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.

Authors

Uffe Haagerup

Department of Mathematics and Computer Science
University of Southern Denmark
5230 Odense
Denmark

Steen Thorbjørnsen

Department of Mathematics and Computer Science
University of Southern Denmark
5230 Odense
Denmark