On C⁎-algebras associated to product systems

Abstract

Let P be a unital subsemigroup of a group G. We propose an approach to C⁎-algebras associated to product systems over P. We call the C⁎-algebra of a given product system E its covariance algebra and denote it by A×EP, where A is the coefficient C⁎-algebra. We prove that our construction does not depend on the embedding P↪G and that a representation of A×EP is faithful on the fixed-point algebra for the canonical coaction of G if and only if it is faithful on A. We compare this with other constructions in the setting of irreversible dynamical systems, such as Cuntz–Nica–Pimsner algebras, Fowler's Cuntz–Pimsner algebra, semigroup C⁎-algebras of Xin Li and Exel's crossed products by interaction groups.