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

2011/11/30 by Farzad Fathizadeh, Masoud Khalkhali · 1 citation
Mathematics · #math.DG #math.OA #math.QA

paper · pdf · doi:10.1007/s11005-012-0593-2

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

arxiv created 2012/11/06 · arxiv updated 2015/06/03

Abstract

We prove the analogue of Weyl's law for a noncommutative Riemannian manifold, namely the noncommutative two torus \mathbbT_θ² 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 \mathbbT_θ². 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 \mathbbT_θ².

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