On Summability of Distributions and Spectral Geometry

Abstract

Modulo the moment asymptotic expansion, the Cesàro and parametric behaviours of distributions at infinity are equivalent. On the strength of this result, we construct the asymptotic analysis for spectral densities, arising from elliptic pseudodifferential operators. We show how Cesàro developments lead to efficient calculations of the expansion coefficients of counting number functionals and Green functions. The bosonic action functional proposed by Chamseddine and Connes can more generally be validated as a Cesàro asymptotic development.

On the Ultraviolet Behavior of Quantum Fields Over Noncommutative Manifolds

“…Our results in [31] extend the validity of (3) to R n , for compactly supported functions. Therefore, in the commutative case, the integral identity (2) subsumes the formula given by Langmann.…”

Moyal Planes are Spectral Triples

“…This holds in the ordinary asymptotic sense, and not merely the Cesàro sense, by the "sandwich" argument used in the proof of [34,Cor. 4.1].…”

“…(Here ′′ of course denotes the strong bidual space, not a bicommutant.) As distributions, the elements of (M θ L ) ′ and (M θ R ) ′ belong to O ′ C , and a fortiori they are Cesàro summable [34].…”

Spectral action on noncommutative torus

“…More precisely, given a spectral triple .A; H; D/ where A is an algebra acting on the Hilbert space H and D is a Dirac-like operator (see [7], [23]), Chamseddine and Connes proposed a physical action depending only on the spectrum of the covariant Dirac operator whereˆis any even positive cut-off function which could be replaced by a step function up to some mathematical difficulties investigated in [16]. This means that S counts the spectral values of jD A j that are less than the mass scale ƒ (note that the resolvent of D A is compact since, by assumption, the same is true for D; see Lemma 3.1 below).…”