On commutative, operator amenable subalgebras of finite von Neumann algebras

TL;DR

This paper proves that any closed, commutative, operator amenable subalgebra of a finite von Neumann algebra is similar to a commutative C*-subalgebra within the same algebra, using measurable operators.

Contribution

It establishes a similarity result for commutative, operator amenable subalgebras of finite von Neumann algebras, advancing understanding of their structure.

Findings

Any such subalgebra is similar to a C*-subalgebra within the algebra.

The similarity is implemented by an element of the von Neumann algebra.

The proof utilizes the algebra of measurable operators affiliated to the von Neumann algebra.

Abstract

An open question, raised independently by several authors, asks if a closed amenable subalgebra of ${\mathcal B}({\mathcal H})$ must be similar to an amenable C*-algebra; the question remains open even for singly-generated algebras. In this article we show that any closed, commutative, operator amenable subalgebra of a finite von Neumann algebra ${\mathcal M}$ is similar to a commutative C*-subalgebra of ${\mathcal M}$, with the similarity implemented by an element of ${\mathcal M}$. Our proof makes use of the algebra of measurable operators affiliated to ${\mathcal M}$.