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

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.

arXiv commentAMSLaTeX; 30 pages; v4: final version to appear in the Journal of Fourier Analysis and Applications

References in corpus (9)

• Group algebras acting on $L^p$-spaces
• Nilpotent group C*-algebras as compact quantum metric spaces
• Rigidity of ${\ell }^p$ Roe-type algebras
• L^p operator algebras with approximate identities I
• Banach algebras generated by an invertible isometry of an $L^p$-space
• Dimension and dynamical entropy for metrized C*-algebras
• Rigidity results for $L^p$-operator algebras and applications
• Nonclassifiability of UHF $L^p$-operator algebras
• Directly finite algebras of pseudofunctions on locally compact groups

Cited by in corpus (1)

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