Almost everywhere convergence for noncommutative spaces

Abstract

Almost everywhere convergence is an essential part of classical measure theory. However, when passing to the quantum setting of noncommutative $L^p$-spaces, the absence of an explicit measure space makes it very difficult to give expression to notions like almost everywhere convergence. There is a rich literature devoted to different ways of circumventing this challenge, positing various notions of “measure theoretic” convergence in the noncommutative case. However, not many of these seem to be suited to dealing with Haagerup $L^p$-spaces. In this paper we review several noncommutative notions of convergence before proposing versions of these notions which have been recast in terms of spectral projections. The harmony of exisiting notions with these revised notions is then investigated in the semifinite setting, at which point we also demonstrate the efficacy of the “new” approach by establishing a matching noncommutative monotone convergence theorem. On the basis of the theory achieved in the semifinite setting, we then show how this “reshaped” theory may be lifted to the setting of Haagerup $L^p$-spaces. In closing we show that even here a monotone convergence theorem based on these notions is valid.