On commutative, operator amenable subalgebras of finite von Neumann algebras

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 $\mathcal{M}$.