Perturbation of l-copies and measure convergence in preduals of von Neumann algebras

The present article deals with convergence in probability in L-spaces from a functional analytic point of view. The L-spaces in question are the preduals of von Neumann algebras with finite faithful normal traces. To consider an easy example we look at the commutative case: Let (Ω,Σ, μ) be a finite measure space, let (fn) be a bounded sequence in L (Ω,Σ, μ). If (appropriately chosen representatives of) the fn have pairwise disjoint supports then clearly (fn) converges to 0 in measure. From the functional analytic point of view such a sequence, up to normalization, is the canonical basis of an isometric copy of l. If one perturbes (fn) by a norm null sequence (gn) then (fn + gn) still μ-converges to 0 and spans l 1 almost isometrically (in a sense to be made precise below in §2). It has been known [10, Th. 2] (see also [19, Th. 3, Rem. 6bis]) for quite a time that, roughly speaking, these are essentially the only examples of μ-null sequences. Theorem 1 contains the analogous statement for the predual of a von Neumann algebra with finite faithful normal trace. (For notation and definitions see §2.) Theorem 1 Let (xn) be a bounded sequence in L 1(N , τ) = N∗ where (N , τ) is a von Neumann algebra with a finite normal faithful trace τ . Then the following assertions are equivalent.