Abstract
In the process of developing the theory of free probability and free entropy, Voiculescu introduced in 1991 a random matrix model for a free semicircular system. Since then, random matrices have played a key role in von Neumann algebra theory (cf. [V8], [V9]). The main result of this paper is the following extension of Voiculescui¯s random matrix result: Let (X(n) 1 , . . . , X(n) r ) be a system of r stochastically independent n i? n Gaussian self-adjoint random matrices as in Voiculescui¯s random matrix paper [V4], and let (x1, . . . , xr) be a semi-circular system in a C.-probability space. Then for every polynomial p in r noncommuting variables lim niaeiApX(n) 1 (¥o), . . . , X(n) r (¥o) = p(x1, . . . , xr), for almost all ¥o in the underlying probability space. We use the result to show that the Ext-invariant for the reduced C.-algebra of the free group on 2 generators is not a group but only a semi-group. This problem has been open since Anderson in 1978 found the first example of a C.-algebra A for which Ext(A) is not a group.
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