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[55, odzicki , T cltoso differential of tools ical f ∆ P ´ g L n 2 ´ ş L ϕ 1 uaieintegral mutative , T , i result his 8 f 1 hni ranges it when Z , ste defined then is 8 P smeasurable is ntecircle the on A Important . C θ 8 es f “ p R M n C q p onto we , T n θ rd g q 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). To relate our description p q p q g Ñ to the scalar curvaturep to previousp descriptions we use the results of [36] on complex powers of positive elliptic ΨDOs on NC tori. Namely, we re-express κ g in terms of the 2nd sub-leading ´1 p q symbol σ´4 ξ; λ of the resolvent ∆g λ (Proposition 11.7). This provides us with a solid notion of scalarp q curvature for curvedp ´ NCq tori of dimension 3. In addition, we recover the ě modular scalar curvature of NC 2-torip of [14, 15] by analytic continuation of the dimension (see Remark 11.10). This paper is organized as follows. In Section 2, we review the main facts on Connes’ quantized calculus and the noncommutative integral. In Section 3, we review the main definitions and prop- erties of noncommutative tori. In Section 4, we further review the main definitions and properties of pseudodifferential operators on noncommutative tori. In Section 5, we survey the construction of the noncommutative residue trace for ΨDOs on NC tori. ln Section 6, we establish that any ΨDO agrees with a scalar-symbol ΨDO modulo the closure of the weak trace class commutator subspace. In Section 7, we establish a Weyls law for positive ΨDOs with scalar symbols. In Section 8, we prove the version for noncommutative tori of Connes’ trace theorem and give a couple of applications. In Section 9, we review the main definitions and properties of Riemannian metrics and Laplace-Beltrami operators. In Section 10, we prove the curved integration formula and present a few consequences. In Section 11, we set up a natural notion of scalar curvature for curved NC tori. In Appendix A we sketch a proof of (1.5). In Appendix B we compare our integration formulas with the curved integration formula of [42].

Acknowlegements. My warmest thanks go Edward McDonald, Fedor Sukochev, and Dmitriy Zanin for stimulating and passionate discussions on Connes’ integration formula. I thank Galina Levitina and Alex Usachev for providing a reference for Proposition 2.2. I also thank University of New South Wales (Sydney, Australia) and University of Qu´ebec (Montr´eal, Canada) for their hospitality during the preparation of the paper.

2. Quantized Calculus and Noncommutative Integral In this section, we review the main facts on Connes’ quantized calculus and the construction of the noncommutative integral.

2.1. Quantized calculus. The main goal of the quantized calculus of Connes [8] is to translate into a the Hilbert space formalism of quantum mechanics the main tools of the infinitesimal and integral calculi. In what follows, we let H be a (separable) Hilbert space. We mention the first few lines of Connes’ dictionary. 3 Classical Quantum Complex variable Operator on H Real variable Selfadjoint operator on H Infinitesimal variable Compact operator on H Infinitesimal of order α 0 Compact operator T such that ą ´α µk T O k p q“ p q The first two lines arise from quantum mechanics. Intuitively speaking, an infinitesimal is meant to be smaller than any real number. For a bounded operator the condition T ǫ for all ǫ 0 gives T 0. This condition can be relaxed into } }ă ą “ ǫ 0 E H such that dim E and T K ǫ. @ ą D Ă ă8 } |E }ă This condition is equivalent to T being a compact. Denote by K the closed two-sided ideal of compact operator. The order of compactness of an operator T K is given by its characteristic values (a.k.a. singular values), P µk T inf T K ; dim E k p q“ } |E } “ k 1 -th eigenvalue of( T ?T ˚T. “ p ` q | |“ The last line is the min-max principle. Each eigenvalue is counted according to multiplicity. We record the main properties of the characteristic values (see, e.g., [27, 53]),

˚ (2.1) µk T µk T µk T , µk l S T µk S µl T , p q“ p q“ p| |q ` p ` qď p q` p q ˚ (2.2) µk AT B A µk T B , µk U TU µk T , p qď} } p q} } p q“ p q where S,T K , A, B L H and U L H is any unitary operator. P P p q P p q ´α An infinitesimal operator of order α 0 is any compact operator such that µk T O k . For p 0 the weak Schatten class L p,8ąis defined by p q“ p q ą 1 p,8 ´ L : T K ; µk T O k p . “ P p q“ p q ! ) Thus, T is an infinitesimal operator of order α if and only if T L p,8 with p α´1. We refer to [40], and the references therein, for background onP the weak Schatten“ classes L p,8. In particular, they are quasi-Banach ideals (see, e.g., [52]) with quasi-norms,

1 p,8 T p,8 sup k 1 p µk T ,T L . } } “ kě0p ` q p q P For p 1 this quasi-norm is equivalent to a norm, and so L p,8 is actually a Banach ideal. In generalą (see, e.g., [52]), we have

1 p,8 (2.3) S T p, 2 p S p, T p, , S,T L . } ` } 8 ď p} } 8 `} } 8q P For p 1 the weak trace class L 1,8 is strictly contained in the Dixmier-Macaev ideal, “

p1,8q L : T K ; µk T O log N . “ P p q“ p q kăN " ÿ * The Dixmier-Macaev ideal is a Banach ideal. It is also called the weak trace class by various authors. We follow the convention of [40] where, under Calkin’s correspondance, L p,8 is the 1 p p,8 N0 ´ image of the weak-ℓ sequence space ℓ ak C ; ak O k p . We also record the inclusions, “ p qP | |“ p q ( L p L p,8 L 1 L 1,8 L p1,8q, p 1. Ă Ă Ă Ă ą All these inclusions are strict. 4 2.2. Noncommutative integral. We seek for an analogue of the notion of integral in the setting of the quantized calculus. As an Ansatz this should be a linear functional satisfying the following conditions: (I1) It is defined on all infinitesimal operators of order 1. (I2) It vanishes on infinitesimal operators of order 1. (I3) It takes non-negative values on positive operators.ą (I4) It is invariant under Hilbert space isomorphisms. The conditions (I1) and (I4) lead us to seek for a trace on L 1,8, where by a trace we mean any linear form that is annihilated by the commutator subspace, Com L 1,8 : Span A, T ; A L H ,T L 1,8 . “ r s P p q P 1 It can be shown that any` trace˘ on L ,8 vanishes on trace-class operators( (see Proposition 2.1 below). As every infinitesimal operators of order 1 is trace-class, the condition (I2) is always satisfied by traces on L 1,8 (see Proposition 2.1 below).ą The condition (I3) further requires the NC integral to be positive. Therefore, we are seeking for a positive trace on the quasi-Banach space L 1,8. There is a whole zoo of traces on L 1,8. We refer to [40, 41], and the references therein, for a detailed account on traces on L 1,8 and the commutator subspace Com L 1,8 . We will only need the following two results. p q Proposition 2.1 ([19, 40]). Every trace-class operator is contained in Com L 1,8 . In particular, every trace on L 1,8 is annihilated by L 1. p q Proposition 2.2. Every positive trace on L 1,8 is continuous. Remark 2.3. Any positive linear form on a C˚-algebra is continuous (see [44, Theorem 3.3.1]). The same type argument shows that on any positive linear form on L 1,8 is continuous. Remark 2.4. Proposition 2.2 has a converse. Namely, every continuous trace on L 1,8 is a linear combination of 4 positive traces (see [11, Corollary 2.2]). Therefore, the positive traces on L 1,8 span the space of continuous traces. A well-known example of positive trace is the Dixmier trace [17]. We briefly present its con- L 1,8 struction by following [13, Appendix A]. In what follows we denote by ` the cone of positive 1 operators in L 1,8. Given any T L ,8, for u 0 and λ e, we set P ` ą ě s λ 1 σu T du σu T : µrss t ds, τλ T p q . p q “ 0 p q p q“ log λ log u u ż że L 1,8 Note that σN T kăN µk T for every integer N 1. The fact that T ` ensures us that p q“ p q ˚ ě P the function λ τλ T is in the C -algebra Cb e, of bounded continuous functions on e, . Ñ řp q r 8q r 8q Denote by C0 e, the (closed) ideal of continuous functions vanishing at . It can be shown r 8q 1, 8 that, for all S,T L 8, we have P ` (2.4) τλ S T τλ T τλ S mod C0 e, . p ` q“ p q` p q r 8q ˚ Let ω be a state on the quotient C -algebra A : Cb e, C0 e, (i.e., ω is a positive linear functional such that ω 1 1). Then (2.4) ensures“ usr 8q{ that wer define8q an additive functional 1,8 p q “ ϕω : L 0, by letting ` Ñr 8q 1,8 ϕw T ω τ T ,T L , p q“ p q P ` where τ T is the class of λ τλ T in`“ A. This‰˘ uniquely extends to a positive linear trace r1,8p qs Ñ p q ϕω : L C. This trace is called the Dixmier trace associated with ω. We note that, for every 1,8 Ñ T L , we have P ` 1 (2.5) lim µk T L ϕω T L. NÑ8 log N p q“ ùñ p q“ kăN ÿ In particular, in this case the value of ϕω T is independent of the choice of ω. p q 5 1,8 Definition 2.1 (Connes [8]). An operator T L is called measurable when the value of ϕω T P p q is independent of the choice of the state ω. We then define the noncommutative integral ´T by

