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arXiv · 2302.03989

KK-duality for self-similar groupoid actions on graphs

Nathan Brownlowe·Alcides Buss·Daniel Gonçalves·Jeremy B. Hume·Aidan Sims·Michael F. Whittaker
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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.

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Nathan Brownlowe, Alcides Buss, Daniel Gonçalves, Jeremy B. Hume, Aidan Sims, Michael F. Whittaker. 2023-02-08. KK-duality for self-similar groupoid actions on graphs. https://arxiv.org/abs/2302.03989

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