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

We study a canonical C${}^{\ast }$-algebra, $\mathcal{S}(\mathrm{\Gamma },\mu )$, that arises from a weighted graph $(\mathrm{\Gamma },\mu )$, 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 $\mathcal{S}(\mathrm{\Gamma },\mu )$, and study the structure of its positive cone. We then study the $\ast$-algebra, $\mathcal{A}$, generated by the generators of $\mathcal{S}(\mathrm{\Gamma },\mu )$, 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.