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Isometries of Kellendonk–Savinien spectral triples and Connes metrics

Roberto Conti

Department SBAI, Sapienza Università di Roma, 00161 Rome, Italy

E-mail Address: roberto.conti@sbai.uniroma1.it

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and

Carla Farsi

https://orcid.org/0000-0002-9761-7212

Department of Mathematics, University of Colorado at Boulder, Boulder, CO 80309-0395, USA

E-mail Address: carla.farsi@colorado.edu

Corresponding author.

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https://doi.org/10.1142/S0129167X22500847Cited by:1 (Source: Crossref)

Abstract

We study the large and the small isometry groups of Kellendonk–Savinien spectral triples associated to a choice function for a self-similar compact ultrametric Cantor set; in particular, we show that under reasonable assumptions they coincide. To characterize these isometry groups, we use the Michon rooted weighted tree associated to an ultrametric Cantor set: the small and large isometry groups turn out to be equal to the subgroup of the automorphism group of the tree consisting of elements that commute with the choice function in a suitable sense. When the rooted tree is binary, these isometry groups are equal to $ℤ_{2}$ . We also examine examples of Connes metrics associated to Pearson–Bellissard spectral triples, presenting a gamut of cases in where it is infinite.

Communicated by Yasuyuki Kawahigashi

Keywords:

AMSC: 58B34, 46L40

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