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

Abstract: In this paper 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, $p \in [1, \infty)$. Specifically, we construct $L^p$-spectral triples for reduced group $L^p$-operator algebras and for $L^p$ UHF-algebras of infinite tensor product type. Furthermore, inspired by Christensen and Ivan's construction of a Dirac operator on AF C*-algebras, we provide an $L^p$ version of this construction for $L^p$ UHF-algebras. This leads to the construction of a $p$-quantum compact metric space structure on the state space of the $L^p$ UHF-algebra.