Camila Fabre Sehnem: On C*-algebras associated to product systems

Abstract: Many examples of product systems arise from semigroup actions by endomorphisms of a $C^*$-algebra. In this talk, assuming that $P$ is a unital subsemigroup of a group $G$, we will define and discuss some properties of the covariance algebra of a product system $\mathcal{E}=(\mathcal{E}_p)_{p\in P}$, which we denote by $A\times_\mathcal{E}P$, where $A$ is the coefficient $C^*$-algebra. Under the appropriate assumptions, a representation of $A\times_\mathcal{E}P$ in a $C^*$-algebra is injective if and only if it is injective on $A$. In particular, this may be viewed as a generalisation of a Cuntz--Pimsner algebra of a single correspondence.