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

19 Nov 2018
•
Sehnem Camila F.

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.(read more)