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

TL;DR

This paper introduces a new approach to constructing $ ext{C}^*$-algebras from product systems over semigroups, establishing independence from embeddings and characterizing faithfulness of representations.

Contribution

It defines the covariance algebra for product systems over semigroups, proves its independence from group embeddings, and compares it with existing constructions in dynamical systems.

Findings

Construction is independent of the embedding $P o G$.

Faithfulness on the fixed-point algebra characterizes faithfulness of representations.

Connects the new construction with existing $ ext{C}^*$-algebra frameworks.

Abstract

Let $P$ be a unital subsemigroup of a group $G$. We propose an approach to $\mathrm{C}^*$-algebras associated to product systems over $P$. We call the $\mathrm{C}^*$-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 $\mathrm{C}^*$-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 $\mathrm{C}^*$-algebras of Xin Li and Exel's crossed products by interaction groups.