T : ϕω T , ω any state on Q e, . ş ´ “ p q r 8q ż L 1,8 ´1 Let T0 be any operator on ` such that µk T0 k 1 for all k 0, i.e., there is an p q “ p´1` q ě orthonormal basis ξk kě0 of H such that T0ξk k 1 ξk for all k 0. It follows from (2.5) p q “ p ` q ě 1,8 that ϕω T0 1 for every state ω. More generally, we say that a trace ϕ on L is normalized p q“ when ϕ T0 1. This condition does not depend on the choice of T0. Therep areq“ positive normalized traces on L 1,8 that are not Dixmier traces. In fact, there are even positive normalized traces on L 1,8 that do not extend to the Dixmier-Macaev ideal L p1,8q (see [51, Theorem 4.7]). Therefore, it stands for reason to consider a stronger notion of measurability. Definition 2.2. An operator T L 1,8 is called strongly measurable when there is L C such that ϕ T L for every normalizedP positive trace ϕ on L 1,8. We then define the noncommutativeP p q“ integral ´T by

1 ş T : ϕ T , ϕ any normalized positive trace on L ,8. ´ “ p q ż Remark 2.5. The class of strongly measurable operators is strictly contained in the class of mea- surable operators. We refer to [51, Theorem 7.4] for an example of measurable operator that is not strongly measurable. 2.3. Connes’ trace theorem. Let M n,g be a closed Riemannian manifold. We refer to [35, Chapter 16] for background on smoothp densitiesq on manifolds and their integrals. The Riemannian 2 density ν g is given in local coordinates by integration against g x . Let Lg M be Hilbert p2 q p q p q space of L -functions on M equipped with the inner product defined by ν g , i.e., a p q u v uvν g , u,v L2 M . x | yg “ p q P gp q żM Let Ψm M , m Z, be the space of (classical) m-th order pseudodifferential operators P : C8 M p Cq8 M P. Any P Ψm M with m 0 uniquely extends to a bounded operator p 2 q Ñ p2 q P p q mď L p,8 P : Lg M Lg M . When m 0, any P Ψ M is in the weak Schatten class with p q1 Ñ p q ă P p q 1 p : n m ´ , i.e., P is an infinitesimal operator of order m n´ . In particular, we get weak trace-class“ | | operators when m n. ě | | Z m “´ L 8 Set Ψ M mPZ Ψ M ; this is a subalgebra of C M . The noncommutative residue p q“Z p q p p qq trace Res : Ψ M C of Guillemin [30] and Wodzicki [55] is defined by p Ťq Ñ Z (2.6) Res P cP x , P Ψ M , p q“ p q P p q żM where cP x is the smooth density on M given in local coordinates by p q ´n n´1 cP x 2π p´n x, ξ d ξ, p q “ p q Sn´1 p q ż where p n x, ξ is the homogeneous symbol of degree n of P . ´ p q ´ Theorem 2.6 (Connes’ trace theorem [7, 34]). Every operator P Ψ´n M is strongly measur- able, and we have P p q 1 (2.7) P Res P . ´ “ n p q ż Remark 2.7. Connes [7] only considered Dixmier traces, so he only established the measurablity of operators in P Ψ´n M along with the trace formula (2.7). The strong measurability is a consequence of theP resultsp ofq [34], where the trace formula is established for every trace ϕ on L 1,8. In fact, only a slight elaboration of Connes’ original proof is needed in order to get strong measurability. 6 Remark 2.8. We refer to [34, Corollary 7.23] for an example of non-classical pseudodifferential operator that is not measurable. Together with the formula (2.6) the trace formula (2.7) allows us to compute the NC integral of any P Ψ´n M from the sole knowledge of its principal symbol in local coordinates. As an applicationP thisp allowsq us to recover the Riemannian density ν g x as follows. 8 8 p qp q Let ∆g : C M C M be the (positive) Laplace-Beltrami operator. This is an selfadjoint n p q Ñ p q ´ 2 elliptic 2nd order differential operator with non-negative spectrum. The power ∆g is an operator ´n ´n in Ψ M with principal symbol ξ g , where g is the Riemannian metric on the cotangent p q | | | ¨ | n ˚ 2 ij ´ 2 8 bundle T M (i.e., ξ g g ξiξj ). By applying the trace formula (2.7) to f∆g , f C M , we arrive at the following| | “ result. P p q ř n 8 ´ 2 Theorem 2.9 (Connes integration formula [7]). For every f C M , the operator f∆g is strongly measurable, and we have P p q

n ´ 2 1 ´n n´1 (2.8) f∆g cn fν g , where cn : 2π S . ´ “ p q “ np q | | ż żM Remark 2.10. Connes’ integration formula even holds for any f x L2 M (cf. [34, 39]). p qP gp q One upshot of the trace formula (2.7) is the extension of the NC integral to all ΨDOs, including ΨDOs that need not be weak trace class or even bounded. Namely, as the noncommutative residue is defined for all ΨDOs, it is natural to use the r.h.s. of (2.7) as a definition of the NC integral for ΨDOs. That is, we set 1 (2.9) P : Res P for any P ΨZ M . ´ “ n p q P p q ż The integration formula (2.8) shows that the NC integral recaptures the Riemannian density. n ´1 ´ 2 Equivalentely, this allows us to interpret cn ∆g a quantum volume element. Thinking of the length element as the n-th root of the volume element, this leads us to interpret the operator n 1 1 1 ´1 ´ 2 ´ n ´ 2 k ds : cn ∆g n cn ∆g as a quantum length element. It is then natural to interpret ´ds as a“k-dimensional p q “ volume for k 1,...,n. Note this uses the extension of the NC integral to all ΨDOs since dsk is a ΨDO of order“ k n when k n. We refer to [45] for an elaborationş of this line of thought in terms of the Dirac´ ą´ operator. ă When k n 2 and n 3 we obtain “ ´ ě (2.10) fdsn´2 c1 fκ g ν g for all f C8 M , ´ “´ n p q p q P p q ż żM where κ g is the scalar curvature and c1 is a universal constant given by p q n ´2 1 n n n n π 2 c n . (2.11) 1 3 4 n Γ 1 cn “ p q 2 ´ This result is part of folklore. For reader’s convenience` a proof˘ is included in Appendix A. When f x 1 the r.h.s. of (2.10) agrees up to a constant with the Einstein-Hilbert action. This providesp q us ” with quantum interpretations of the Einstein-Hilbert action and of the scalar cur- vature. These interpretations lie at the very roots of the spectral action principal of Chamseddine- Connes [3] and the concept of modular scalar curvature of NC tori by Connes-Tretkoff [15] and Connes-Moscovici [14] (see also [4, 9, 23] for recent surveys of those topics).

