Spectral triples for noncommutative solenoids and a Wiener’s lemma
Carla Farsi
University of Colorado Boulder, USATherese Landry
University of California, Santa Barbara, USANadia S. Larsen
University of Oslo, NorwayJudith Packer
University of Colorado Boulder, USA
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Abstract
In this paper, we construct odd finitely summable spectral triples based on length functions of bounded doubling on noncommutative solenoids. Our spectral triples induce a Leibniz Lip-norm on the state spaces of the noncommutative solenoids, giving them the structure of Leibniz quantum compact metric spaces. By applying methods of R. Floricel and A. Ghorbanpour, we also show that our odd spectral triples on noncommutative solenoids can be considered as inductive limits of spectral triples on rotation algebras. In the final section, we prove a noncommutative version of Wiener’s lemma and show that our odd spectral triples can be defined to have an associated smooth dense subalgebra which is stable under the holomorphic functional calculus, thus answering a question of B. Long and W. Wu. The construction of the smooth subalgebra also extends to the case of nilpotent discrete groups.
Cite this article
Carla Farsi, Therese Landry, Nadia S. Larsen, Judith Packer, Spectral triples for noncommutative solenoids and a Wiener’s lemma. J. Noncommut. Geom. (2024), published online first
DOI 10.4171/JNCG/557