Approximate Equivalence in von Neumann Algebras

TL;DR

This paper investigates conditions under which unital *-homomorphisms from certain C*-algebras into von Neumann algebras are approximately unitarily equivalent, extending results to general finite von Neumann algebras.

Contribution

It establishes approximate unitary equivalence of *-homomorphisms under Murray-von Neumann equivalence of range projections in specific von Neumann algebra settings.

Findings

Proves approximate unitary equivalence modulo compact operators in sigma-finite II_infinity factors.

Extends results to arbitrary finite von Neumann algebras.

Provides a general framework for 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 $\pi,\rho :\mathcal{A}\rightarrow\mathcal{R}$ 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.