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

#### Authors

Yemon Choi, Ilijas Farah, Narutaka Ozawa*

* Corresponding author

MSC 2010: 47L30 (primary); 03E75 46L05 (secondary)

DOI: 10.1017/fms.2013.6

#### Status

Preprint version available at arXiv 1309.2145

#### Reviews

[ Math Review (summary) | Zentralblatt ]

#### 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.

The original preprint of this paper was written by IF and NO, with the original counterexample containing all of $K(H)$ and hence not being contained in a subhomogeneous von Neumann algebra. YC's contributions, among them the observation that the original method could be adapted to get counterexamples inside $\ell^\infty\overline{\otimes}{\bf M}_2$, came in later versions.