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

By Uffe Haagerup, S. Thorbjørnsen. 2002

353 citations recorded by the source corpus.

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 Voiculescui¯s random matrix result: Let (X(n) 1 , . . . , X(n) r ) be a system of r stochastically independent n i? n Gaussian self-adjoint random matrices as in Voiculescui¯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 lim niaeiApX(n) 1 (¥o), . . . , X(n) r (¥o) = p(x1, . . . , xr), for almost all ¥o 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.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/a7b2cba1165e7b9032727b29865a5c50ff995b0d

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Citations in the corpus

On the Asymptotic Spectrum of Products of Independent Random Matrices. (2010, background): “ger–Verlag, Berlin 1995 - 2nd ed., 316 pp. [9] Go¨tze, F. and Tikhomirov, A. N. The Circular Law for Random Matrices. Annals of Probability (2010), vol. 58, N 4, 1444-1491, Preprint: arxiv:0709.3995. [10] U. Haagerup and S. Torbjørnsen A new application of random matrices: Ext(C∗ red(F2)) is not a g”
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Eigenvalues of random lifts and polynomials of random permutation matrices (2018, methodology): “It was developed simultaneously in various areas of mathematics, and applied in operator algebras by Haagerup and Thorbjørnsen and subsequently improved by Anderson [6].”
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