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 Voiculescu’s random matrix result: Let (X(n)1,…,X(n)r) be a system of r stochastically independent n×n Gaussian self-adjoint random matrices as in Voiculescu’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
for almost all ω 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 for which Ext() is not a group.