A new family of ambient C*-algebras around graph products yields universal properties, nuclearity/exactness characterizations, a maximal ideal, and new simplicity and trace-uniqueness criteria for graph product C*-algebras.
On the structure of graph product von Neumann algebras
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abstract
We undertake a comprehensive study of structural properties of graph products of von Neumann algebras equipped with faithful, normal states, as well as properties of the graph products relative to subalgebras coming from induced subgraphs. Among the technical contributions in this paper include a complete bimodule calculation for subalgebras arising from subgraphs. As an application, we obtain a complete classification of when two subalgebras coming from induced subgraphs can be amenable relative to each other. We also give complete characterizations of when the graph product can be full, diffuse, or a factor. Our results are obtained in a broad generality, and we emphasize that they are new even in the tracial setting. They also allow us to deduce new results about when graph products of groups can be amenable relative to each other.
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Universal C$^{\ast}$-Algebras from Graph Products: Structure and Applications
A new family of ambient C*-algebras around graph products yields universal properties, nuclearity/exactness characterizations, a maximal ideal, and new simplicity and trace-uniqueness criteria for graph product C*-algebras.