C*-algebras and Equivalences for C*-correspondences

TL;DR

This paper explores shift equivalence in C*-correspondences and its impact on associated C*-algebras, establishing new Morita equivalence results and addressing open problems in the theory.

Contribution

It proves that shift equivalent, regular, full C*-correspondences have Morita equivalent Pimsner dilations and Cuntz-Pimsner algebras, extending previous results and proposing an analogue of the Shift Equivalence Problem.

Findings

Shift equivalent, regular, full C*-correspondences have Morita equivalent Pimsner dilations.

Shift equivalent correspondences imply Morita equivalent Cuntz-Pimsner algebras.

Subshifts of finite type that are eventually conjugate have Morita equivalent Cuntz-Krieger algebras.

Abstract

We study several notions of shift equivalence for C*-correspondences and the effect that these equivalences have on the corresponding Pimsner dilations. Among others, we prove that non-degenerate, regular, full C*-correspondences which are shift equivalent have strong Morita equivalent Pimsner dilations. We also establish that the converse may not be true. These results settle open problems in the literature. In the context of C*-algebras, we prove that if two non-degenerate, regular, full C*-correspondences are shift equivalent, then their corresponding Cuntz-Pimsner algebras are strong Morita equivalent. This generalizes results of Cuntz and Krieger and Muhly, Tomforde and Pask. As a consequence, if two subshifts of finite type are eventually conjugate, then their Cuntz-Krieger algebras are strong Morita equivalent. Our results suggest a natural analogue of the Shift Equivalence Problem in the context of C*-correspondences. Even though we do not resolve the general Shift Equivalence Problem, we obtain a positive answer for the class of imprimitivity bimodules.