Splittings for C*-correspondences and strong shift equivalence
Abstract
We present an extension of the notion of in-splits from symbolic dynamics to topological graphs and, more generally, to C*-correspondences. We demonstrate that in-splits provide examples of strong shift equivalences of C*-correspondences. Furthermore, we provide a streamlined treatment of Muhly, Pask, and Tomforde's proof that any strong shift equivalence of regular C*-correspondences induces a (gauge-equivariant) Morita equivalence between Cuntz-Pimsner algebras. For topological graphs, we prove that in-splits induce diagonal-preserving gauge-equivariant *-isomorphisms in analogy with the results for Cuntz-Krieger algebras. Additionally, we examine the notion of out-splits for C*-correspondences.
Keywords
Cite
@article{arxiv.2305.01917,
title = {Splittings for C*-correspondences and strong shift equivalence},
author = {Kevin Aguyar Brix and Alexander Mundey and Adam Rennie},
journal= {arXiv preprint arXiv:2305.01917},
year = {2025}
}
Comments
After this article was published, Adam Dor-On informed us of a gap in our Theorem 3.5. The gap is rectified by a recent result of Boris Bilich, Adam Dor-On, and Efren Ruiz. We have added Remark 3.7 which explains the gap and its fix. We thank Adam Dor-On for bringing this to our attention