3. Noncommutative Tori In this section, we review the main definitions and properties of noncommutative n-tori, n 2. We refer to [8, 31, 47], and the references therein, for a more comprehensive account. ě Throughout this paper, we let θ θjk be a real anti-symmetric n n-matrix. Denote by “ p q 2 n ˆ 2 θ1,...,θn its column vectors. We also let L T be the Hilbert space of L -functions on the ordinary torus Tn Rn 2πZn equipped with thep innerq product, “ { (3.1) ξ η 2π ´n ξ x η x dx, ξ, η L2 Tn . x | y “ p q Tn p q p q P p q ż 7 2 n 2 n For j 1,...,n, let Uj : L T L T be the unitary operator defined by “ p q Ñ p q ixj 2 n Ujξ x e ξ x πθj , ξ L T . p q p q“ p ` q P p q We then have the relations,

2iπθjk (3.2) UkUj e UjUk, j, k 1, . . . , n. “ “ Tn ˚ The noncommutative torus Aθ C θ is the C -algebra generated by the unitary operators “ p ˚ q n U1,...,Un. For θ 0 we obtain the C -algebra C T of continuous functions on the ordinary n “ p q 2 n n-torus T . Note that (3.2) implies that Aθ is the closure in L L T of the linear span of the unitary operators, p p qq k k1 kn n U : U1 U , k k1,...,kn Z . “ ¨ ¨ ¨ n “ p qP Let τ : L L2 Tn C be the state defined by the constant function 1, i.e., p p qq Ñ τ T T 1 1 T 1 x dx, T L L2 Tn . p q“x | y“ Tn p qp q P p q ż ˚ ` ˘ k This induces a continuous tracial state on the C -algebra Aθ such that τ 1 1 and τ U 0 p q “ p q “ for k 0. The GNS construction then allows us to associate with τ a -representation of Aθ as follows.‰ ˚ Let be the sesquilinear form on Aθ defined by x¨|¨y ˚ (3.3) u v τ uv , u,v Aθ. x | y“ p q P Note that the family U k; k Zn is orthonormal with respect to this sesquilinear form. We 2 n t P u 0 let Hθ L T the Hilbert space arising from the completion of A with respect to the pre- “ p θ q θ inner product (3.3). The action of Aθ on itself by left-multiplication uniquely extends to a - 2 n ˚ representation of Aθ in Hθ. When θ 0 we recover the Hilbert space L T with the inner “ n p q k product (3.1) and the representation of C T by bounded multipliers. In addition, as U k Zn p q p q P is an orthonormal basis of Hθ, every u Hθ can be uniquely written as P k k (3.4) u ukU , uk : u U , “ “ | kPZn ÿ @ D where the series converges in Hθ. When θ 0 we recover the Fourier series decomposition in L2 Tn . “ pTheq natural action of Rn on Tn by translation gives rise to an action on L L2 Tn . This p p qq induces a -action s,u αs u on Aθ given by ˚ p q Ñ p q k is¨k k n n αs U e U , for all k Z and s R . p q“ P P ˚ n This action is strongly continuous, and so we obtain a C -dynamical system Aθ, R , α . We are A 8 Tn p ˚ q especially interested in the subalgebra θ C θ of smooth elements of this C -dynamical system (a.k.a. smooth noncommutative torus“). Namely,p q

8 n Aθ : u Aθ; αs u C R ; Aθ . “ P p qP p q " * k n The unitaries U , k Z , are contained in Aθ, and so Aθ is a dense subalgebra of Aθ. Denote by S Zn the space ofP rapid-decay sequences with complex entries. In terms of the Fourier series decompositionp q (3.4) we have

k n Aθ u ukU ; uk k Zn S Z . “ “ p q P P p q kPZn " ÿ * When θ 0 we recover the algebra C8 Tn of smooth functions on the ordinary torus Tn and the Fourier-series“ description of this algebra.p q For j 1,...,n, let δj : Aθ Aθ be the derivation defined by “ Ñ A δj u Dsj αs u s“0, u θ, p q“ p q|8 P 1 1 B where we have set Ds s . When θ 0 the derivation δj is just the derivation Dx j “ i B j “ j “ i Bxj on C8 Tn . In general, for j, l 1,...,n, we have p q “ Uj if l j, δj Ul “ p q“ 0 if l j. # ‰ n More generally, given any multi-order β N0 , define P β β β1 βn δ u D αs u s 0 δ1 δ u , u Aθ. p q“ s p q| “ “ ¨ ¨ ¨ n p q P We endow Aθ with the locally convex topology defined by the semi-norms, β n (3.5) Aθ u δ u , β N0 . Q ÝÑ p q P With the involution inherited from Aθ this turns› A›θ into a (unital) Fr´echet -algebra. The Fourier › › ˚ series (3.4) of every u Aθ converges in Aθ with respect to this topology. In addition, it can be P shown that Aθ is closed under holomorphic functional calculus (see, e.g., [6, 31]).

4. Pseudodifferential Operators on Noncommutative Tori In this section, we review the main definitions and properties of pseudodifferential operators on noncommutative tori.

4.1. Symbol classes. There are various classes of symbols on noncommutative tori.

m n Definition 4.1 (Standard Symbols; see [2, 5]). S R ; Aθ , m R, consists of maps ρ ξ 8 n p q P p q P C R ; Aθ such that, for all multi-orders α and β, there exists Cαβ 0 such that p q ą α β m´|β| n δ ρ ξ Cαβ 1 ξ ξ R . } Bξ p q} ď p ` | |q @ P n 8 n Definition 4.2. S R ; Aθ consists of maps ρ ξ C R ; Aθ such that, for all N 0 and p q p q P p q ě multi-orders α, β, there exists CNαβ 0 such that ą α β ´N n δ ρ ξ CNαβ 1 ξ ξ R . } Bξ p q} ď p ` | |q @ P n m n Remark 4.1. S R ; Aθ R S R ; Aθ . p q“ mP p q n 8 n Definition 4.3 (HomogeneousŞ Symbols). Sq R ; Aθ , q C, consists of maps ρ ξ C R 0; Aθ that are homogeneous of degree q, i.e., ρ tξ p tqρ ξq forP all ξ Rn 0 and t p0.qP p z q p q“ p q P z ą Rn A 8 Rn Remark 4.2. If ρ ξ Sq ; θ and χ ξ Cc is such that χ ξ 1 near ξ 0, then pℜqq P n p q p q P p q p q “ “ 1 χ ξ ρ ξ S R ; Aθ . p ´ p qq p qP p q q n 8 n Definition 4.4 (Classical Symbols; see [2]). S R ; Aθ , q C, consists of maps ρ ξ C R ; Aθ that admit an asymptotic expansion, p q P p qP p q n ρ ξ ρq´j ξ , ρq´j Sq´j R ; Aθ , p q„ jě0 p q P p q ÿ where means that, for all N 0 and multi-orders α, β, there exists CNαβ 0 such that, for all ξ Rn„with ξ 1, we have ě ą P | |ě α β ℜq´N´|β| (4.1) δ ρ ρq j ξ CNαβ ξ . Bξ ´ ´ p q ď | | › jăN › › ` ÿ ˘ › q n ℜq n Remark 4.3. S R ; Aθ › S R ; Aθ . › p qĂ p q α m n Example 4.4. Any polynomial map ρ ξ aαξ , aα Aθ, m N0, is in S R ; Aθ . p q“ |α|ďm P P p q Remark 4.5. It is also convenient to considerř scalar-valued symbols. In this case we denote by m n n n q n S R , S R , Sq R and S R the corresponding classes of scalar-valued symbols. We will p q p q p q p q consider them as sub-classes of the Aθ-valued symbol classes via the natural embedding of C into Aθ. 9 m n 4.2. Pseudo-differential operators. Given ρ ξ S R ; Aθ , m R. We let Pρ : Aθ Aθ be the linear operator defined by p q P p q P Ñ

