$$L^p$$-spectral Triples and p-Quantum Compact Metric Spaces | CU Experts

Abstract; ; For; ; ; $$p in [1, infty )$$; ; ; p; ∈; [; 1; ,; ∞; ); ; ; ; ; , we generalize the concept of classical spectral triples by extending the framework from Hilbert spaces to; ; ; $$L^p$$; ; ; L; p; ; ; ; ; -spaces, and from C*-algebras to; ; ; $$L^p$$; ; ; L; p; ; ; ; ; -operator algebras. In addition, we define an; ; ; $$L^p$$; ; ; 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; ; ; $$L^p$$; ; ; L; p; ; ; ; ; -spectral triples for reduced; ; ; $$L^p$$; ; ; L; p; ; ; ; ; -group algebras of countable discrete groups with proper length functions and also for; ; ; $$L^p$$; ; ; 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; ; ; $$L^p$$; ; ; L; p; ; ; ; ; -spectral triples associated with; ; ; $$L^p$$; ; ; L; p; ; ; ; ; -group algebras (provided that the length function is of bounded doubling) and those associated with; ; ; $$L^p$$; ; ; L; p; ; ; ; ; UHF-algebras are always metric.;

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