Cuntz-Pimsner Algebras Associated to Tensor Products of Correspondences

TL;DR

This paper investigates the structure of Cuntz-Pimsner algebras associated with tensor products of correspondences, revealing their embedding properties and connections to gauge actions, with implications for various algebraic constructions.

Contribution

It establishes an embedding of the Cuntz-Pimsner algebra of a tensor product into a tensor product of Cuntz-Pimsner algebras and describes this subalgebra via gauge actions, extending understanding of these structures.

Findings

The Cuntz-Pimsner algebra of a tensor product embeds into the tensor product of individual Cuntz-Pimsner algebras.

The subalgebra can be characterized using gauge actions on the Cuntz-Pimsner algebras.

Applications include results for graph algebras, crossed products, and coactions on correspondences.

Abstract

Given two correspondences X and Y, we show that (under mild hypotheses) the Cuntz-Pimsner algebra of the tensor product of X and Y embeds as a certain subalgebra of the tensor product of the Cuntz-Pimsner algebra of X and the Cuntz=Pimsner algebra of Y. Furthermore, this subalgebra can be described in a natural way in terms of the gauge actions on the Cuntz-Pimsner algebras. We explore implications for graph algebras, crossed products by the integers, and crossed products by completely positive maps. We also give a new proof of a result of Kaliszewski and Quigg related to coactions on correspondences.