Tensor products of measurable operators

Abstract

We introduce and study a stability property for submodules of measurable operators and Calkin spaces and characterize the tensor stable singly generated Calkin spaces. Given semifinite von Neumann algebras $(\mathcal{M},\tau)$, $(\mathcal{N},\sigma)$ and corresponding measurable operators $S$, $T$, we provide a necessary and sufficient condition for the operator $S\otimes T$ to be measurable with respect to $(\mathcal{M}\otimes\mathcal{N},\tau\otimes\sigma)$.