand-Naimark-Segal representation theorem; bundle convergence; almost sure convergence; orthogonal double sequence of vectors in, ; Rademacher-Menshov inequality; strong law of large numbers, \phi )L_2

Received August 12, 1999, and in revised form October 17, 2000. (Registered under 2784/2009.)