[Paper Review] A Random Matrix Approach to the Lack of Projections in C*_red(F_2)

Uffe Haagerup, Hanne Schultz+1 more|ArXiv.org|Dec 30, 2004

Geometric and Algebraic Topology4 citations

TL;DR

This paper presents a novel random matrix-based proof that the reduced C*-algebra of the free group on two generators, $C^*_{ ext{red}}(\mathbb{F}_2)$, is projectionless—meaning it contains no nontrivial projections. By extending spectral distribution results for polynomials of independent GUE, GOE, and GSE random matrices to arbitrary selfadjoint polynomials, the authors establish eigenvalue concentration and gap separation phenomena, which are then used to show that any putative projection in the algebra must have trace zero, contradicting the existence of nontrivial projections.

ABSTRACT

In 1982 Pimsner and Voiculescu computed the K_0- and K_1-groups of the reduced group C*-algebra C*_red(F_k) of the free group F_k on k generators and settled thereby a long standing conjecture: C*_red(F_k) has no projections except for the trivial projections 0 and 1. Later simpler proofs of this conjecture were found by methods from K-theory or from non-commutative differential geometry. In this paper we provide a new proof of the fact that C*_red(F_k) is projectionless. The new proof is based on random matrices and is obtained by a refinement of the methods recently used by the first and the third named author to show that the semigroup Ext(C*_red(F_k)) is not a group for k >= 2. By the same type of methods we also obtain that two phenomena proved by Bai and Silverstein for certain classes of random matrices: ``no eigenvalues outside (a small neighbourhood of) the support of the limiting distribution'' and ``exact separation of eigenvalues by gaps in the limiting distribution'' also hold for arbitrary non-commutative selfadjoint polynomials of independent GUE, GOE or GSE random matrices with matrix coefficients.

Motivation & Objective

• To provide a new, random matrix-based proof of the projectionless property of $C^*_{\text{red}}(\mathbb{F}_2)$, resolving a long-standing conjecture.
• To extend spectral distribution results—previously known for linear polynomials in random matrices—to arbitrary selfadjoint polynomials with matrix coefficients.
• To establish eigenvalue concentration and exact gap separation in the limiting spectral distribution for non-commutative selfadjoint polynomials of independent GUE, GOE, and GSE random matrices.
• To use these spectral results to derive a contradiction from the assumption of a nontrivial projection in $C^*_{\text{red}}(\mathbb{F}_2)$, thereby proving its projectionless nature.

Proposed method

• A refined random matrix approach is developed, generalizing earlier results on convergence of operator norms of polynomials in independent GUE matrices to arbitrary selfadjoint polynomials with matrix coefficients.
• The method relies on a new algebraic proof of the linearization trick, replacing earlier $C^*$-algebraic tools like Stinespring’s and Arveson’s theorems.
• Mean value and variance estimates for spectral traces are extended: for any smooth test function $\varphi$, $\mathbb{E}[({\text{tr}}_m \otimes {\text{tr}}_n)\varphi(Q_n)] = ({\text{tr}}_m \otimes \tau)\varphi(q) + \frac{1}{n}\Lambda(\varphi) + O(1/n^2)$, with $\text{supp}(\Lambda) \subseteq \sigma(q)$, valid for GUE, GOE, and GSE ensembles.
• A continuous path of random matrices is constructed via rotation in the space of Gaussian matrices, connecting GUE/GOE matrices to SGRM (spherically Gaussian random matrices), preserving spectral properties.
• The Stieltjes transform of the spectral measure is used to show continuity of the trace functional along the path, enabling topological arguments on the spectrum.
• A contradiction is derived by showing that the expected spectral trace of a putative projection must be an integer multiple of $n$, but the limiting trace is zero, forcing the integer to be zero and thus ruling out nontrivial projections.

Experimental results

Research questions

• RQ1 Can the projectionless property of $C^*_{\text{red}}(\mathbb{F}_2)$ be proven using random matrix theory rather than K-theory or noncommutative geometry?
• RQ2 Do the spectral distribution phenomena—no eigenvalues outside a neighborhood of the limiting support and exact gap separation—extend beyond linear polynomials to arbitrary selfadjoint polynomials of independent GUE, GOE, and GSE matrices?
• RQ3 Can the trace of a putative projection in $C^*_{\text{red}}(\mathbb{F}_2)$ be shown to be zero via random matrix approximation, thereby ruling out nontrivial projections?
• RQ4 Is the spectral measure of a non-commutative selfadjoint polynomial of independent Gaussian random matrices stable under continuous deformation of the matrix ensemble?
• RQ5 Can the continuity of the spectral trace functional along a path of random matrices be established using Stieltjes transform techniques?

Key findings

• The spectral distribution of any selfadjoint polynomial $q(X_1^{(n)}, \dots, X_r^{(n)})$ with matrix coefficients converges almost surely to the spectral distribution of $q(x_1, \dots, x_r)$ in the sense that $\sigma(q(X^{(n)})) \subseteq \sigma(q(x)) + (-\varepsilon, \varepsilon)$ eventually as $n \to \infty$, for any $\varepsilon > 0$.
• For all three ensembles (GUE, GOE, GSE), the mean spectral trace satisfies $\mathbb{E}[({\text{tr}}_m \otimes {\text{tr}}_n)\varphi(Q_n)] = ({\text{tr}}_m \otimes \tau)\varphi(q) + \frac{1}{n}\Lambda(\varphi) + O(1/n^2)$ with $\text{supp}(\Lambda) \subseteq \sigma(q)$, extending prior results to arbitrary-degree polynomials.
• The variance of the spectral trace is $O(1/n^2)$, and if $\varphi'$ vanishes near $\sigma(q)$, the variance is $O(1/n^4)$, enabling almost sure convergence via Borel-Cantelli and Chebyshev.
• The Stieltjes transform of the spectral measure is continuous in time along a continuous path of random matrices, ensuring continuity of the trace functional along the path.
• The expected spectral trace of a putative projection in $C^*_{\text{red}}(\mathbb{F}_2)$ is shown to be zero in the limit, contradicting the requirement that a nontrivial projection must have trace $k/n$ for some integer $k \geq 1$, thus ruling out such projections.
• The paper establishes that the semigroup $\mathrm{Ext}(C^*_{\text{red}}(\mathbb{F}_k))$ is not a group for $k \geq 2$, via the same random matrix framework, reinforcing the nonexistence of projections.