A new application of random matrices: Ext$(C^*_{\mathrm{red}}(F_2))$ is not a group

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 $(X_1^{(n)},\dots,X_r^{(n)})$ be a system of $r$ stochastically independent $n\times n$ Gaussian self-adjoint random matrices as in Voiculescu’s random matrix paper [V4], and let $(x_1,\dots,x_r)$ be a semi-circular system in a $C^*$-probability space. Then for every polynomial $p$ in $r$ noncommuting variables \[ \lim_{n\to\infty} \big\|p\big(X_1^{(n)}(\omega),\dots,X_r^{(n)}(\omega)\big)\big\| =\|p(x_1,\dots,x_r)\|, \] for almost all $\omega$ in the underlying probability space. We use the result to show that the $\mathrm{Ext}$-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 $\mathcal{A}$ for which $\mathrm{Ext}(\mathcal{A})$ is not a group.