Connes’s 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 ě 2. This allows us to recover and improve earlier versions of this result in dimension n " 2 and n " 4 by Fathizadeh-Khalkhali [21,22]. We also recover the Connes integration formula for flat noncommutative tori of McDonald-Sukochev-Zanin [43]. 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.

Dixmier Trace Formulas and Negative Eigenvalues of Schroedinger Operators on Curved Noncommutative Tori

“…Regarding general ΨDOs, the Cwikel-type estimate (1.3) shows that if x ∈ L 2 (T n θ ) and P is a (classical) ΨDO of order −n, then the operator λ(x)P is in the weak-trace class L 1,∞ . In fact, the results of [64] and the Cwikel-type estimate (1.3) enable us to show further that λ(x)P is strongly measurable, and we have (1.9) − λ(x)P = 1 n τ xν P , where ν P :=…”

“…In this follow up paper we extend the results of [60] to pseudodifferential operators and to curved noncommutative tori, where the role of the usual Laplacian is played by Laplace-Beltrami operators associated with arbitrary densities and Riemannian metrics. Furthermore, we obtain L 2 and L 1 + versions of the integration formulas of [62,64].…”

“…Connes' integration formulas (1.7)-(1.8) were extended to NC tori in [64]. A further goal of this paper is to obtain L 2 and L 1 + versions of these results, where L 1 + (T n θ ) = p>1 L p (T n θ ).…”

Noncommutative Residue and Canonical Trace on Noncommutative Tori. Uniqueness Results

“…where θ = (θ jl ) is a given real anti-symmetric matrix. The pseudodifferential calculus on NC tori of Connes [3,4,5,28,29] has been receiving a great deal of attention recently thanks to its role in the spectral theoretic approach to curvature on noncommutative tori following the seminal work of Connes-Tretkoff [13] and Connes-Moscovici [12] (see, e.g., [14,16,17,18,19,23,30,36,41,42,43,49,53]; see also [9,20,37] for recent surveys).…”

“…It was extended to any dimension n ≥ 2 by Lévy-Neira-Paycha [40] who also constructed the canonical trace. We refer to [16,49] for the relationship between noncommutative residue and scalar curvature on NC tori.…”