Noncommutative Geometry, the Spectral Standpoint

Connes, Alain

doi:10.1017/9781108854399.003

Cited by 14 publications

(20 citation statements)

References 175 publications

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“…Appearence of the factor ν in the inequality above, which is closely related to the density of spectral geometry, is not incidental. There is a similar factor in the Connes-Sullivan formula [Co,IV,Theorem 17] and other formulae from [Co].…”

Section: Dixmier Trace and Hausdorff Measuresmentioning

confidence: 77%

Spectral geometries on a compact metric space

Buyalo

2019

St. Petersburg Math. J.

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The notion of a spectral geometry on a compact metric space X is introduced. This notion serves as a discrete approximation of X motivated by the notion of a spectral triple from noncommutative geometry. A set of axioms charaterizing spectral geometries is given. Bounded deformations of spectral geometries are studied and the relationship between the dimension of a spectral geometry and more traditional dimensions of metric spaces is investigated.2 Axioms of spectral geometries 2.1 The main constructionThen we define a spectral triple M (B) = {A, H, ds} as follows. The Hilbert space H consists of functions ξ : B → C satisfying the conditionThe algebra A = C(X) of all continuous functions X → C is represented by the multiplication operators (π(a)(ξ)(y, y ′ ) = a(y)ξ(y, y ′ ) for all a ∈ A, ξ ∈ H, (y, y ′ ) ∈ B. Clearly, this represention is involutive, π(a * ) = π(a) * , andwhere Y is the projection of B on X (by (a) it is the same whether one takes the projection on the first or on the second factor).The unit length operator ds is defined by ds(ξ)(y, y ′ ) = |yy ′ |ξ(y ′ , y).This operator is injective and self-adjoint. Its eigenvalues are ±|yy ′ |, (y, y ′ ) ∈ B. It follows from (b) that ds is compact. The operator D = ds −1 is defined onand called the Dirac operator. The set dom D contains all functions ξ ∈ H with finite support and whence is dense in H. Furthermore, dom D = dom D * and D = D * . Thus D is an unbounded self-adjoint operator with discrete spectrum {±|yy ′ | −1 : (y, y ′ ) ∈ B} ⊂ R.The set dom D is invariant under every π(a), a ∈ A, and we have ([D, π(a)]ξ)(y, y ′ ) = a(y ′ ) − a(y) |yy ′ | ξ(y ′ , y).

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Section: Dixmier Trace and Hausdorff Measuresmentioning

confidence: 77%

Spectral geometries on a compact metric space

Buyalo

2019

St. Petersburg Math. J.

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“…As a special case of scalar-tensor-fourth-order gravity theory is the Non-Commutative Spectral Geometry (NCSG) [65,66]. At a given cutoff scale (e.g.…”

Section: Extended Gravity Modelsmentioning

confidence: 99%

Photon frequency shift in curvature based Extended Theories of Gravity

Capozziello¹,

Lambiase²,

Stabile³

2021

Preprint

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We study the frequency shift of photons generated by rotating gravitational sources in the framework of curvature based Extended Theories of Gravity. The discussion is developed considering the weak-field approximation. Following a perturbative approach, we analyze the process of exchanging photons between Earth and a given satellite, and we find a general relation to constrain the free parameters of gravitational theories. Finally, we suggest the Moon as a possible laboratory to test theories of gravity by future experiments which can be, in principle, based also on other Solar System bodies.

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“…SPOT and CH. Connes (see for example [6], pp. 20-21) objects to the use of ultrafilters but approves of the Continuum Hypothesis (CH).…”

Section: Final Remarksmentioning

confidence: 99%

Infinitesimal analysis without the Axiom of Choice

Hrbacek,

Katz

2020

Preprint

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It is often claimed that analysis with infinitesimals requires more substantial use of the Axiom of Choice than traditional elementary analysis. The claim is based on the observation that the hyperreals entail the existence of nonprincipal ultrafilters over N, a strong version of the Axiom of Choice, while the real numbers can be constructed in ZF. The axiomatic approach to nonstandard methods refutes this objection. We formulate a theory SPOT which suffices to carry out infinitesimal arguments, and prove that SPOT is a conservative extension of ZF. Thus the methods of Calculus with infinitesimals are just as effective as those of traditional Calculus. This result and conclusion extend to large parts of ordinary mathematics and beyond. We also develop a stronger axiomatic system SCOT, conservative over ZF + ADC, which is suitable for handling such features as an infinitesimal approach to the Lebesgue measure. Proofs of the conservativity results combine and extend the methods of forcing developed by Enayat and Spector.

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Noncommutative Geometry, the Spectral Standpoint

Cited by 14 publications

References 175 publications

Spectral geometries on a compact metric space

Spectral geometries on a compact metric space

Photon frequency shift in curvature based Extended Theories of Gravity

Infinitesimal analysis without the Axiom of Choice

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