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CONNES’ TRACE THEOREM FOR CURVED NONCOMMUTATIVE TORI. APPLICATION TO SCALAR CURVATURE RAPHAEL¨ PONGE 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. 1. Introduction The main goal of noncommutative geometry is to translate the classical tools of differential geometry in the Hilbert space formalism of quantum mechanics [8]. In this framework the role of the integral is played by positive (normalized) traces on the weak trace class L 1,8. Important examples of such traces are provided by Dixmier traces [17]. An operator T L 1,8 is measurable (resp., strongly measurable) when the value ϕ T is independent of the traceP ϕ when it ranges p q throughout Dixmier traces (resp., positive normalized traces). The NC integral ´T is then defined as the single value ϕ T . Given any closed Riemannianp q manifold M n,g , Connes’ trace theorem [7, 34]ş asserts that every pseudodifferential operator P of order npon M qis strongly measurable, and we have ´ 1 (1.1) P Res P , ´ “ n p q ż where Res is the noncommutative residue trace of Guillemin [30] and Wodzicki [55, 56]. Let ∆g n ´ 2 8 be the Laplace-Beltrami operator on functions. For operators of the form f∆g , f C M , we further have Connes integration formula [7], P p q n ´ 2 1 ´n n´1 (1.2) f∆g cn fν g , cn : 2π S , ´ “ p q “ np q | | ż żM where ν g is the Riemannian density. In particular, this shows that the noncommutative integral recapturesp q the Riemannian density. arXiv:1912.07113v2 [math.OA] 17 Feb 2020 This paper deals with analogues for noncommutative tori of the trace theorem (1.1) and the integration formula (1.2). Noncommutative 2-tori naturally arise from actions of Z on the circle S1 Tθ by irrational rotations. More generally, an n-dimensional noncommutative torus Aθ C n ˚ “ p q is a C -algebra generated by unitaries U1,...,Un subject to relations, 2iπθjl UlUj e Uj Ul, j,l 1, . , n, “ “ n where θ θjl is some given real anti-symmetric matrix. There is a natural action of R onto “ p q A 8 Tn Aθ. The subalgabra of smooth vectors θ C θ is the smooth noncommutator tori. The 2 n “ p q Hilbert space Hθ L T is obtained as the space of the GNS representation for the standard “ p θ q normalized trace τ of Aθ. The research for this article was partially supported by NSFC Grant 11971328 (China). 1 The pseudodifferential calculus on NC tori of Connes [5] has been receiving a great deal of attention recently due to its role in the early development of modular geometry on NC tori by Connes-Tretkoff [15] and Connes-Moscovici [14] (see also [9, 23] for surveys). There is a noncom- Tn mutative residue trace for integer order ΨDOs on θ (see [24, 22, 38, 46]). This is even the unique trace up to constant multliple [46]. The trace theorem of this paper (Theorem 8.1) states that any pseudodifferential operator P of order n on Tn is strongly measurable, and we have ´ θ 1 (1.3) P Res P , ´ “ n p q ż where on the r.h.s. we have the noncommutative residue trace for NC tori. This refines earlier results of Fathizadeh-Khalkhali [21, 22] in dimensions n 2 and n 4, where measurability and a trace formula are established, but strong measurability“ is not. Th“e trace formula (1.3) can be extended to all operators aP with a Aθ and P as above (see Corollary 8.3). Applying this to n ´ 2 P Tn P ∆ , where ∆ is the flat Laplacian of θ , immediately gives back the Connes integration formula“ for flat NC tori of McDonald-Sukochev-Zanin [43] (see Corollary 8.4). We seek for a curved version of the flat integration formula of [43], i.e., an extension involving Tn the Laplace-Beltrami operator ∆g associated with an arbitrary Riemannian metric g on θ which was recently constructed in [33]. We refer to Section 9 for the precise definition