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 {{T}_{θ}^{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 {{T}_{θ}^{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 {{T}_{θ}^{2}}.
 Publication:

Letters in Mathematical Physics
 Pub Date:
 January 2013
 DOI:
 10.1007/s1100501205932
 arXiv:
 arXiv:1111.1358
 Bibcode:
 2013LMaPh.103....1F
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematics  Differential Geometry;
 Mathematics  Operator Algebras
 EPrint:
 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