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.
The atoms of graph product von Neumann algebras
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abstract
We completely classify the atomic summands in a graph product $(M,\varphi) = *_{v \in \mathcal{G}} (M_v,\varphi_v)$ of von Neumann algebras with faithful normal states. Each type I factor summand $(N,\psi)$ is a tensor product of type I factor summands $(N_v,\psi_v)$ in the individual algebras. The existence of such a summand and its weight in the direct sum can be determined from the $(N_v,\psi_v)$'s using explicit polynomials associated to the graph.
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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.