SMALL C*-ALGEBRAS THAT FAIL TO HAVE SEPARABLE REPRESENTATIONS

It is shown that any -finite, wild AW*-algebra of type II₁ failsto have non-trivial separable representations. Here a wild AW*-algebrameans an AW*-algebra with no direct summand, which is *-isomorphicto a von Neumann algebra. Let be the -finite, wild type II₁AW*-algebra formed by taking the monotone complete tensor productof the Dixmier algebra and a factor of type II₁ acting ona separable Hilbert space. As both and act on separable Hilbertspaces, it seems natural to describe as small and to ask thefollowing 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.