  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 ${\mathcal M}$.
