
Free product C*algebras associated to graphs, free differentials, and laws of loops
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.
Keywords:free probability, C*algebra Category:46L09 