Compact ideals and rigidity of representations for amenable operator algebras

TL;DR

This paper explores the rigidity of representations of amenable operator algebras with compact ideals, proving generalized conjectures and showing how certain representations relate to isometries, with implications for non-$C^*$-algebra examples.

Contribution

It establishes a generalized Kadison's conjecture for abelian quotients and demonstrates that injective completely bounded representations are similar to complete isometries.

Findings

Generalized Kadison's conjecture holds for abelian quotients.

Injective completely bounded representations are similar to complete isometries.

The results apply even under the weaker total reduction property.

Abstract

We examine rigidity phenomena for representations of amenable operator algebras which have an ideal of compact operators. We establish that a generalized version of Kadison's conjecture on completely bounded homomorphisms holds for the algebra if the associated quotient is abelian. We also prove that injective completely bounded representations of the algebra are similar to complete isometries. The main motivating example for these investigations is the recent construction of Choi, Farah and Ozawa of an amenable operator algebra that is not similar to a $C^*$-algebra, and we show how it fits into our framework. All of our results hold in the presence of the total reduction property, a property weaker than amenability.