Spectral Asymptotics for Quantized Derivatives on Quantum Euclidean Spaces

TL;DR

This work proves sharp spectral asymptotics for quantized derivatives on quantum Euclidean spaces by combining a noncommutative Wiener–Ikehara Tauberian approach with a C*-algebraic pseudo-differential framework. For det$(\theta)\neq0$, the authors show that, on the homogeneous Sobolev space $\\dot{W}^1_d({\mathbb{R}_\theta^d})$, the singular values of the quantized derivative satisfy $\\lim_{n\to\infty} n^{1/d}\\mu(n, {\\char'26\\mkern-12mu d}x)=\\kappa_d \\|||x|||_{\\dot{W}^1_d({\mathbb{R}_\theta^d})}}$, with a newly defined semi-norm controlling the limit. They also extend the asymptotics to a class of pseudo-differential-type operators in the algebra $\\Pi$, showing that for operators compactly supported on the right, $\\lim_{n\to\infty} n^{1/d}\\mu(n, T(1-\\Delta_\theta)^{-1/2})=\\kappa_d \\|\\operatorname{sym}(T)\\|_{L_d( L_\infty({\mathbb{R}_\theta^d}) \\bar{\\otimes} L_\infty(\\mathbb{S}^{d-1}))}$. The paper develops a mixed-trace machinery and an operator zeta-function analysis to realize these results, situating them as a noncommutative analogue of endpoint Sobolev/Schatten-type spectral asymptotics. The outcomes refine the understanding of infinitesimals in Connes’ noncommutative geometry for quantum Euclidean spaces and provide a blueprint for spectral asymptotics in the C*-algebraic calculus on noncommutative manifolds.

Abstract

We obtain spectral asymptotics for the quantized derivatives of elements from the first-order homogeneous Sobolev space on the quantum Euclidean space, extending an earlier result of McDonald, Sukochev and Xiong (Commun. Math. Phys. 2020). Our approach is based on a noncommutative Wiener-Ikehara Tauberian theorem and a recently developed $C^\ast$-algebraic version of pseudo-differential operator theory.

Theorems & Definitions (94)

• Theorem 1.1
• Remark 1.1
• Definition 1.2
• Theorem 1.3
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6