A new application of Random Matrices: Ext(C*_{red}(F_2)) is not a group

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_1^(n),...,X_r^(n) be a system of r stochastically independent n by 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->oo}||p(X_1^(n),...,X_r^(n))|| = ||p(x_1,...,x_r)||, 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.

65 pages, published version

References in corpus (1)

• A simple proof of a theorem of Kirchberg and related results on $C^{\ast }$-norms

Cited by in corpus (19)

• The largest eigenvalues of finite rank deformation of large Wigner matrices: convergence and nonuniversality of the fluctuations
• Improved subspace estimation for multivariate observations of high dimension: the deterministic signals case
• Convergence of the largest singular value of a polynomial in independent Wigner matrices
• Gaussian fluctuations for linear spectral statistics of large random covariance matrices
• Products of independent elliptic random matrices
• Large deviations and stochastic calculus for large random matrices
• Random quantum channels II: Entanglement of random subspaces, Renyi entropy estimates and additivity problems
• On the convergence of the extremal eigenvalues of empirical covariance matrices with dependence
• On the operator norm of non-commutative polynomials in deterministic matrices and iid GUE matrices
• Polynomials in Asymptotically Free Random Matrices
• Power law decay for systems of randomly coupled differential equations
• The free field: realization via unbounded operators and Atiyah property
• Shape theory and extensions of C*-algebras
• On the reduction criterion for random quantum states
• On the convergence of output sets of quantum channels
• On the separability of unitarily invariant random quantum states - the unbalanced regime
• An l^{p}-Version of von Neumann Dimension for Banach Space Representation of Sofic Groups II
• Norm of matrix-valued polynomials in random unitaries and permutations
• Operator-Valued Matrices with Free or Exchangeable Entries