Free Product C* -algebras Associated with Graphs, Free Differentials, and Laws of Loops

Abstract

Abstract. We study a canonical C * -algebra, S(Γ, µ), that arises from a weighted graph (Γ, µ), speci c cases of which were previously studied in the context of planar algebras. We discuss necessary and su cient conditions of the weighting which ensure simplicity and uniqueness of trace of S(Γ, µ), and study the structure of its positive cone. We then study the * -algebra, A, generated by the generators of S(Γ, µ), and use a free di erential 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. A er modifying techniques of Shlyakhtenko and Skoufranis to show that self adjoint elements x ∈ M n (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.

On finite free Fisher information for eigenvectors of a modular operator

“…Thus if t(e m ) = s(e 1 ) has minimal µ-weight amongst all vertices traversed by the loop, then c(e 1 ) · · · c(e m ) is diffuse by Corollary 5.10. A more direct proof of this result as well as a broader consideration of graphs and weightings µ can be found in [15].…”

“…by (15). So by Corollary 5.7, T y = 0 for each y ∈ G and hence (q 1 ⊗ p)#δ y (E λ1 (x)) = 0 for each y ∈ G. Let x 0 = y 1 • • • y m for y 1 , .…”

“…, x n is non-singular [3]. These techniques have since been applied by Hartglass to show that certain elements in C * -algebras associated to weighted graphs have diffuse spectrum [15]. In this paper, these techniques are brought to bear on non-tracial von Neumann algebras.…”

Compact Quantum Metric Spaces from Free Graph Algebras

“…Here, each a ∈ (0, ∞) depends on the weighting of the source and target of , which is chosen so that S(Γ, µ) has a semifinite trace Tr. By [Har17], S(Γ, µ) is simple exactly when µ(v) < ∈ E s( )=v µ(t( ));…”

“…We consider the free loop algebra S 0 = S 0 (Γ, µ) := p v 0 S(Γ, µ)p v 0 , which can be described as generated by loops on Γ based at v 0 . Under condition (1), S 0 (Γ, µ) also has unique trace [Har17].…”