[Paper Review] Connes Integration Formula without singular traces

Fedor Sukochev, Dmitriy Zanin|arXiv (Cornell University)|Mar 16, 2021

Advanced Operator Algebra Research25 references4 citations

TL;DR

This paper establishes a new version of Connes' integration formula that derives the asymptotic eigenvalue behavior of pseudodifferential operators on compact Riemannian manifolds without relying on singular traces or ultrafilters. By proving a sharp weak Schatten norm estimate for the operator $(1- abla_g)^{-d/4}M_f(1- abla_g)^{-d/4}$, the authors extend the class of integrable functions to the Orlicz space $L_M(X, \mathrm{vol}_g)$, showing that the trace asymptotics match the Riemannian volume integral via eigenvalue decay rates.

ABSTRACT

A version of Connes Integration Formula which provides concrete asymptotics of the eigenvalues is given. This radically extending the class of quantum-integrable functions on compact Riemannian manifolds.

Motivation & Objective

• To resolve Connes' open question on deriving the noncommutative integration formula without using ultrafilters or singular traces.
• To extend the class of quantum-integrable functions beyond $L^2$ to the Orlicz space $L_M(X, \mathrm{vol}_g)$, where $M(t) = t\log(e+t)$.
• To establish a direct asymptotic formula for the eigenvalues of the symmetric operator $(1-\Delta_g)^{-d/4}M_f(1-\Delta_g)^{-d/4}$ that recovers the Riemannian volume integral.

Proposed method

• Prove a sharp weak Schatten norm estimate $\|M_f(1-\Delta_g)^{-d/4}\|_{2,\infty} \leq C_{X,g}\|f\|_{L_M^{(2)}}$ for $f \in L_M^{(2)}(X, \mathrm{vol}_g)$, extending Solomyak's result to compact manifolds.
• Use the Birman-Solomyak theory of spectral asymptotics and recent advances in quasi-Banach ideal norms to analyze the singular value behavior of the operator.
• Apply the Birman-Koplienko-Solomyak inequality in the quasi-Banach ideal setting to control the difference between $ABA_+$ and $AB_+A$ for self-adjoint operators.
• Establish convergence of the singular value function $\mu(t,T)$ in $\mathcal{L}_{1,\infty}$-norm for approximating sequences of smooth functions.
• Leverage the fact that $[M_f, (1-\Delta_g)^{-d/4}] \in (\mathcal{L}_{2,\infty})_0$ for $f \in L_M(X, \mathrm{vol}_g)$ to control error terms in the asymptotic expansion.
• Use the limit $\lim_{t\to\infty} t \mu(t, T_+) = \frac{\mathrm{Vol}(\mathbb{S}^{d-1})}{d(2\pi)^d} \int_X f_+ \, d\mathrm{vol}_g$ to recover the integral via eigenvalue decay.

Experimental results

Research questions

• RQ1 Can the Connes integration formula be derived directly from eigenvalue asymptotics without invoking singular traces or ultrafilters?
• RQ2 What is the maximal class of functions $f$ on a compact Riemannian manifold for which the operator $(1-\Delta_g)^{-d/4}M_f(1-\Delta_g)^{-d/4}$ admits a trace asymptotics matching the Riemannian volume integral?
• RQ3 Does the weak Schatten norm estimate $\|M_f(1-\Delta_g)^{-d/4}\|_{2,\infty} \leq C_{X,g}\|f\|_{L_M^{(2)}}$ hold on compact manifolds, extending Solomyak's result from the torus?
• RQ4 Can the asymptotic eigenvalue behavior of the symmetric operator be used to define a noncommutative integral for functions outside $L^2$?
• RQ5 Is the difference between the positive parts of $ABA$ and $AB_+A$ in the $(\mathcal{L}_{1,\infty})_0$ ideal when $[A,B] \in (\mathcal{L}_{2,\infty})_0$?

Key findings

• The asymptotic eigenvalue behavior of $(1-\Delta_g)^{-d/4}M_f(1-\Delta_g)^{-d/4}$ satisfies $\lim_{t\to\infty} t \mu(t, T_+) = \frac{\mathrm{Vol}(\mathbb{S}^{d-1})}{d(2\pi)^d} \int_X f_+ \, d\mathrm{vol}_g$ for all real-valued $f \in L_M(X, \mathrm{vol}_g)$.
• The result holds for the negative part as well: $\lim_{t\to\infty} t \mu(t, T_-) = \frac{\mathrm{Vol}(\mathbb{S}^{d-1})}{d(2\pi)^d} \int_X f_- \, d\mathrm{vol}_g$, completing the symmetric formula.
• The class of integrable functions is extended beyond $L^2$ to the Orlicz space $L_M(X, \mathrm{vol}_g)$ with $M(t) = t\log(e+t)$, which is strictly larger than $L^2$.
• The weak Schatten norm estimate $\|M_f(1-\Delta_g)^{-d/4}\|_{2,\infty} \leq C_{X,g}\|f\|_{L_M^{(2)}}$ is established for all compact $d$-dimensional Riemannian manifolds.
• The commutator $[M_f, (1-\Delta_g)^{-d/4}]$ belongs to the quasi-Banach ideal $(\mathcal{L}_{2,\infty})_0$ for $f \in L_M(X, \mathrm{vol}_g)$, enabling the use of spectral approximation techniques.
• The proof avoids ultrafilters by using direct spectral analysis and the Birman-Solomyak inequality in quasi-Banach ideals, providing a constructive derivation of the Connes formula.