of Riemannian Tn metrics on θ in the sense of [33, 48]. They are given by symmetric positive invertible matrices g gij with entries in Aθ. Any Riemannian metric g defines Riemannian density ν g , which is “ p q p q a positive invertible element of Aθ. The original setting of [14, 15] corresponds to the special case 2 of conformally flat metrics gij k δij , k Aθ, k 0. “ P ą n The Riemannian density ν g defines a weight ϕg a : 2π τ aν g , a Aθ. The very datum of this weight producesp aq non-trivial modular geometry,p q “ p eveq nr whenp qsg isP conformally flat. In particular, the GNS construction for ϕg produces two non-isometric -representations of Aθ ˝ ˚ and its opposite algebra Aθ. The corresponding Tomita involution and modular operator Jg a : 1 1 1 p q “ ´ 2 ˚ 2 ´ H ˝ ν g a ν g and ∆ a : ν g aν g , a Aθ. Thus, if we let g be the Hilbert space p q p q p q “ p˝q p q P ˝ H ˝ of the GNS representation of Aθ, then have a -representation a a of Aθ in θ , where ˝ ˚ ´ 1 1 ˚ Ñ a Jga Jg ν g 2 aν g 2 acts by left multiplication. “ “ p q p q The Laplace-Beltrami operator ∆g : Aθ Aθ of [33] is an elliptic 2nd order differential operator 1 1 Ñ 1 ´ 2 2 2 ij 2 ´1 ij with principal symbol ν g ξ gν g , where ξ g : g ξiξj and g g is the inverse matrix of g. This is alsop anq essentially| | p q selfadjoint| | operator“ p on Hq ˝ with non-negative“ p q spectrum. ř g Our curved integration formula (Theorem 10.6) then states that, for every a Aθ, the operator n ˝ ´ 2 P a ∆g is strongly measurable, and we have n ˝ ´ 2 1 n´1 (1.4) a ∆g cˆnτ aν˜ g , cˆn : S , ´ “ p q “ n| | ż n´1 ´1 ´n n´1 “ ‰ whereν ˜ g : S ´1 ξ d ξ is the so-called spectral Riemannian density (see Sec- p q “ | | Sn | |g tion 10). ş The spectral Riemannian densityν ˜ g enjoys many of the properties of the Riemannian density ν g (see Proposition 10.1). In general,p qν g andν ˜ g need not agree, but they do agree when g p q p q p q is self-compatible, i.e., the entries gij pairwise commute with each other (loc. cit.). For instance, conformally flat metrics and the functional metrics of [28] are self-compatible. In the self-compatible case, the integration formula (1.4) shows that the NC integral recaptures the Riemannian weight ϕg (see Corollary 10.8). This provides us with a neat link between the NC integral in the sense of noncommutative geometry, and the noncommutative measure theory in the sense of operator algebras, where the role of Radon measures is played by weights on C˚-algebras. In the same way as with ordinary (compact) manifolds the trace formula (1.3) allows us to Tn extend the NC integral to all ΨDOs on θ even those that are not weak trace class or even bounded (see Definition 11.1). Together with the curved integration formula this allows us to n 1 1 1 ´1 ´ 2 ´ n ´ 2 setup a quantum length element ds : cn ∆g n cn ∆g and spectral k-dimensional volumes pkq “ p q “ Tn k Volg θ : ´ds for k 1,...,n (see Section 11). p q “ “ 2 ş Ą For any closed Riemannian manifold M n,g , n 3, we have p q ě (1.5) fdsn´2 c1 κ g ν g f C8 M , ´ “´ n p q p q @ P p q ż żM 1 where κ g is the scalar curvature and cn is some (positive) universal constant. Given any curved p qn NC tori T ,g , n 3, the scalar curvature is the unique ν g Aθ such that p θ q ě p qP 1 n´2 ads τ aκ g a Aθ. ´c1 ´ “ p q @ P n ż “ ‰ This is consistent with previous approaches to the scalar curvature for NC tori. When θ 0 we recover the usual notion of scalar curvature. “ The scalar curvature κ g is naturally expressed in terms of some integral of the symbol of n 1 1 1 ´ 2 ` p q ´ degree n of ∆g , where ∆g ν g 2 ∆ν g 2 is the Laplace-Beltrami operator ∆g under the ´ n 1“ p q˝ p q unitary isomorphism 2π 2 ν g 2 : H Hθ (see Proposition 11.2).

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