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A NONSEPARABLE AMENABLE OPERATOR ALGEBRA WHICH IS NOT ISOMORPHIC TO A $C^*$ -ALGEBRA

Part of: Set theory Selfadjoint operator algebras Linear spaces and algebras of operators

Published online by Cambridge University Press:  10 March 2014

YEMON CHOI ,
ILIJAS FARAH  and
NARUTAKA OZAWA
YEMON CHOI
Affiliation:
Department of Mathematics and Statistics, University of Saskatchewan, McLean Hall, 106 Wiggins Road, Saskatoon, Saskatchewan, Canada S7N 5E6choi@math.usask.ca
ILIJAS FARAH
Affiliation:
Department of Mathematics and Statistics, York University, 4700 Keele Street, North York, Ontario, Canada M3J 1P3ifarah@mathstat.yorku.ca Matematicki Institut, Kneza Mihaila 35, Belgrade, Serbia
NARUTAKA OZAWA
Affiliation:
RIMS, Kyoto University, 606-8502 Japan; narutaka@kurims.kyoto-u.ac.jp

Abstract

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

MSC classification

Secondary: 47L30: Abstract operator algebras on Hilbert spaces 46L05: General theory of $C^*$-algebras 03E75: Applications of set theory

Information

Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 2 , 01 July 2014 , e2
Creative Commons
Creative Common License - CCCreative Common License - BY
The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence .
Copyright
© The Author(s) 2014

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