Pseudodifferential calculus on noncommutative tori, II. Main properties

Abstract

This paper is the 2nd part of a two-paper series whose aim is to give a detailed description of Connes' pseudodifferential calculus on noncommutative n-tori, n ě 2. We make use of the tools introduced in the 1st part to deal with the main properties of pseudodifferential operators on noncommutative tori of any dimension n ě 2. This includes the main results mentioned in [2,5,11]. We also obtain further results regarding action on Sobolev spaces, spectral theory of elliptic operators, and Schatten-class properties of pseudodifferential operators of negative order, including a trace-formula for pseudodifferential operators of order ă´n.1¨¨¨δ αn n (see [5,8]). For instance, the Laplacian ∆ " δ 2 1`¨¨¨`δ 2 n is such an operator. In the 1st part [35] (referred thereafter as Part I) we have introduced an oscillating integral for A θ -amplitudes and have used it to give a precise explanation of the definition of pseudodifferential 2010 Mathematics Subject Classification. 58B34, 58J40.

A $$C^*$$ C ∗ -algebraic approach to the principal symbol II

“…From there it is easy to see that h ∈ ℓ 1 (L ∞ )(R d ). This completes the proof of (18). As ϕ vanishes on L 1 , we may use (18) with y = W * g xW g to obtain in (17) to obtain, τ θ (x)l(b) = ϕ(π 1 (W * g xW g )π 2 (V g b)(1 − ∆) −d/2 ) so by Lemma 6.13: τ θ (x)l(b) = τ θ (W * g xW g )l(V g b).…”

Connes’s trace theorem for curved noncommutative tori: Application to scalar curvature

“…In this paper we will only need the following results. Proposition 4.12 (see [32]). Let ρpξq P S m pR n ; A θ q, m ď 0.…”

“…Proposition 4.10 (see [2,5,32]). Let ρ 1 pξq P S m1 pR n ; A θ q and ρ 2 pξq P S m2 pR n ; A θ q, m 1 , m 2 P R.…”

“…A detailed account on spectral theoretic properties of ΨDOs is given in [32]. In this paper we will only need the following results.…”

“…As stated above Proposition 4.13 is part of the contents of[32, Proposition 13.8] for p ą 1 only. However, as pointed out in[32, Remark 13.12], the result holds verbatim for any quasi-Banach ideal I such that ∆m 2 P I , where ∆ " δ 2 1`¨¨¨`δ is the flat Laplacian. As µ k p∆ m 2 q " Opk m 2 q " Opk´1 p q, the result holds for L p,8 with p ď 1.…”