On strong shift equivalence for row-finite graphs and C*-algebras

TL;DR

This paper explores the relationship between strong shift equivalence of row-finite graphs and Morita equivalence of their C*-algebras, extending known theorems and examining the role of insplits and outsplits.

Contribution

It strengthens Bates' theorem linking strong shift equivalence and Morita equivalence, and investigates whether this relation is generated by insplits and outsplits.

Findings

Strong shift equivalent graphs have Morita equivalent C*-algebras.

Morita equivalences induced by insplits and outsplits respect weighted gauge actions.

Open questions about generating strong shift equivalence from insplits and outsplits.

Abstract

This note extends and strengthens a theorem of Bates that says that row-finite graphs that are strong shift equivalent have Morita equivalent graph C*-algebras. This allows us to ask whether our stronger notion of Morita equivalence does in fact characterise strong shift equivalence. We believe this will be relevant for future research on infinite graphs and their C*-algebras. We also study insplits and outsplits as particular examples of strong shift equivalences and show that the induced Morita equivalences respect a whole family of weighted gauge actions. We then ask whether strong shift equivalence is generated by (generalised) insplits and outsplits.