L^p-spectral triples and p-quantum compact metric spaces

2024/11/20 by Delfín, Alonso, Farsi, Carla, Packer, Judith · 1 citation
#46H35 #46L89 #58B34 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Primary 46H15 #Secondary 43A15

paper · doi:10.48550/arxiv.2411.13735

Abstract

For p ∈ [1, ∞), we generalize the concept of classical spectral triples by extending the framework from Hilbert spaces to L^p-spaces, and from C*-algebras to L^p-operator algebras. In addition, we define an L^p-spectral triple to be metric when the state space of the algebra has a p-quantum compact metric space structure. Specifically, we construct L^p-spectral triples for reduced L^p-group algebras of countable discrete groups with proper length functions and also for L^p UHF-algebras of infinite tensor product type, the latter inspired by E. Christensen and C. Ivan's construction of a Dirac operator on AF C*-algebras. We prove that L^p-spectral triples associated with L^p-group algebras (provided that the length function is of bounded doubling) and those associated with L^p UHF-algebras are always metric.

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• C*-like modules and matrix p-operator norms

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