The amenability–similarity conjecture for commutative operator algebras

Let $\mathfrak{H}$ be a Hilbert space and let $\mathfrak{B}(\mathfrak{H})$ denote the bounded operators on $\mathfrak{H}$. A subalgebra of $\mathfrak{B}(\mathfrak{H})$ is amenable when it is amenable as a Banach algebra, and two operator algebras are similar when one is the image of the other under conjugation by an invertible bounded operator.

Amenability–similarity conjecture. A commutative Banach subalgebra of $\mathfrak{B}(\mathfrak{H})$ is amenable if and only if it is similar to a $C^*$-algebra.

This conjecture asks whether amenability completely characterizes, among commutative operator algebras, those that are similar to $C^*$-algebras. The supplied text presents it as a long-standing conjecture, and gives no resolution.