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KK-duality for self-similar groupoid actions on graphs.
- Source :
-
Transactions of the American Mathematical Society . Aug2024, Vol. 377 Issue 8, p5513-5560. 48p. - Publication Year :
- 2024
-
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. [ABSTRACT FROM AUTHOR]
- Subjects :
- *DIRECTED graphs
*C*-algebras
*SOLENOIDS
*ALGEBRA
Subjects
Details
- Language :
- English
- ISSN :
- 00029947
- Volume :
- 377
- Issue :
- 8
- Database :
- Academic Search Index
- Journal :
- Transactions of the American Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 178359022
- Full Text :
- https://doi.org/10.1090/tran/9183