Cuntz-Pimsner Algebras and Twisted Tensor Products

TL;DR

This paper establishes an isomorphism between the Cuntz-Pimsner algebra of a twisted tensor product of correspondences and a balanced twisted tensor product of their individual Cuntz-Pimsner algebras, generalizing known constructions.

Contribution

It introduces a new framework connecting twisted tensor products of correspondences with their Cuntz-Pimsner algebras, extending existing theories.

Findings

Isomorphism between $ ext{O}_{Xoxtimes Y}$ and $ ext{O}_X oxtimes_ ext{T} ext{O}_Y$

Generalizes tensor and crossed product constructions

Connects to existing results on crossed and tensor product Cuntz-Pimsner algebras

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.