MPS-RR 1999-40
October 1999

Random Matrices and Non-Exact C*-algebras

Uffe Haagerup

Steen Thorbjørnsen

Abstract

From the introduction:

In the paper [HT2], we gave new proofs based on random matrix methods of the following two results:

(1) Any unital exact stably finite C*-algebra has a tracial state.
(2) If A is a unital exact C*-algebra, then any state on K0(A) comes from a tracial state on A.

For each of the results (1) and (2), one may ask whether or not it holds without the assumption that the C*-algebra be exact. These two problems are still open, and both problems are equivalent to Kaplansky's famous problem, whether all AW*-factors of type II1 are von Neumann algebras (cf. [Ha] and [BR]).

In the present note, we provide examples which show that the method used in [HT2] cannot be employed to show that (1) and (2) hold for all C*-algebras.

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