´n is¨ξ (4.2) Pρu 2π e ρ ξ α s u dsdξ, u Aθ. “ p q p q ´ p q P ij The above integral is meant as an oscillating integral (see [31]). Equivalently, for all u k A “ kPZn ukU in θ, we have k (4.3)ř Pρu ukρ k U . “ p q kPZn ÿ In any case, this defines a continuous linear operator Pρ : Aθ Aθ (see [31]). Ñ q Tn C A A Definition 4.5. Ψ θ , q , consists of all linear operators Pρ : θ θ with ρ ξ in q n p q P Ñ p q S R ; Aθ . p q q n Remark 4.6. If P Pρ with ρ ξ in S R ; Aθ , ρ ξ ρq´j ξ , then ρ ξ is called a symbol “ p q p q p q „ n p q p q n for P . This symbol is not unique, but its restriction to Z and its class modulo S R ; Aθ are ř p q unique (see [31]). As a result, the homogeneous symbols ρq j ξ are uniquely determined by P . ´ p q The leading symbol ρq ξ is called the principal symbol of P . p q A α A Example 4.7. A differential operator on θ is of the form P |α|ďm aαδ , aα θ (see [5, 8]). α “ P This is a ΨDO of order m with symbol ρ ξ aαξ (see [31]). p q“ ř A 1 A A Let θ be the topological dual of θ equippedř with its strong dual topology. Note that θ 1 1 embeds into A as a dense subspace (see, e.g., [31]). A linear operator R : Aθ A is called θ Ñ θ smoothing when its range is contained in Aθ and it uniquely extends to a continuous linear map 1 ´8 n R : A Aθ. We denote by Ψ T the space of smoothing operators. θ Ñ p θ q Proposition 4.8 ([31, Proposition 6.30]). A linear operator R : Aθ Aθ is smoothing if and n Ñ only if this the ΨDO associated with some symbol in S R ; Aθ . p q n q n ´8 n q n Remark 4.9. As S R ; Aθ C S R ; Aθ , we see that Ψ T C Ψ T . p q“ qP p q p θ q“ qP p θ q m1 n 4.3. Composition of ΨDOs.ŞSuppose we are given symbols ρ1 ξ S ŞR ; Aθ , m1 R, and Sm2 Rn A R p q PA p q P ρ2 ξ ; θ , m2 . As Pρ1 and Pρ2 are linear operators on θ, the composition Pρ1 Pρ2 p qP p q P n makes sense as such an operator. In addition, we define the map ρ1 ρ2 : R Aθ by 7 Ñ ´n it¨η n (4.4) ρ1 ρ2 ξ 2π e ρ1 ξ η α t ρ2 ξ dtdη, ξ R , 7 p q “ p q p ` q ´ r p qs P ij where the above integral is meant as an oscillating integral (see [2, 32]).

m1 n m2 n Proposition 4.10 (see [2, 5, 32]). Let ρ1 ξ S R ; Aθ and ρ2 ξ S R ; Aθ , m1,m2 R. p qP p q p qP p q P m1`m2 n 1 α α (1) ρ1 ρ2 ξ S R ; Aθ , and we have ρ1 ρ2 ξ ρ1 ξ δ ρ2 ξ . 7 p qP p q 7 p q„ α! Bξ p q p q (2) The operators Pρ1 Pρ2 and Pρ1 ρ2 agree. 7 ř qi Corollary 4.11. For i 1, 2, let Pi Ψ Aθ , qi C, have principal symbol ρi ξ . Then the “ q1P`q2 p q P p q composition P1P2 is an operator in Ψ Aθ and has principal symbol ρ1 ξ ρ2 ξ . p q p q p q 4.4. Boundedness and infinitesimalness. A detailed account on spectral theoretic properties of ΨDOs is given in [32]. In this paper we will only need the following results. m n Proposition 4.12 (see [32]). Let ρ ξ S R ; Aθ , m 0. Then the operator Pρ uniquely p q P p q ď extends to a bounded operator Pρ : Hθ Hθ. We obtain a compact operator when m 0. Ñ ă q n This result allows us to identify Ψ T with a subspace of L Hθ when ℜq 0. p θ q p q ď Proposition 4.13 ([32]). Let m 0 and set p n m ´1. m n ă “ | | p,8 (1) For every ρ ξ S R ; Aθ the operator Pρ is in L , i.e., this is an infinitesimal of 1 p q P p q order p . ď m n p,8 (2) This provides us with a continuous linear map S R ; Aθ ρ ξ Pρ L . 10 p qQ p q Ñ P Remark 4.14. As stated above Proposition 4.13 is part of the contents of [32, Proposition 13.8] for p 1 only. However, as pointed out in [32, Remark 13.12], the result holds verbatim for any m 2 2 ą I 2 I quasi-Banach ideal such that ∆ , where ∆ δ1 δn is the flat Laplacian. As 1 m m ´ P p,8“ `¨¨¨` µk ∆ 2 O k 2 O k p , the result holds for L with p 1. p q“ p q“ p q ď Corollary 4.15. Every operator P Ψq Tn with ℜq n is trace-class. P p θ q ă´ Corollary 4.16. Every operator P Ψ´n Tn is in the weak trace-class L 1,8. P p θ q 5. Noncommutative Residue In this section, we survey the construction of the noncommutative residue trace for ΨDOs on NC tori. Z n q n Z n The noncommutative residue is defined on Ψ T : Z Ψ T . Note that Ψ T is a p θ q “ qP p θ q p θ q sub-algebra of L Aθ , and so by a trace we shall mean any linear functional that vanishes on Z Ť commutators P,Qp , P,Qq Ψ Tn . r s P p θ q Definition 5.1. The noncommutative residue is the linear functional Res : ΨZ Tn C given by p θ q Ñ n´1 Z (5.1) Res P τ ρ´n ξ d ξ, P Ψ Aθ , p q“ Sn´1 r p qs P p q ż where ρ n ξ is the symbol of degree n of P . ´ p q ´ Z n Remark 5.1. If P Ψ T , then the homogeneous symbol ρ n ξ is uniquely determined by P P p θ q ´ p q (cf. Remark 4.6). We make the convention that ρ n ξ 0 when P has order n. ´ p q“ ă´ Remark 5.2. It is immediate from the above definition that Res P 0 when its symbol of degree n is zero. In particular, the noncommutative residue vanishesp q on “ the following classes of operators:´ (i) ΨDOs of order n, including smoothing operators. (ii) Differential operatorsă´ (since for such operators the symbols are polynomials without any homogeneous component of negative degree). Remark 5.3. In dimension n 2 (resp., n 4) we recover the noncommutative residue of [24] (resp., [22]). “ “ Remark 5.4. Under the equivalence between our classes of ΨDOs and the classes of toroidal ΨDOs (see [31]) the above noncommutative residue agrees with the noncommutative residue for toroidal ΨDOs introduced in [38]. q q1 1 Proposition 5.5 ([22, 24, 38]). Let P Ψ Aθ and Q Ψ Aθ be such that q q Z. Then P p q P p q ` P Res PQ Res QP . p q“ p q Z In particular, the noncommutative residue Res is a trace on the algebra Ψ Aθ . p q By a well-known result of Wodzicki [55], on connected closed manifolds of dimension n 2 the noncommutative residue is the unique trace up to constant multiple. The following is the analogueě of this result for NC tori (see also [24, 22, 38]). Z Tn Proposition 5.6 ([46]). Every trace on Ψ θ is a constant multiple of the noncommutative residue. p q We also observe that the formula (5.1) can be rewritten in the form,

