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 \(X\boxtimes Y\) 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, \(\mathcal O_{X\boxtimes Y}\), is isomorphic to a "balanced" twisted tensor product \(\mathcal O_X\boxtimes_\mathbb T\mathcal O_Y\) 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.