Approximate Equivalence in von Neumann Algebras

Abstract

Suppose $\mathcal{A}$ is a separable unital ASH C*-algebra, $\mathcal{R}$ is a sigma-finite II${}_{\infty }$ factor von Neumann algebra, and $\pi ,\rho :\mathcal{A}\to \mathcal{R}$ are unital $\ast$-homomorphisms such that, for every $a\in \mathcal{A}$, the range projections of $\pi \left(a\right)$ and $\rho \left(a\right)$ are Murray von Neuman equivalent in $\mathcal{R}$. We prove that $\pi$ and $\rho$ are approximately unitarily equivalent modulo ${\mathcal{K}}_{\mathcal{R}}$, where ${\mathcal{K}}_{\mathcal{R}}$ is the norm closed ideal generated by the finite projections in $\mathcal{R}$. We also prove a very general result concerning approximate equivalence in arbitrary finite von Neumann algebras.