AMS :: Transactions of the American Mathematical Society

$KK$-duality for self-similar groupoid actions on graphs
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by Nathan Brownlowe, Alcides Buss, Daniel Gonçalves, Jeremy B. Hume, Aidan Sims and Michael F. Whittaker;

Trans. Amer. Math. Soc. 377 (2024), 5513-5560

DOI: https://doi.org/10.1090/tran/9183

Published electronically: June 11, 2024

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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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