CUNTZ–PIMSNER ALGEBRAS ASSOCIATED TO TENSOR PRODUCTS OF -CORRESPONDENCES

Given two $C^{\ast }$ -correspondences $X$ and $Y$ over $C^{\ast }$ -algebras $A$ and $B$ , we show that (under mild hypotheses) the Cuntz–Pimsner algebra ${\mathcal{O}}_{X\otimes Y}$ embeds as a certain subalgebra of ${\mathcal{O}}_{X}\otimes {\mathcal{O}}_{Y}$ and that this subalgebra can be described in a natural way in terms of the gauge actions on ${\mathcal{O}}_{X}$ and ${\mathcal{O}}_{Y}$ . We explore implications for graph algebras, crossed products by $\mathbb{Z}$ , crossed products by completely positive maps, and give a new proof of a result of Kaliszewski, Quigg, and Robertson related to coactions on correspondences.