A characterization of purely atomic von Neumann algebras

Abstract

Let $M$ be a semifinite von Neumann algebra with a faithful normal semifinite trace $τ$, and let $Φ$ be an Orlicz function. We prove that the noncommutative Orlicz space $L^Φ(M,τ)$ is contained in $M$ if and only if $M$ is purely atomic and there is $δ>0$ such that $τ(e)\geqδ$ for every nonzero projection $e\in M$. We also prove that every left $M$-module map on $L^Φ(M,τ)$ is bounded. As a consequence, a left ideal of $M$ contained in $L^Φ(M,τ)$ and dense in it admits an $M$-invariant algebraic complement if and only if it is equal to $L^Φ(M,τ)$. These results extend a theorem of Fidaleo and Zsidó from $L^2$ to all noncommutative Orlicz spaces.