A new application of random matrices: Ext(C_(red)~*(F_2)) is not a group

摘要

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. The main result of this paper is the following extension of Voiculescu's random matrix result: Let (X_1~((n)),...,X_r~((n)) ) be a system of r stochastically independent n x n Gaussian self-adjoint random matrices as in Voiculescu's random matrix paper, and let (x_1,..., x_r) be a semi-circular system in a C~*-probability space. Then for every polynomial p in r noncommuting variables lim_(n → ∞)‖p(X_1~((n))(ω),...,X_r~((n))(ω))‖= ‖p(x_1,...,x_r)‖, 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 A for which Ext (A) is not a group.