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Annals of Mathematics
A new application of random
matrices: Ext(C red(F2)) is not a group
By Uffe Haagerup and Steen Thorbjørnsen
Annals of Mathematics, 162 (2005), 711–775
A new application of random matrices: Ext(Cred(F2)) is not a group
By Uffe Haagerup and Steen Thorbjørnsen*
Dedicated to the memory of Gert Kjærg˚ard 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 follow-ing extension of Voiculescu’s random matrix result: Let (X(n),...,X(n)) 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
n→∞ °p¡X(n)(ω),...,X(n)(ω)¢° = kp(x1,...,xr)k,
for almost all ω 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 ran-dom variables on a probability space (Ω,F,P). As in [T], we denote by SGRM(n,σ2) the class of complex self-adjoint n ×n random matrices
X = (Xij)i,j=1,
for which (Xii)i, ( 2ReXij)i 0 with the following property: There exists a sequence of natural numbers (n(m))m∈N and a sequence of r-tuples (u(m),...,u(m))m∈N of n(m) ×n(m) unitary matrices, such that
°Xu(m) ⊗u¯(m0)° ≤ C, i=1
whenever m,m0 ∈ N and m = m0. Then C(r) = 2 r −1.
Pisier proved in [P3] that C(r) ≥ 2√r −1 and Valette proved subsequently in [V] that C(r) = 2 r −1, when r is of the form r = p + 1 for an odd prime number p.
We end Section 9 by using Theorem A to prove the following result on powers of “circular” random matrices (cf. §9):
Corollary 3. Let Y be a random matrix in the class GRM(n, 1 ), i.e., the entries of Y are independent and identically distributed complex Gaussian random variables with density z → ne−n|z|2, z ∈ C. Then for every p ∈ N and almost all ω ∈ Ω,
lim °Y (ω)p° = µ(p + 1)p+1 ¶2 .
Note that for p = 1, Corollary 3 follows from Geman’s result [Ge].
In the remainder of this introduction, we sketch the main steps in the
proof of Theorem A. Throughout the paper, we denote by Asa the real vector space of self-adjoint elements in a C∗-algebra A. In Section 2 we prove the
following “linearization trick”:
Let A,B be unital C∗-algebras, and let x1,...,xr and y1,...,yr be opera-tors in Asa and Bsa, respectively. Assume that for all m ∈ N and all matrices
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