A NONSEPARABLE AMENABLE OPERATOR ALGEBRA WHICH IS NOT ISOMORPHIC to A

Abstract

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