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

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.