TAILIEUCHUNG - Đề tài " A new application of random matrices: Ext(C red(F2)) is not a group "

Dedicated to the memory of Gert Kjærg˚ Pedersen ard 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 follow(n) (n) ing extension of Voiculescu’s random matrix result: Let (X1 , . . . , Xr ) be a system of r stochastically independent n × n Gaussian self-adjoint random matrices as in Voiculescu’s random matrix paper. | Annals of Mathematics A new application of random matrices Ext C red F2 is not a group By Uffe Haagerup and Steen Thorbjornsen Annals of Mathematics 162 2005 711 775 A new application of random matrices Ext C ed F2 is not a group By Uffe Haagerup and Steen THORBJ0RNSEN Dedicated to the memory of Gert Kjmrgdrd Pedersen 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 . X n 1 be a system of r stochastically independent n X 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 lim p X n H . Xrn H p xi . Xr n -tt for almost all w 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. 1. Introduction A random matrix X is a matrix whose entries are real or complex random variables on a probability space Q F P . As in T we denote by SGRM n Ơ2 the class of complex self-adjoint n X n random matrices X Xij j 1 for which Xii i y 2ReXij i j y 2Im Xij i j are n2 independent identically distributed . Gaussian random variables with mean value 0 and vari- This work was carried out while the first author was a member of the MaPhySto -Centre for Mathematical Physics and Stochastics funded by the Danish National Research Foundation. The second author was supported by The Danish National Science Research Council. 712 UFFE HAAGERUP AND STEEN THORBJORNSEN ance Ơ2. In the .

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