<p>For <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$p \in [1, \infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, we generalize the concept of classical spectral triples by extending the framework from Hilbert spaces to <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-spaces, and from C-algebras to <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-operator algebras. In addition, we define an <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-spectral triple to be metric when the state space of the algebra has a <i>p</i>-quantum compact metric space structure. Specifically, we construct <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-spectral triples for reduced <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-group algebras of countable discrete groups with proper length functions and also for <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-spectral triples associated with <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-group algebras (provided that the length function is of bounded doubling) and those associated with <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> UHF-algebras are always metric.</p>

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

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2026年1月8日

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Alonso Delfín,
Carla Farsi,
Judith Packer

摘要

For $p \in [1, \infty )$ $p \in [1, \infty)$ , 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.

\(L^p\)-spectral Triples and p-Quantum Compact Metric Spaces

摘要