n´1 (5.2) Res P τ cP , where cP : ρ´n ξ d ξ. p q“ “ Sn´1 p q ż m n Note that if P Ψ T , m Z“, has‰ symbol ρ ξ ρm j ξ , then, for every a Aθ, the P p θ q P p q „ ´ p q P operator aP has symbol aρ ξ aρm´j ξ . In particular, its symbol of degree n is aρ´n ξ , p q „ n´1 p q ř A ´ p q and so we have caP n´1 aρ´n ξ d ξ acP . Thus, for all a θ, we have “ S přq “ P (5.3)ş Res aP τ caP τ acP . p q“ “ This shows that the noncommutative residue is“ a local‰ functional.“ ‰ 11 6. Derivatives and Sums of Commutators In this section, we establish that any ΨDO agrees with a scalar-symbol ΨDO modulo the closure of the weak trace class commutator subspace. The approach is based on the following observation. m n Proposition 6.1 ([46]). Let ρ ξ S R ; Aθ , m R. Then there are symbols ρ1 ξ ,...,ρn ξ m n p q P p q P p q p q in S R ; Aθ such that p q (6.1) ρ ξ τ ρ ξ δ1ρ1 ξ δnρn ξ . p q“ p q ` p q`¨¨¨` p q ´1 ´1 Remark 6.2. For instance in (6.1) we“ may‰ take ρj ξ δj∆ ρ ξ , where ∆ is the partial 2 2 p q “ p q m Rn A inverse of the (flat) Laplacian ∆ : δ1 δn . Furthermore, if ρ ξ S ; θ , then the m n “ p `¨¨¨` q p q P p q above symbols are in S R ; Aθ as well (see [46]). p q Sm Rn A It can be shown that if ρ ξ ; θ , then Pδj ρ δj , Pρ (see [46]). When m n p q P p q “ r s 1,8 ď ´ this does not imply that the operators Pδ ρ lie in the commutator space Com L , because j p q the derivatives δj are unbounded operators. Nevertheless, as the following result asserts, those operators are contained in the L 1,8-closure Com L 1,8 of Com L 1,8 . p q p q ´n n Proposition 6.3. Let ρ ξ S R ; Aθ . p qP p q (1) For j 1,...,n, we have “ 1 ´1 L 1,8 Pδj ρ lim αtej Pραte Pρ in . “ tÑ0 t j ´ ´ ¯ L 1,8 (2) The operators Pδ1ρ,...,Pδ ρ are contained in Com . n p q Rn R A Proof. Let j 1,...,n . Given any ξ , the map t αtej ρ ξ θ is smooth and dℓ P tℓ ℓu P Q Ñ r p qs P ℓ αte ρ ξ i αte δ ρ ξ for all ℓ 1. Thus, by the Taylor formula for maps with values in dt j r p qs “ j r j p qs ě locally convex spaces (see, e.g., [31, Proposition C.15]), for all t R, we have 1 P 2 2 αtej ρ ξ ρ ξ tδj ρ ξ t 1 s αsej δj ρ ξ ds. p q ´ p q´ p q“´ 0 p ´ q p q ż n “ ‰ ´1 “ ‰ For t 0 and ξ R set ρt ξ t αte ρ ξ ρ ξ . We have ‰ P p q“ p j r p qs ´ p qq 1 2 ρt ξ δj ρ ξ t 1 s αsej δj ρ ξ ds p q´ p q ď 0 p ´ q p q ż › › 1 › “ ‰› › › › 2 › t 1 s δj ρ ξ ds ď 0 p ´ q p q ż 1 › › t δ2ρ ξ ›. › ď 2 j p q ´n n 2 m n As ρ ξ S R ; Aθ there is C 0 such that› δ ρ ξ› C 1 ξ for all ξ R . Thus, there p qP p q ą › j p ›q ď p ` | |q P is C 0 such that, for all ξ Rn and t R 0, we have ą P P z › › › › ´n ρt ξ δj ρ ξ Ct 1 ξ . p q´ p q ď p ` | |q α β m´|β| n Likewise, given any multi-orders› α and β, as›δ ρ ξ S R ; Aθ , there is Cαβ 0 such › › Bξ p qP p q ą that, for for all ξ Rn and t R 0, we have P P z α β 1 α β α β α β δ ρt ξ δj ρ ξ αte δ ρ ξ δ ρ ξ δj δ ρ ξ Bξ r p q´ p qs “ t j Bξ p q ´ Bξ p q ´ Bξ p q › › › › › ´ “ ´n´|β| ‰ ¯ › › › ›Ct 1 ξ . › ď › p ` | |q › This implies that, for all multi-orders α and β, we have lim sup 1 ξ n`|β| δα β ρ ξ δ ρ ξ 0. 0 ξ t j tÑ ξPRnp ` | |q B r p q´ p qs “ ´n n › › This shows that ρt ξ δj ρ ξ in S R ; A›θ as t 0. Combining› this with Proposition 4.13 shows that, in L 1,8p ,q we Ñ havep q p q Ñ 1 Pδj ρ lim Pρt lim Pα pρq Pρ . “ tÑ0 “ tÑ0 t tej ´ 12 ` ˘ We claim that, for all t R, we have P ´1 (6.2) P αte Pρα . αspρq “ j tej k To see this set s tej , and let u Zn ukU be in Aθ. As αs is a continuous automorphism of “ “ kP Aθ we have ř ´1 ´1 k (6.3) αsPρα u uk αsPρα U . p s q “ p s q kPZn ÿ ´1 k k ´is¨k k As α U α s U e U we get s p q“ ´ p q“ ´1 k ´is¨k k ´is¨k k ´is¨k k ´1 k Pρ α U e Pρ U e Pρ U e ρ k U ρ k α U . s “ p q“ p q“ p q “ p q s Thus, “ ` ˘‰ ` ˘ ´1 k ´1 k k αsPρα U αs ρ k α U αs ρ k U . s “ p q s “ r p qs Combining this with (6.3)` gives ˘` ˘ “ ` ˘‰ ´1 k αsPρα u ukαs ρ k U P u. s “ r p qs “ αspρq kPZn ` ˘ ÿ This proves (6.2). Combining (6) and (6.2) proves the first part of the proposition. The 2nd part follows from the ´1 ´1 1,8 fact that αte Pρα Pρ αte , Pρα Com L . The proof is complete.  j tej ´ “r j tej sP p q 7. Weyls law for Scalar-Valued Symbols In this section, we establish a Weyls law for positive ΨDOs with scalar symbols. This will allows us to get a preliminary version of the trace theorem for ΨDOs with scalar-symbols. n Given any homogeneous symbol ρ ξ Sm R ; Aθ , m R, we define the linear operator p q P p q P Pρ : Aθ Aθ by Ñ k k Pρ ukρ k U , u ukU Aθ. “ p q “ P k Zn 0 Pÿz ÿ m n 8 n In fact, Pρ Pρ˜, withρ ˜ ξ 1 χ ξ ρ ξ S R ; Aθ , where χ ξ is any function in C R “ p q “ p ´ p qq p qP p q p q c p q such that χ ξ 1 near ξ 0 and χ ξ 1. In particular, we see that Pρ is an operator in m p q “ “ p q ě Ψ Aθ and has ρ ξ as principal symbol. p q p q n Suppose now that ρ ξ has scalar values, i.e., ρ ξ Sm R . In this case, the operator Pρ is p q p q P k p q even diagonal with respect to the orthonormal basis U k Zn . Namely, p q P k k n Pρ 1 0, Pρ U ρ k U , k Z 0. p q“ p q“ p q P z Therefore, the operator is (formally) normal and its (non-zero) eigenvalues are given by the values of the symbol ρ ξ on Zn 0. p q z Assume further that m 0 and ρ ξ 0. This implies that Pρ is elliptic and ρ k 0 for all k Zn 0. It is convenientą to extendpρqξ ąby continuity to ξ 0 by setting ρ 0 p 0.qą Then, the P z p q “ n p q “ operator Pρ is a selfadjoint Fredholm operator with spectrum ρ Z . Note also that the nullspace p q of Pρ is just C 1. Therefore, we can list the eigenvalues of Pρ as a non-decreasing sequence, ¨ 0 λ0 Pρ λ1 Pρ λ2 Pρ , “ p qă p qď p q﨨¨ where each eigenvalue is repeated according to multiplicity. We also define the counting function,

