A note on the Voiculescu's theorem for commutative C$^*$-algebras in semifinite von Neumann algebras

TL;DR

This paper extends Voiculescu's theorem to commutative C*-algebras within semifinite von Neumann algebras, broadening the understanding of approximate equivalence of *-homomorphisms.

Contribution

It generalizes the compact operator aspect of Voiculescu's theorem for commutative C*-algebras in semifinite von Neumann algebras and extends Hadwin's results on approximate summands.

Findings

Generalization of Voiculescu's theorem to semifinite von Neumann algebras

Extension of Hadwin's results on approximate summands

Broader framework for *-homomorphism equivalence

Abstract

In the current paper, we generalize the "compact operator" part of the Voiculescu's non-commutative Weyl-von Neumann theorem on approximate equivalence of unital $*$-homomorphisms of an commutative C$^*$ algebra $\mathcal{A}$ into a semifinite von Neumann algebra. A result of D. Hadwin for approximate summands of representations into a finite von Neumann factor $\mathcal{R}$ is also extended.