[Paper Review] Matrices of unitary moments

Ken Dykema, Kate Juschenko|ArXiv.org|Jan 2, 2009

Advanced Operator Algebra Research12 references3 citations

TL;DR

This paper investigates the set of matrices formed by second-order mixed moments of unitary operators in C*-algebras with tracial states, establishing that Connes' embedding problem is equivalent to the equality of two such matrix sets: one derived from finite-dimensional unitary matrices ($\mathcal{F}_n$) and another from general C*-algebraic unitaries ($\mathcal{G}_n$). The key contribution is a new lower bound on the scaling constant $c_X$ for matrices in $\mathcal{G}_n$, showing $c_X \geq \sqrt{2}/(1+\sqrt{2}) \approx 0.586$ for a specific 4x4 matrix, providing a concrete example on the boundary of $\mathcal{F}_4$.

ABSTRACT

We investigate certain matrices composed of mixed, second-order moments of unitaries. The unitaries are taken from C*-algebras with moments taken with respect to traces, or, alternatively, from matrix algebras with the usual trace. These sets are of interest in light of a theorem of E. Kirchberg about Connes' embedding problem.

Motivation & Objective

• To investigate the structure of matrices formed by second-order mixed moments of unitary operators in C*-algebras with tracial states.
• To clarify the relationship between the sets $\mathcal{F}_n$ (matrices from finite-dimensional unitary matrices) and $\mathcal{G}_n$ (matrices from general C*-algebraic unitaries), particularly in relation to Connes' embedding problem.
• To establish quantitative lower bounds on the scaling constant $c_X$ for matrices in $\mathcal{G}_n$, which determines whether a matrix lies in $\mathcal{F}_n$.
• To provide explicit examples and bounds that help characterize the boundary of $\mathcal{F}_n$ within the set of correlation matrices $\Theta_n$.

Proposed method

• The paper defines $\mathcal{G}_n$ as the set of $n \times n$ matrices of the form $(\tau(U_i^*U_j))_{i,j=1}^n$, where $U_i$ are unitaries in C*-algebras with a faithful tracial state $\tau$, and $\mathcal{F}_n$ as the closure of matrices from $k \times k$ unitary matrices with normalized trace $\mathrm{tr}_k$.
• It uses the Gelfand-Naimark-Segal construction to represent $\mathcal{G}_n$ as limits of positive tracial functionals on the universal $*$-algebra generated by $n$ unitaries.
• The authors analyze the convex and compact structure of $\mathcal{F}_n$ and $\mathcal{G}_n$, showing invariance under conjugation by diagonal unitary and permutation matrices.
• They introduce the scaling constant $c_X$ for $X \in \Theta_n$, defined as the largest $c$ such that $cX + (1-c)I \in \mathcal{F}_n$, and derive bounds on $c_X$ using permutation averaging and Schur products.
• A key technique involves conjugating matrices to simplify the imaginary part of $X - \overline{X}$, enabling the use of known results on extreme correlation matrices.
• For a specific 4x4 matrix $X$ from Corollary 2.11, the authors apply Lemma 4.2 with $d = \sqrt{2}$ to derive the bound $c_X \geq \sqrt{2}/(1 + \sqrt{2}) \approx 0.586$.

Experimental results

Research questions

• RQ1 Is the set $\mathcal{F}_n$ equal to $\mathcal{G}_n$ for all $n$, and how does this relate to Connes' embedding problem?
• RQ2 What is the precise value of the scaling constant $c_X$ for a given correlation matrix $X$ in $\mathcal{G}_n$, and how does it determine membership in $\mathcal{F}_n$?
• RQ3 Can explicit lower bounds on $c_X$ be derived using permutation averaging and unitary conjugation?
• RQ4 How do the eigenvalues and off-diagonal structure of $X$ influence the value of $c_X$?
• RQ5 What is the exact boundary behavior of $\mathcal{F}_n$ within $\Theta_n$, and can concrete examples be constructed?

Key findings

• The paper establishes that $c_X \geq \frac{6}{n^2 - n}$ for any $X \in \Theta_n$, providing a universal lower bound on the scaling constant $c_X$ for matrices in $\mathcal{G}_n$.
• For a specific 4x4 matrix $X$ derived from Corollary 2.11, the authors prove $c_X \geq \frac{\sqrt{2}}{1 + \sqrt{2}} \approx 0.586$, offering a concrete example on the boundary of $\mathcal{F}_4$.
• By conjugating $X$ with a diagonal unitary matrix, the authors reduce the norm of the imaginary part of $X - \overline{X}$, improving the bound on $d$ and thus tightening the lower bound on $c_X$.
• The set $\mathcal{F}_n$ is shown to be closed under Schur products and invariant under conjugation by permutation and diagonal unitary matrices, reflecting its convex and compact structure.
• The paper confirms that $\mathcal{F}_n \subseteq \mathcal{G}_n \subseteq \Theta_n$, and that Connes' embedding problem is equivalent to the equality $\mathcal{F}_n = \mathcal{G}_n$ for all $n$, as established by Kirchberg.
• The authors derive a second bound: $c_X \geq \min\left(\frac{6}{(n^2 - n)(1 - \lambda_0)}, 1\right)$, where $\lambda_0$ is the smallest eigenvalue of $X$, strengthening the earlier bound when $\lambda_0 < 1$.