N Pρ; λ : # ℓ N0; λℓ Pρ λ , λ 0. p q “ t P p qď u ě Note that we have n (7.1) N Pρ; λ : # k Z 0; ρ k λ . p q “ t P z p qď u n Proposition 7.1 (Weyls law). Let ρ ξ S1 R , ρ ξ 0. p qP p q p qą (1) As λ , we have Ñ8 n ´1 1 ´ 1 n´1 N Pρ; λ c ρ λ 1 O λ , c ρ : ρ ξ n d ξ 0. p q“ p q ` p q p q “ n Sn´1 p q ą ż ` ˘ 13 (2) As ℓ 0, we have Ñ 1 ´1 n (7.2) λℓ Pρ c ρ ℓ O 1 . p q“ p q ` p q Proof. The proof follows along similar lines` as that˘ of the classical proof of the Weyl law for the Laplacian on an ordinary torus. For k Zn set I k k 0, 1 n. Note that P p q“ `r s ξ k ?n for all ξ I k . | ´ |ď P p q For λ 0, we set ě A λ ξ Rn; ρ ξ 1 , B λ I k . p q“t P p qă u p q“ p q k A λ Zn P pďqX We observe that by (7.1) we have n (7.3) N Pρ; λ # A λ Z I k B λ . p q“ p p qX q“ p q “ | p q| k A λ Zn P pÿqX Rn Rn As ρ ξ S1 , the partial derivatives ξ1 ρ ξ ,..., ξn ρ ξ are symbols in S0 , and so they arep boundedq P p onq Rn 0. Therefore, there isB C p q0 suchB thatp q p q z ą (7.4) ρ ξ ρ η C ξ η ξ, η Rn 0. | p q´ p q| ď | ´ | @ P z Let λ 0 and k A λ Zn. The inequality (7.4) for ξ I k and η k gives ě P p qX P p q “ ρ ξ ρ k C ξ k λ C?n. | p q| ď | p q| ` | ´ |ď ` Therefore, we see that B λ A λ C?n . Conversely, suppose thatp qĂλ pC?`n andq let ξ A λ C?n and k Zn be such that ξ I k . Then by using (7.4) we get ě P p ´ q P P p q ρ k ρ ξ C ξ k λ C?n C?n λ. | p q| ď | p q| ` | ´ | ď p ´ q` “ Thus, k A λ Zn, and so ξ B λ . It then follows that A λ C?n B λ . CombiningP p qX the inclusions A Pλ pCq?n B λ A λ C?p n´with (7.3)qĂ givesp q p ´ qĂ p qĂ p ` q A λ C?n N Pρ; λ A λ C?n λ C?n. p ´ q ď p qď p ` q @ ě Moreover, the homogeneityˇ of ρ ˇξ implies that,ˇ for λ 0, weˇ have ˇ pˇ q ˇ ą ˇ A λ ξ Rn; ρ λ´1ξ 1 λA 1 λn A 1 . | p q| “ P p qă “ | p q| “ | p q| Therefore, for λ C?n we haveˇ (ˇ ě ˇ ˇ n n λ C?n A 1 N Pρ; λ λ C?n A 1 . p ´ q | p q| ď p q ď p ´ q | p q| As λ C?n n λn O λn´1 as λ , we then deduce that p ˘ q “ ` p q Ñ8 n ´1 (7.5) N Pρ; λ A 1 λ 1 O λ . p q “ | p q| ` p q It remains to compute A 1 . Thanks to the homogeneity` of ˘ρ ξ , in polar coordinates ξ rη the inequality ρ ξ 1 gives| p q| 1 ρ rη rρ η , i.e., r ρ η ´p1.q Therefore, by integrating“ in polar coordinatesp q we ď get ě p q “ p q ă p q 1 ρpηq´ A 1 dξ rn´1dr dn´1η c ρ , | p q| “ 1 “ Sn´1 0 “ p q żρpξqď ż ˜ż ¸ 1 1 1 1 ´ n n´ Sn´ where we have set c ρ n Sn´1 ρ η d η. Note that c ρ 0 since ρ ξ 0 on . Combining this withp (7.5)q “ gives the Weylsp q law, p q ą p q ą ş n ´1 (7.6) N Pρ; λ c ρ λ 1 O λ as λ . p q“ p q ` p q Ñ8 It is routine to deduce from (7.6) the Weyls` law (7.2)˘ for the eigenvalues λℓ Pρ . Namely, n ´1 1 p q as (7.6) implies that N Pρ; λ c ρ λ , we see that λ c ρ N Pρ; λ n . This allows us to rewrite (7.6) in the form,p q “ p q „ p p q p qq

n ´1 ´ 1 λ c ρ N Pρ; λ 1 O N Pρ; λ n as λ . “ p q p q ` p q Ñ8 ” ´14 ¯ı Equivalently, as λ , we have Ñ8 1 1 1 ´1 n ´ ´1 n (7.7) λ c ρ N Pρ; λ 1 O N Pρ; λ n c ρ N Pρ; λ O 1 . “ p q p q ` p q “ p q p q ` p q ” ´ ¯ı Note that, for` all ℓ 0 and ˘ǫ 0, we have N Pρ; λℓ Pρ ` ǫ ℓ N Pρ˘; λℓ P . Combining this with (7.7 ) we seeě that, as ℓ ą , we have p p q´ qď ď p p qq Ñ8 1 1 ´1 n ´1 n λℓ Pρ c ρ N Pρ; λℓ Pρ O 1 c ρ ℓ O 1 , p q“ p q p p qq ` p qď 1 p q ` p q 1 ´1 n ´1 n λℓ Pρ λℓ Pρ ǫ “O 1 c ρ N Pρ‰; λℓ Pρ ǫ ` O 1˘ c ρ ℓ O 1 . p q“ p q´ ` p q“ p q p p q´ q ` p qě p q ` p q This gives the Weyls law (7.2). The“ proof is complete. ‰ ` ˘ 

