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

• Autores: Uffe Haagerup, Steen Thorbjornsen
• Localización: Annals of mathematics, ISSN 0003-486X, Vol. 162, Nº 2, 2005, págs. 711-775
• Idioma: inglés
• DOI: 10.4007/annals.2005.162.711
• Texto completo no disponible (Saber más ...)
• Resumen

  • 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 lim n¡æ¡Ä
    
    pX(n) 1 (¥ø), . . . , X(n) r (¥ø)
    
    = p(x1, . . . , xr), for almost all ¥ø 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.