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

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(n)1,…,X(n)r) be a system of r stochastically independent n×n Gaussian self-adjoint random matrices as in Voiculescu’s random matrix paper [V4], and let (x1,…,xr) be a semi-circular system in a C∗-probability space. Then for every polynomial p in r noncommuting variables limn→∞∥∥p(X(n)1(ω),…,X(n)r(ω))∥∥=∥p(x1,…,xr)∥, 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  for which Ext() is not a group.

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### 数学年刊（Annals of Mathematics）

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