1, In what follows we let T0 be a selfadjoint operator in L 8 whose eigenvalue sequence is the ´1 harmonic sequence, i.e., there is an orthonormal basis eℓ ℓď0 of Hθ such that T0eℓ ℓ 1 eℓ for all ℓ 0. p q “ p ` q ě n Proposition 7.2. For every ρ ξ S n R , we have p qP ´ p q 1 1,8 (7.8) Pρ Res Pρ T0 mod Com L . “ n p q p q n ´n Proof. Let ρ ξ S´n R . As mentioned above, Pρ is an operator in Ψ Aθ with principal p q P p q n´1 p q symbol ρ ξ , and so Res Pρ Sn´1 ρ ξ d ξ. We observep q that bothp sidesq“ of (7.8)p dependsq linearly on the symbol ρ ξ . Upon writing ρ ξ 1 1 ş p q p q“ 2 ρ ξ ρ ξ 2i ρ ξ ρ ξ we see it is enough to establish (7.8) when ρ ξ is real-valued. p Assumep q` p thatqq `ρ ξp pisq´ real-valuedp qq and is not identically zero. Set c sup pρqξ ; ξ 1 . By homogeneity ρ ξ p q ξ ´nρ ξ ´1ξ c ξ ´n for all ξ Rn 0. In particular,“ t| wep q| see| that| “ cu 0, and so ρ ξ | 2pc ξq|´ “n | | c ξ ´p|n | 0q for ď ξ| | 0. As ρ ξ P ρzξ 2c ξ ´n 2c ξ ´n, we seeą that p q` | | ě | | ą ‰ p q “n p p q` | | q´ | | ρ ξ is a linear combination of positive symbols in S´n R . Together with the linearity of (7.8) thisp q further reduces the verification of (7.8) to the casep of positivq e symbols. ´ 1 n ´n Suppose now that ρ ξ 0. Set σ ξ ρ ξ n . Then σ ξ S1 R and ρ ξ σ ξ . pk q ą ´n kp q “ p q Zn p q P p ´nq p q “ p q This implies that Pρ U σ k U for all k 0, and so Pρ Pσ . It then follows that ´np q “ p q P z “ µℓ Pρ λℓ`1 Pρ for all ℓ 0. Combining this with the Weyls law (7.2) then shows that, as ℓ p q“, we havep q ě Ñ8 ´n ´n ´1 1 ´ 1 1 ´ 1 µℓ Pρ λℓ 1 Pρ c σ ℓ n 1 O ℓ n c σ 1 O ℓ n . p q“ ` p q “ p q ` p q “ p q ℓ ` p q 1 ´n n´1 ” 1 ´ n´1 ¯ı ´ ¯ Here c σ n´1 σ ξ d ξ n´1 ρ ξ d ξ Res Pρ . Thus, p q“ n S p q “ n S p q “ p q ş ş 1 1 ´p1` 1 q (7.9) µℓ Pρ Res Pρ O ℓ n . p q“ n p q ℓ ` ´ ¯ ´1 Recall there is an orthonormal basis eℓ ℓď0 of Hθ such that T0eℓ ℓ 1 eℓ for all ℓ 0. p q k “ p ` q ě Let vℓ ℓ 0 be a re-arrangement of the sequence U k Z such that Pρvℓ µℓ Pρ vℓ for all ℓ 0. p q ě p q P “ p q ě In addition, let V L H be the unitary operator such that V vℓ eℓ for all ℓ 0, and set 1 ˚ P p q “ ě Tρ Res Pρ V T0V . We have “ n p q 1 1 ˚ 1 ˚ 1,8 (7.10) Tρ Res Pρ T0 Res Pρ V T0V V Res Pρ V ,T0V Com L . ´ n p q “ n p qp ´ q“ n p qr sP p q 1 ˚ 1 ´1 Moreover, Tρvℓ Res Pρ V T0eℓ Res Pρ ℓ 1 vℓ. It then follows that the operator “ n p q “ n p qp ` q Pρ Tρ is diagonal with respect to the orthonormal basis vℓ ℓě0 with eigenvalues µℓ Pρ 1 ´ ´1 p q p q ´ Res Pρ ℓ 1 , ℓ 0. Combining this with (7.9) gives n p qp ` q ě 1 Tr Pρ Tρ µℓ Pρ Res Pρ . p| ´ |q “ p q´ n ℓ 1 p q ă8 ℓě0 ˇ ˇ ÿ ˇ p ` q ˇ ˇ ˇ L 1,8 This shows that Pρ Tρ is a trace-classˇ operator, and hence is containedˇ in Com by ´ p1,8 q Proposition 2.1. Combining this with (7.10) we then deduce that Pρ Tρ Com L . This proves (7.8) when ρ ξ is positive. The proof is complete. ´ P p q  p q 15 8. Connes’ trace theorem for Noncommutative Tori In this section, we prove the version for noncommutative tori of Connes’ trace theorem and give a couple of applications. ´n Theorem 8.1. Every operator P Ψ Aθ is strongly measurable, and we have P p q 1 (8.1) P Res P . ´ “ n p q ż ´n ´n n n Proof. Let P Ψ Aθ have symbol ρ ξ S R ; Aθ and principal symbol ρ´n ξ S´n R ; Aθ . P p q 8 Rn p qP p q p qP p q Setρ ˜´n ξ 1 χ Cc is equal to 1 near ξ 0 and vanishes for ξ 1. Then p q´ “n pn ´ q P p q ´n´1 n “ ´n´1 | | ě ρ˜´n ξ S R and ρ ξ ρ˜´n ξ S R . As operators in Ψ Aθ are trace-class, andp henceq P arep containedq inp q´ Com Lp1,q8 P by Propositionp q 2.1, we deduce that p q p q 1,8 (8.2) P Pρ Pρ˜ mod Com L . “ “ ´n p q By applying Proposition 6.1 and Proposition 6.3 toρ ˜´n ξ we see that there are symbols ´n n p q ρ1 ξ ,...,ρn ξ in S R ; Aθ such that p q p q p q L 1,8 (8.3) Pρ˜ P ˜ Pδ1ρ1 Pδ ρ P ˜ mod Com , ´n “ τrρ´ns ` `¨¨¨` n n “ τrρ´ns p q 1,8 1,8 1,8 where Com L is the closure in L of Com L . Note that τ ρ˜ n ξ 1 χ ξ τ ρ n ξ , p q p q r ´ p qs “ p ´ p qq r ´ p qs and so P ˜ Pτ ρ n . Combining this with (8.2) and (8.3) we get τrρ´ns “ r ´ s (8.4) P P mod Com L 1,8 . “ τrρ´ns p q ´n Rn n´1 Note that τ ρ´n ξ S and Res Pτrρ´ns Sn´1 τ ρ´n ξ d ξ Res P . Therefore, r p qs P p q p q“1 r p qs 1“, p q by using Proposition 7.2 we see that P n´ Res P T0 Com L 8 . Combining this τrρ´ns ş with (8.4) then gives ´ p q P p q

