Cuntz-Pimsner Algebras and Twisted Tensor Products

Abstract

Given two correspondences $X$ and $Y$ and a discrete group $G$ which acts on $X$ and coacts on $Y$, one can define a twisted tensor product $X⊠Y$ which simultaneously generalizes ordinary tensor products and crossed products by group actions and coactions. We show that, under suitable conditions, the Cuntz-Pimsner algebra of this product, ${\mathcal{O}}_{X⊠Y}$, is isomorphic to a "balanced" twisted tensor product ${\mathcal{O}}_X{⊠}_{\mathbb{T}}{\mathcal{O}}_Y$ of the Cuntz-Pimsner algebras of the original correspondences. We interpret this result in several contexts and connect it to existing results on Cuntz-Pimsner algebras of crossed products and tensor products.