On ${\mathrm{C}}^{\ast }$-algebras associated to product systems

Abstract

Let $P$ be a unital subsemigroup of a group $G$. We propose an approach to ${\mathrm{C}}^{\ast }$-algebras associated to product systems over $P$. We call the ${\mathrm{C}}^{\ast }$-algebra of a given product system $\mathcal{E}$ its covariance algebra and denote it by $A{\times }_{\mathcal{E}}P$, where $A$ is the coefficient ${\mathrm{C}}^{\ast }$-algebra. We prove that our construction does not depend on the embedding $P\hookrightarrow G$ and that a representation of $A{\times }_{\mathcal{E}}P$ 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 ${\mathrm{C}}^{\ast }$-algebras of Xin Li and Exel's crossed products by interaction groups.

27 pages; Proposition 4.8 was added

References in corpus (1)

• C*-algebras associated to product systems of Hilbert bimodules

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• Groupoid algebras as covariance algebras
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• Boundary quotient C*-algebras of semigroups