C*-algebras and Equivalences for C*-correspondences
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.
Keywords
Cite
@article{arxiv.1210.6066,
title = {C*-algebras and Equivalences for C*-correspondences},
author = {Evgenios T. A. Kakariadis and Elias G. Katsoulis},
journal= {arXiv preprint arXiv:1210.6066},
year = {2022}
}
Comments
3+33pp. We add a corrigendum to the original entry, in compliance to the arXiv policy, concerning a gap in the proof of Theorem 5.8, that results to a weaker version of SE than the one sought. In the corrigendum we show where the problem lies, adjust the main results --which still tackle the two conjectures of the introduction in the original entry-- and make connections with current literature