Weyl's Law and Connes' Trace Theorem for Noncommutative Two Tori

Abstract

We prove the analogue of Weyl's law for a noncommutative Riemannian manifold, namely the noncommutative two torus ${\mathbb{T}}_{\theta }^2$ equipped with a general translation invariant conformal structure and a Weyl conformal factor. This is achieved by studying the asymptotic distribution of the eigenvalues of the perturbed Laplacian on ${\mathbb{T}}_{\theta }^2$. We also prove the analogue of Connes' trace theorem by showing that the Dixmier trace and a noncommutative residue coincide on pseudodifferential operators of order -2 on ${\mathbb{T}}_{\theta }^2$.

17 pages. Derivation of small time heat kernel expansion and the construction of the Dixmier trace is expanded, one reference added, to appear in Letters in Mathematical Physics

Cited by in corpus (4)

• The Theory of Pseudo-differential Operators on the Noncommutative n-Torus
• Laplace-Beltrami Operators on Noncommutative Tori
• Wodzicki residue and minimal operators on a noncommutative 4-dimensional torus
• Connes Trace Theorem for Curved Noncommutative Tori. Application to Scalar Curvature