A new application of random matrices: Ext(C*red(F2)) is not a group

• 成果类型: Article
• 署名作者: Haagerup, U; Thorbjornsen, S
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2005.162.711
• 发表日期: 2005
• 页码: 711-775
• 关键词: c-star-algebras; asterisk-algebras; tensor-products; cstar-algebras; Operators; EXTENSIONS; INEQUALITY; EIGENVALUE; THEOREM; forms

摘要

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)),..., X-r((n))) 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 (x(1),..., x(r)) be a semi-circular system in a C*-probability space. Then for every polynomial p in r noncommuting variables lim(n ->infinity) parallel to p(X-1((n))(omega),..., X-r((n))(omega)) parallel to = parallel to p(x(1),..., x(r)) parallel to' for almost all omega 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.