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The Quarterly Journal of Mathematics Advance Access originally published online on July 28, 2006
The Quarterly Journal of Mathematics 2007 58(1):91-105; doi:10.1093/qmath/hal014
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© 2006. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

SMALL C*-ALGEBRAS THAT FAIL TO HAVE SEPARABLE REPRESENTATIONS

Kazuyuki Saitô{dagger}

Mathematical Sciences, King's College, University of Aberdeen, Aberdeen AB24 3UE, Scotland

{dagger} 2-7-5 Yoshinari, Aoba-ku, Sendai 989-3205, Japan; E-mail: k.saito{at}maths.abdn.ac.uk

Received 27 June 2005; revised 30 January 2006 revised 31 May 2006
  Abstract

It is shown that any {sigma}-finite, wild AW*-algebra of type II1 fails to have non-trivial separable representations. Here a wild AW*-algebra means an AW*-algebra with no direct summand, which is *-isomorphic to a von Neumann algebra. Let A be the {sigma}-finite, wild type II1 AW*-algebra formed by taking the monotone complete tensor product of the Dixmier algebra D and a factor M of type II1 acting on a separable Hilbert space. As both D and M act on separable Hilbert spaces, it seems natural to describe A as small and to ask the following question. Does A have a non-trivial separable representation? The above described result answers negatively to this question. That is, A fails to have non-trivial separable representations.


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