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

TL;DR

This paper extends the concept of spectral triples from Hilbert spaces and C*-algebras to $L^p$-spaces and $L^p$-operator algebras, establishing a framework for $p$-quantum metric spaces and constructing examples for group and UHF algebras.

Contribution

It introduces $L^p$-spectral triples and demonstrates their metric properties for specific classes of $L^p$-algebras, broadening noncommutative geometry tools.

Findings

$L^p$-spectral triples can be constructed for $L^p$-group algebras with bounded doubling length functions.

$L^p$-spectral triples for $L^p$ UHF-algebras are always metric.

The framework generalizes classical spectral triples to a wider $L^p$ setting.

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.