Graph products of operator algebras

TL;DR

This paper introduces graph product constructions for operator algebras and quantum groups, establishing their stability and permanence properties, thereby generalizing classical product concepts in operator algebra theory.

Contribution

It defines graph products for C*-algebras, von Neumann algebras, and quantum groups, and proves key stability and permanence properties.

Findings

Graph products for operator algebras are well-defined and preserve key properties.

Stability properties like the Haagerup property and exactness are maintained under graph products.

Under certain conditions, the property of Rapid Decay is also preserved.

Abstract

Graph products for groups were defined by Green in her thesis as a generalization of both Cartesian and free products. In this paper we define the corresponding graph product for reduced and maximal C*-algebras, von Neumann algebras and quantum groups. We prove stability properties including permanence properties of II_1-factors, the Haagerup property, exactness and under suitable conditions the property of Rapid Decay for quantum groups.