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Search: MSC category 46L09 ( Free products of $C^$-algebras *$-algebras * )

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1. CJM 2016 (vol 69 pp. 548)

Hartglass, Michael
Free Product C*-algebras Associated with 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

2. CJM 2005 (vol 57 pp. 1056)

Ozawa, Narutaka; Rieffel, Marc A.
Hyperbolic Group $C^*$-Algebras and Free-Product $C^*$-Algebras as Compact Quantum Metric Spaces
Let $\ell$ be a length function on a group $G$, and let $M_{\ell}$ denote the operator of pointwise multiplication by $\ell$ on $\bell^2(G)$. Following Connes, $M_{\ell}$ can be used as a ``Dirac'' operator for $C_r^*(G)$. It defines a Lipschitz seminorm on $C_r^*(G)$, which defines a metric on the state space of $C_r^*(G)$. We show that if $G$ is a hyperbolic group and if $\ell$ is a word-length function on $G$, then the topology from this metric coincides with the weak-$*$ topology (our definition of a ``compact quantum metric space''). We show that a convenient framework is that of filtered $C^*$-algebras which satisfy a suitable ``Haagerup-type'' condition. We also use this framework to prove an analogous fact for certain reduced free products of $C^*$-algebras.

Categories:46L87, 20F67, 46L09

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