Residual finite dimensionality and representations of amenable operator algebras

TL;DR

This paper investigates whether bounded representations of amenable operator algebras are automatically completely bounded, providing affirmative results for residually finite-dimensional cases and linking certain algebras to $C^*$-algebras.

Contribution

It proves that bounded representations are completely bounded for residually finite-dimensional amenable operator algebras and shows weak-* closed, amenable, residually finite-dimensional algebras are similar to $C^*$-algebras.

Findings

Bounded representations are completely bounded for residually finite-dimensional amenable operator algebras.

Weak-* closed, amenable, residually finite-dimensional operator algebras are similar to $C^*$-algebras.

All bounded representations of these algebras are completely bounded.

Abstract

We consider a version of a famous open problem formulated by Kadison, asking whether bounded representations of operator algebras are automatically completely bounded. We investigate this question in the context of amenable operator algebras, and we provide an affirmative answer for representations whose range is residually finite-dimensional. Furthermore, we show that weak-${}^*$ closed, amenable, residually finite-dimensional operator algebras are similar to $C^*$-algebras, and in particular have the property that all their bounded representations are completely bounded. We prove our results for operator algebras having the so-called total reduction property, which is known to be weaker than amenability.