A nonseparable amenable operator algebra which is not isomorphic to a C*-algebra

TL;DR

This paper constructs a nonseparable amenable operator algebra that is not isomorphic to any C*-algebra, addressing a longstanding open problem and exploring the limitations of current methods.

Contribution

It provides the first known nonseparable counterexample to the conjecture that all amenable operator algebras are isomorphic to C*-algebras.

Findings

Existence of a nonseparable counterexample to the conjecture.

Method cannot produce a separable counterexample.

Initiates study of unitarizability of representations of amenable groups.

Abstract

It has been a longstanding problem whether every amenable operator algebra is isomorphic to a (necessarily nuclear) C*-algebra. In this note, we give a nonseparable counterexample. The existence of a separable counterexample remains an open problem. We also initiate a general study of unitarizability of representations of amenable groups in C*-algebras and show that our method cannot produce a separable counterexample.