KK-duality for self-similar groupoid actions on graphs

TL;DR

This paper generalizes KK-duality for C*-algebras associated with self-similar groupoid actions on graphs, removing recurrence constraints and extending from finite alphabets to finite graphs, linking to Smale space dynamics.

Contribution

It extends Nekrashevych's KK-duality to self-similar groupoid actions on graphs, removing recurrence conditions and broadening the setting from alphabets to graphs.

Findings

Established KK-duality for new class of C*-algebras

Proved Morita equivalence to Ruelle algebras of Smale spaces

Generalized from finite alphabet to finite graph setting

Abstract

We extend Nekrashevych's $KK$-duality for $C^*$-algebras of regular, recurrent, contracting self-similar group actions to regular, contracting self-similar groupoid actions on a graph, removing the recurrence condition entirely and generalising from a finite alphabet to a finite graph. More precisely, given a regular and contracting self-similar groupoid $(G,E)$ acting faithfully on a finite directed graph $E$, we associate two $C^*$-algebras, $\mathcal{O}(G,E)$ and $\widehat{\mathcal{O}}(G,E)$, to it and prove that they are strongly Morita equivalent to the stable and unstable Ruelle C*-algebras of a Smale space arising from a Wieler solenoid of the self-similar limit space. That these algebras are Spanier-Whitehead dual in $KK$-theory follows from the general result for Ruelle algebras of irreducible Smale spaces proved by Kaminker, Putnam, and the last author.