Spectral triples, Coulhon-Varopoulos dimension and heat kernel estimates

C\'edric Arhancet

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Spectral triples, Coulhon-Varopoulos dimension and heat kernel estimates

Cédric Arhancet

Operator Algebras · Submitted 25 Sep 2022

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim More precisely, we prove that the completely bounded local Coulhon-Varopoulos dimension $d$ exceeds the spectral dimension and even implies that the associated Hodge-Dirac operator is $d^+$-summable.

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We investigate the relations between the (completely bounded) local Coulhon-Varopoulos dimension and the spectral dimension of spectral triples associated to sub-Markovian semigroups (or Dirichlet forms) acting on classical (or noncommutative) $\mathrm{L}^p$-spaces associated to finite measure spaces. More precisely, we prove that the completely bounded local Coulhon-Varopoulos dimension $d$ exceeds the spectral dimension and even implies that the associated Hodge-Dirac operator is $d^+$-summable. We explore different settings to compare these two values: compact Riemannian manifolds, compact Lie groups, sublaplacians, metric measure spaces, noncommutative tori and quantum groups. Specifically, we prove that, while very often equal in smooth compact settings, these dimensions can diverge. Finally, we show that the existence of a symmetric sub-Markovian semigroup on a von Neumann algebra with finite completely bounded local Coulhon-Varopoulos dimension implies that the von Neumann algebra is necessarily injective.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We investigate the relations between the (completely bounded) local Coulhon-Varopoulos dimension and the spectral dimension of spectral triples associated to sub-Markovian semigroups (or Dirichlet forms) acting on classical (or noncommutative) $\mathrm{L}^p$-spaces associated to finite measure spaces.	contribution 0.96	0.83	0.08	asserted 0.62
s1	More precisely, we prove that the completely bounded local Coulhon-Varopoulos dimension $d$ exceeds the spectral dimension and even implies that the associated Hodge-Dirac operator is $d^+$-summable.	result 0.99	0.95	0.66	asserted 1.00
s2	We explore different settings to compare these two values: compact Riemannian manifolds, compact Lie groups, sublaplacians, metric measure spaces, noncommutative tori and quantum groups.	contribution 0.60	0.78	0.04	asserted 0.74
s3	Specifically, we prove that, while very often equal in smooth compact settings, these dimensions can diverge.	result 0.97	0.88	0.11	asserted 0.94
s4	Finally, we show that the existence of a symmetric sub-Markovian semigroup on a von Neumann algebra with finite completely bounded local Coulhon-Varopoulos dimension implies that the von Neumann algebra is necessarily injective.	result 1.00	0.93	0.31	asserted 1.00

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Original abstract

We investigate the relations between the (completely bounded) local Coulhon-Varopoulos dimension and the spectral dimension of spectral triples associated to sub-Markovian semigroups (or Dirichlet forms) acting on classical (or noncommutative) $\mathrm{L}^p$-spaces associated to finite measure spaces. More precisely, we prove that the completely bounded local Coulhon-Varopoulos dimension $d$ exceeds the spectral dimension and even implies that the associated Hodge-Dirac operator is $d^+$-summable. We explore different settings to compare these two values: compact Riemannian manifolds, compact Lie groups, sublaplacians, metric measure spaces, noncommutative tori and quantum groups. Specifically, we prove that, while very often equal in smooth compact settings, these dimensions can diverge. Finally, we show that the existence of a symmetric sub-Markovian semigroup on a von Neumann algebra with finite completely bounded local Coulhon-Varopoulos dimension implies that the von Neumann algebra is necessarily injective.

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2 arXiv papers cite this in 8 sentences

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In the next result which is a $Ł^p$-generalization of [ref] , we connect the norm of the commutators $[D_p,π(f)]$ and the amalgamated $Ł^p$-spaces of [ref].

Curvature, Dolbeault-Dirac operators, and an $\mathrm{L}^p$-index theorem on compact K\"ahler manifolds · The norms of commutators of the full Hodge-Dirac operator · 0.92

Cites it as support 1 paper

Furthermore, it is implicitly proved in [ref] (see also [ref] ), that the semigroup $(T_t)_t ≥0$ satisfies $_Ł^p(Γ_q(H) (ℓ^2_I)(0)$.

Curvature, Dolbeault-Dirac operators, and an $\mathrm{L}^p$-index theorem on compact K\"ahler manifolds · Semigroups of Schur multipliers · 0.98

Cites it as background 1 paper

We refer to [ref] , [ref] and [ref] for this line of research and to [ref] , [ref] , [ref] , [ref] and [ref] for related papers.

Curvature, Dolbeault-Dirac operators, and an $\mathrm{L}^p$-index theorem on compact K\"ahler manifolds · Introduction · 0.97

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Now, for any $t>0$, by a result of [ref] we obtain $φ_t_ℓ^2_G =T_t_,Ł^1((G)) →Ł^2((G)) < ∞$.

Quantum information theory and Fourier multipliers on quantum groups · Dimension vs completely bounded dimension · 0.96

Cites as background 2 papers

Lie groups We refer to our previous paper [ref] for more information.

Sobolev algebras on Lie groups and noncommutative geometry · Examples in various classical contexts · 0.93
We refer to [ref] for a (too) brief history in this topic and to the books [ref] and [ref] and finally to the survey [ref] and references therein for more information.

Gaussian Estimates: A Brief History · Examples in various classical contexts · 1.00

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