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.

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

References in corpus (9)

• Frames in Hilbert C*-modules and C*-algebras
• KMS states of quasi-free dynamics on Pimsner algebras
• A Categorical Approach to Imprimitivity Theorems for C*-Dynamical Systems
• Diagonal-preserving gauge-invariant isomorphisms of graph $C^{\ast }$-algebras
• A class of C^*-algebras generalizing both graph algebras and homeomorphism C^*-algebras I, fundamental results
• Conjugacy of local homeomorphisms via groupoids and C*-algebras
• Refined moves for structure-preserving isomorphism of graph C*-algebras
• Topological graphs and singly generated dynamical systems
• Reconstruction of topological graphs and their Hilbert bimodules