Connes' integration and Weyl's laws

Journal of noncommutative geometry

article

Raphaël Ponge¹ 

[1] Sichuan University

关键词: Noncommutative geometry;    weak Schatten classes;    spectral analysis;   

DOI  :  10.4171/jncg/509

学科分类：神经科学

来源: European Mathematical Society

【 摘 要 】

This paper deals with some questions regarding the notion of integral in the framework of Connes' noncommutative geometry. First, we present a purely spectral theoretic construction of Connes' integral. This answers a question of Alain Connes. We also deal with the compatibility of Dixmier traces with Lebesgue's integral. This answers another question of Alain Connes. We further clarify the relationship of Connes' integration with Weyl's laws for compact operators and Birman–Solomyak's perturbation theory. We also give a “soft proof” of Birman–Solomyak's Weyl's law for negative order pseudodifferential operators on closed manifold. This Weyl's law yields a stronger form of Connes' trace theorem. Finally, we explain the relationship between Connes' integral and semiclassical Weyl's law for Schrödinger operators, including for (fractional) Schrödinger operators on Euclidean spaces and on noncommutative manifolds. We thus get a neat links between noncommutative geometry and semiclassical analysis.

【 授权许可】

CC BY