A new application of Random Matrices: Ext(C*_{red}(F_2)) is not a group

TL;DR

This paper extends Voiculescu's random matrix results to show that the Ext-invariant of the reduced C*-algebra of the free group on two generators is only a semi-group, not a group, resolving a long-standing open problem.

Contribution

It generalizes random matrix convergence results to free semicircular systems and applies this to determine the algebraic structure of the Ext-invariant for a specific C*-algebra.

Findings

Ext-invariant for the reduced C*-algebra of the free group on two generators is a semi-group.

Random matrices approximate free semicircular systems in operator norm.

Addresses a problem open since 1978 regarding the Ext-invariant's structure.

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 Voiculescu's random matrix result: Let X_1^(n),...,X_r^(n) be a system of r stochastically independent n by 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 variables lim_{n->oo}||p(X_1^(n),...,X_r^(n))|| = ||p(x_1,...,x_r)||, for almost all omega 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.