Graph products of operator algebras

Abstract

Graph products for groups were defined by Green in her thesis [25] as a generalization of both Cartesian and free products. In this paper we define the corresponding graph product for reduced and maximal $C^{\ast }$-algebras, von Neumann algebras and quantum groups. We prove stability properties including permanence of II${}_1$-factors, the Haagerup property, exactness and, under suitable conditions, the property of rapid decay for quantum groups.