Equivariant homotopy classification of graph C*-algebras

TL;DR

This paper establishes a novel connection between shift equivalence of essential adjacency matrices and gauge-equivariant homotopy equivalence of their associated stabilized graph C*-algebras, using bicategory theory.

Contribution

It provides the first formulation of shift equivalence in terms of gauge actions on graph C*-algebras, avoiding K-theory classification methods.

Findings

Shift equivalence coincides with gauge-equivariant homotopy of stabilized graph C*-algebras

Introduces bicategory theory approach to analyze C*-algebras

First formulation of shift equivalence via gauge actions

Abstract

We show that shift equivalence of essential adjacency matrices coincides with gauge-equivariant homotopy equivalence of their stabilized graph C*-algebras. This provide the first equivalent formulation of shift equivalence of essential matrices in terms of gauge actions on graph C*-algebras. Our proof uses bicategory theory for C*-bimodules developed by Meyer and Sehnem, allowing us to avoid the use of K-theory classification of C*-algebras.