Bundle convergence of harmonic means of orthogonal sequences in noncommutative L2-spaces

AbstractThe notion of bundle convergence for sequences in von Neumann algebras and their L2-spaces was introduced by Hensz, Jajte and Paszkiewicz in 1996. Bundle convergence is stronger than almost sure convergence. We prove that the sequence of harmonic means (of first order) of an orthogonal sequence (ξk) in L2 is bundle convergent to the zero vector o of L2 if the classical Kolmogorov condition is satisfied: Σ |ξk|2/k2 < ∞; while the sequence of harmonic means of second order of (ξk) is bundle convergent to o if Σ |ξk|/k2 (ln ln k)2 < ∞. The latter result seems to be new even in the commutative case