A new application of random matrices: Ext(Cred∗(F2)) is not a group

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 (Xr ) be a system of r stochastically independent n × n Gaussian self-adjoint random matrices as in Voiculescu's random matrix paper [V4], and let (x 1 , . . ., x r ) be a semi-circular system in a C * -probability space.Then for every polynomial p in r noncommuting variablesfor 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 A for which Ext(A) is not a group.