$L^p$-spectral triples and $p$-quantum compact metric spaces
Alonso Delfín, Carla Farsi, Judith Packer
TL;DR
The paper develops a systematic $L^p$-version of spectral triples by replacing Hilbert spaces and C*-algebras with $L^p$-spaces and $L^p$-operator algebras, and defines $p$-quantum compact metric spaces via Connes–Rieffel–type metrics. It constructs Dirac operators on reduced $L^p$-group algebras and on $L^p$ UHF-algebras, proving that the resulting $L^p$-spectral triples are metric when the underlying length function has bounded doubling. It extends Christensen–Ivan’s AF-algebra Dirac-operator framework to the $L^p$ setting and verifies metricity through a Banach-geometry adaptation of Rieffel’s conditions. The results provide a unified approach to quantum metrics on both group and AF-type $L^p$-operator algebras, broadening noncommutative geometry tools beyond the Hilbert-space/C*-algebra paradigm.
Abstract
For $p \in [1, \infty)$, 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.