SMALL C*-ALGEBRAS THAT FAIL TO HAVE SEPARABLE REPRESENTATIONS

It is shown that any σ-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 be the σ-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 as small and to ask the following question. Does have a non-trivial separable representation? The above described result answers negatively to this question. That is, fails to have non-trivial separable representations.