Free product C$^{*}$-algebras associated to graphs, free differentials, and laws of loops

TL;DR

This paper investigates a canonical C*-algebra derived from weighted graphs, analyzing its structure, spectral properties of loop elements, and applications to eigenvalues in random matrices and von Neumann algebras.

Contribution

It establishes conditions for simplicity and trace uniqueness of the graph-based C*-algebra and applies free differential calculus to spectral analysis and eigenvalue problems.

Findings

Conditions for simplicity and unique trace of the algebra

Loop elements have no atoms in their spectral measure

Self-adjoint elements have algebraic Cauchy transforms

Abstract

We study a canonical C$^*$-algebra, $\mathcal{S}(\Gamma, \mu)$, that arises from a weighted graph $(\Gamma, \mu)$, specific cases of which were previously studied in the context of planar algebras. We discuss necessary and sufficient conditions of the weighting which ensure simplicity and uniqueness of trace of $\mathcal{S}(\Gamma, \mu)$, and study the structure of its positive cone. We then study the $*$-algebra, $\mathcal{A}$, generated by the generators of $\mathcal{S}(\Gamma, \mu)$, and use a free differential calculus and techniques of Charlesworth and Shlyakhtenko, as well as Mai, Speicher, and Weber to show that certain "loop" elements have no atoms in their spectral measure. After modifying techniques of Shlyakhtenko and Skoufranis to show that self adjoint elements $x \in M_{n}(\mathcal{A})$ have algebraic Cauchy transform, we explore some applications to eigenvalues of polynomials in Wishart matrices and to diagrammatic elements in von Neumann algebras initially considered by Guionnet, Jones, and Shlyakhtenko.