A nonseparable amenable operator algebra which is not isomorphic to a {$C^*$}-algebra

Choi, Yemon and Farah, Ilijas and Ozawa, Narutaka (2014) A nonseparable amenable operator algebra which is not isomorphic to a {$C^*$}-algebra. Forum of Mathematics, Sigma, 2: e2. ISSN 2050-5094

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Abstract

It has been a long-standing question whether every amenable operator algebra is isomorphic to a (necessarily nuclear) C*-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 C*-algebras and show that our method cannot produce a separable counterexample.