Connes Trace Theorem for Curved Noncommutative Tori. Application to Scalar Curvature

Abstract

In this paper we prove a version of Connes' trace theorem for noncommutative tori of any dimension~$n\ge 2$. This allows us to recover and improve earlier versions of this result in dimension $n=2$ and $n=4$ by Fathizadeh-Khalkhali. We also recover the Connes integration formula for flat noncommutative tori of McDonald-Sukochev-Zanin. As a further application we prove a curved version of this integration formula in terms of the Laplace-Beltrami operator defined by an arbitrary Riemannian metric. For the class of so-called self-compatible Riemannian metrics (including the conformally flat metrics of Connes-Tretkoff) this shows that Connes' noncommutative integral allows us to recover the Riemannian density. This exhibits a neat link between this notion of noncommutative integral and noncommutative measure theory in the sense of operator algebras. As an application of these results, we setup a natural notion of scalar curvature for curved noncommutative tori.

v2: minor changes

References in corpus (8)

• The Gauss-Bonnet Theorem for the noncommutative two torus
• Traces of compact operators and the noncommutative residue
• Noncommutative geometry and lower dimensional volumes in Riemannian geometry
• A survey of spectral models of gravity coupled to matter
• Laplace-Beltrami Operators on Noncommutative Tori
• Noncommutative Residue and Canonical Trace on Noncommutative Tori. Uniqueness Results
• Spectral geometry of functional metrics on noncommutative tori
• Local invariants of non-commutative tori

Cited by in corpus (2)

• Spectral Metric and Einstein Functionals
• Cwikel Estimates and Negative Eigenvalues of Schroedinger Operators on Noncommutative Tori