A noncommutative transfer principle

Abstract

We extend the generalised Calderon Transfer Principle as presented in \cite{dBL} to the setting of trace preserving group actions of $σ$-compact locally compact Hausdorff groups on semifinite von Neumann algebras $\M$. At its essence the transfer principle consists of showing that a large class of regular operators in the group context may be translated to operators in the algebra context by means of this group action. The end result is a protocol for proving ergodic convergence results for group actions on noncommutative Orlicz space of semifinite von Neumann algebras. We start with proving the existence of the transferred operator in the noncommutative context. Two tools are developed for the purpose of actually proving convergence results: (1) a theory of what may be called 2-variable decreasing rearrangements for the algebra $L^\infty \overline{\otimes} \M$ (where ($Γ,ν$) is a Radon measure space) and (2) a concepts of maximal operators suited to the present context. These tools are then used to lift the ergodic convergence results presented in \cite{dBL} to the noncommutative setting. In closing we present examples illustrating the application of the tools