Pro-\(C^{\ast}\)-algebras associated to tensor products of pro-\(C^{\ast}\)-correspondences (Q2014132)

\textit{M. Joita} and \textit{I. Zarakas} [J. Oper. Theory 74, No. 1, 195--211 (2015; Zbl 1399.46078)] introduced the notion of a pro-\(C^{\ast}\)-correspondence and showed how a Cuntz-type pro-\(C^{\ast}\)-algebra can be associated to it. A pro-\(C^{\ast}\)-correspondence \(X\) is a right module over a pro-\(C^{\ast}\)-algebra \(A\) (on a family of \(C^{\ast}\)-seminorms \(\{p_\lambda\}_{\lambda \in \Lambda}\)) together with a left action. The existence of the Cuntz-Pimsner algebra \(\mathcal{O}_X\) requires for each ideal \(\ker p_\lambda\) to be positive invariant, in the sense that it contains \(\left\langle X, \ker p_\lambda \cdot X \right\rangle\). As with \(C^{\ast}\)-correspondences, it is interesting to research the compatibility of these constructions with tensor products and crossed products; for example see [\textit{E. Katsoulis}, ``\(C^{\ast}\) envelopes and the Hao-Ng isomorphism for discrete groups'', Int. Math. Res. Not. IMRN 2017, No. 18, 5751--5768 (2017)]. For pro-\(C^{\ast}\)-algebras one considers the minimal tensor product and for pro-\(C^{\ast}\)-correspondences one considers the exterior tensor product. In the current paper the author shows that the Cuntz-Pimsner algebra \(\mathcal{O}_{X \otimes Y}\) is a pro-\(C^{\ast}\)-subalgebra of \(\mathcal{O}_X \otimes \mathcal{O}_Y\), under the condition that \(X\) and \(Y\) are ideal compatible. This compatibility means that the tensor product of the Katsura ideals of \(X\) and \(Y\) coincides with the Katsura ideal of \(X \otimes Y\) (where one inclusion is shown to always hold). In particular, \(\mathcal{O}_{X \otimes Y}\) is a pro-\(C^{\ast}\)-subalgebra of the \(\mathbb{T}\)-balanced \(\mathcal{O}_X \otimes_{\mathbb{T}} \mathcal{O}_Y\). An isomorphism with the latter is shown to hold when \(X\) and \(Y\) are Katsura non-degenerate, i.e., when Katsura's ideal acts non-degenerately on the right. This covers several examples such as when \(Y\) is a trivial pro-\(C^{\ast}\)-correspondence or when both \(X\) and \(Y\) are Hilbert pro-\(C^{\ast}\)-bimodules.