Covariance algebra of a product system

Camila F. Sehnem (Victoria University of Wellington)

Thursday 17th September, 2020 16:00-17:00 For zoom coordinates contact one of the organisers


Let $P$ be a submonoid of a group $G$. In this talk, we will consider a construction of a $C^*$-algebra associated to a product system $\mathcal{E}$ over $P$. We define it as a quotient of Fowler's Toeplitz algebra of $\mathcal{E}$ by a certain ideal, which is shown to be a maximal gauge-invariant ideal with trivial intersection with the coefficient $C^*$-algebra $A$. We call such a quotient the covariance algebra of $\mathcal{E}$, and denote it by $A\rtimes_\mathcal{E}P$. So we show that a representation of $A\times_\mathcal{E}P$ is injective on the fixed-point algebra for the canonical coaction of $G$ if and only if it is injective on the coefficient $C^*$-algebra $A$. In particular, our construction contains Cuntz--Pimsner algebras of single correspondences and a class of Cuntz--Nica--Pimsner algebras of compactly aligned product systems due to Sims and Yeend. Under the appropriate assumptions, we can describe Fowler's Cuntz--Pimsner algebras, semigroup $C^*$-algebras of Xin Li and Exel's crossed products by interaction groups as covariance algebras of product systems.

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