Refined moves for structure-preserving isomorphism of graph C*-algebras

Abstract

We formalize eight different notions of isomorphism among (unital) graph C*-algebras, and initiate the study of which of these notions may be described geometrically as generated by moves. We propose a list of seven types of moves that we conjecture has the property that the collection of moves respecting one of six notions of isomorphism indeed generate that notion, in the sense that two graphs are equivalent in that sense if and only if one may transform one into another using only these kinds of moves. We carefully establish invariance properties of each move on our list, and prove a collection of generation results supporting our conjecture with an emphasis on the gauge simple case. In two of the six cases, we may prove the conjecture in full generality, and in two we can show it for all graphs defining gauge simple C*-algebras. In the two remaining cases we can show the conjecture for all graphs defining gauge simple C*-algebras provided that they are either finite or have at most one vertex allowing a path back to itself.

In the third version, we have removed Proposition 5.3.4, because we have found a shortcoming in its proof. The unique application (in Lemma 10.3.1) has been replaced with a direct argument. The second version is three times the length of the first and contains more evidence to support the conjectures, along with a development of the two types of isomorphism not considered in the first version

References in corpus (7)

• Continuous orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras
• Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph C*-algebras and Leavitt path algebras
• Geometric classification of graph $C^{\ast }$-algebras over finite graphs
• Diagonal-preserving gauge-invariant isomorphisms of graph $C^{\ast }$-algebras
• Reconstruction of groupoids and C*-rigidity of dynamical systems
• Orbit equivalence of graphs and isomorphism of graph groupoids
• Corners of graph algebras

Cited by in corpus (3)

• Conjugacy of local homeomorphisms via groupoids and C*-algebras
• Geometric classification of isomorphism of unital graph C*-algebras
• Splittings for C*-correspondences and strong shift equivalence