1 1 (8.5) P Res P T0 mod Com L ,8 . “ n p q p q Now, let ϕ be positive normalized trace on L 1,8. By Proposition 2.2 this is a continuous trace, and so it is annihilated by Com L 1,8 . Combining this with (8.5) and the normalization p q ϕ T0 1 we then obtain p q“ 1 1 ϕ P Res P ϕ T0 Res P . p q“ n p q p q“ n p q 1 This proves that P is strongly measurable and ´P Res P . The proof is complete.  “ n p q Remark 8.2. Fathizadeh-Khalkhali [21, 22] obtainedş a version of Theorem 8.1 in dimension n 2 and n 4 by a different approach. They established measurability and derived the trace“ formula (8.1)“ in those dimensions, but they did not establish strong measurability. That is, they established the trace formula (8.1) for Dixmier traces only. We shall now explain how Theorem 8.1 allows us to recover and extend Connes’ integration formula for (flat) NC tori of McDonald-Sukochev-Zanin [43, Theorem 6.15]. First, we have the following consequence of Theorem 8.1. ´n Tn Corollary 8.3. Let P Ψ θ . For every a Aθ, the operator aP is strongly measurable, and we have P p q P 1 aP τ acP , ´ “ n p q ż where cP Aθ is defined as in (5.2). P Proof. We need to show that, for every normalized positive trace ϕ on L 1,8, we have 1 (8.6) ϕ aP τ acP a Aθ, “ n p q @ P When a Aθ, then by combining the` trace˘ formula (8.1) with (5.3) we get P 1 1 ϕ aP aP Res P τ acP . p q“´ “ n p q“ n p q ż 16 This shows that (8.6) holds for all a Aθ. As ϕ is a positive trace, PropositionP 2.2 ensures that ϕ is a continuous linear form on L 1,8. Note 1,8 also that Aθ acts continuously on L by left composition, since this action is the composition 1,8 of the left regular representation of Aθ in L Hθ with the natural left-action of L Hθ on L . p q p q It then follows that both sides of (8.6) are given by continuous linear forms on Aθ. As (8.6) holds on the dense subspace Aθ it holds on all Aθ. The proof is complete.  2 2 A Let ∆ δ1 δn be flat Laplacian on θ. In the notation of Section 7 we have ∆ Pσ “ `¨¨¨`2 ´ n ´n ´ n ´n“ with σ ξ ξ and ∆ 2 Pρ with ρ ξ ξ . In particular, ∆ 2 is contained in Ψ Aθ and hasp principalq “ | | symbol ξ“´n, and so wep q have “ | | p q | | n n 1 n ´ S ´ c∆´ 2 ξ “ Sn´1 | | “ | | ż n Therefore, by specializing Corollary 8.3 to P ∆´ 2 we recover the Connes integration formula for flat NC tori of [43] in the following form. “ ´ n Corollary 8.4 ([43, Theorem 6.15]). For every a Aθ the operator a∆ 2 is strongly measurable, and we have P n 1 ´ 2 n´1 (8.7) a∆ cˆnτ a , cˆn : S . ´ “ p q “ n| | ż Remark 8.5. The apparent discrepancy between the constant cn in (2.8 ) and the constantc ˆn above is due to the fact when θ 0 the trace τ is actually the Radon measure a 2π ´n a x dx. “ Ñ p q Tn p q That is, the overall constant 2π ´n is absorbed into the definition of τ. p q ş Remark 8.6. When θ 0 we recover the integration formula (2.8) for the ordinary torus Tn equipped with its standard“ flat metric.

9. Riemannian Metrics and Laplace-Beltrami Operator In this section, we review the main definitions and properties regarding Riemannian metrics and Laplace-Beltrami operators.

9.1. Riemannian metrics. In what follows we denote by GLn Aθ the group of invertible ma- ` p q trices in Mn Aθ . We also denote by GL Aθ its subset of positive matrices. Here by a positive p q n p q element of Mn Aθ we mean a selfadjoint matrix with non-negative spectrum. As Mn Aθ is closed under holomorphicp q functional calculus we have p q ` ˚ 2 ˚ GL Aθ h h; h h GLn Aθ h ; h GLn Aθ , h h . n p q“ P P p q “ P p q “ ` ´1 ` `` In particular, if h GL A θ , then h and ?h are( in GL Aθ . When n 1 we( denote by A P n p q n p q “ θ the cone of positive invertible elements of Aθ. Let Xθ be the free left-module over Aθ generated by the canonical derivations δ1,...,δn. This plays the role of the module of (complex) vector fields on the NC torus Aθ (cf. [48]). As Xθ is a free module, the Hermitian metrics on Xθ are in one-to-one correspondence with matrices ` A ` A h GLn θ (see [33, 48]). Namely, to any matrix h hij in GLn θ corresponds the HermitianP p metric,q “ p q p q ˚ X, Y h : Xihij Yj , X Xiδi, Y Yj δj . p q “ 1ďi,jďn “ i “ j ÿ ÿ ÿ On a C8-manifold a Riemannian metric is a Hermitian metric on complex vector fields which takes real-values on real vector fields and real differential forms. Equivalently, its matrix and its Tn inverse have real entries. On the NC torus θ the real-valued smooth functions corresponds to A A R A selfadjoint elements of θ. Let θ the real subspace of selfadjoint elements of θ. More generally, A R A for any m 1, we denote by Mn θ the real subspace of Mm θ of matrices with selfadjoint ě p q A R p q entries. Note that Hermitian elements of Mn θ are symmetric matrices. p A Rq A In general the inverse of an element of Mm θ GLm θ need not have selfadjoint entries ` pA RqX p q ` A (see [33]). In what follows we denote by GLm θ the set of matrices g GLm θ such that g and its inverse g´1 have selfadjoint entries. p q P p q 17 Tn X Definition 9.1 ([33, 48]). A Riemmannian metric on θ is a Hermitian metric on θ whose ` R matrix g gij is in GL A . “ p q n p θ q Remark 9.1. As usual we will often identify Riemannian metrics and their matrices.

Example 9.2. The standard flat metric is gij δij . A conformal deformation is of the form 2 `` “ gij k δij with k A . This is the kind of Riemannian metric considered in [14, 15]. “ P θ Example 9.3. A product metric is of the form,

g1 0 g1 0 1 0 ` R A 1 2 (9.1) g , gj GLnj θ , n n n. “ ˜ 0 g2¸ “ ˜ 0 1¸˜0 g2¸ P p q ` “ This includes the kind of Riemannian metrics considered in [10, 16, 18].

In what follows we shall say that matrices a aij Mm Aθ and b bkl Mm1 Aθ “ p q P p q “1 p q P p q are compatible when aij ,bkl 0 for all i, j 1,...,m and k,l 1,...,m . We say that a is self-compatible when itr is compatibles “ with itself.“ “ ` A Example 9.4 (Self-compatible Riemannian metrics [33]). Let g gij GLn θ be self-compatible. ´1 “ p qP ` pA Rq In this case the inverse g has selfadjoint entries as well, and so g GLn θ , i.e., g is a Rie- mannian metric. This example includes the conformally flat metricsP of Ep xampleq 9.2. It also includes the functional metrics of [28]. 9.2. Riemannian density. On a Riemannian manifold M n,g the Riemannian density is given p q in local coordinates by integration against det g x . The volume M,g is then obtained as the integral of the Riemannian density. More generally,p p qq smooth denpsitiesq on M are given by a integration against positive functions in local coordinates. Tn A `` On the NC torus θ the role of positive functions is played by elements of θ . It is natural to think of any ν A `` as a smooth density on T n, or if we think in terms of measures, to think P θ θ as density the corresponding weight ϕν a τ aν , a Aθ (see [33]). p q“ Tr n s P In order to define Riemannian densities on θ we need a notion of determinant. Recall that if h GLn C is positive-definite, then det h exp Tr log h . P p q p q“ r p qs ` Definition 9.2 ([25, 33]). The determinant of any h GL Aθ is defined by P n p q ` `` det h exp Tr log h , h GL Aθ A , p q“ p q P n p qP θ A A where log h is defined by holomorphic“ functional‰ calculus and Tr : Mn θ is the matrix trace. p q Ñ ` Remark 9.5. If h GL Aθ , then log h is a selfadjoint element of Mn Aθ , and so Tr log h is a P n p q p q p q selfadjoint element of Aθ. Therefore, its exponential is selfadjoint and has positive spectrum, i.e., A `` it is contained in θ . ` Proposition 9.6 ([33]). Let h hij be in GL Aθ . “ p q n p q (i) det hs det h s for all s R. p 1 q “ p ` q P (ii) Let h GL Aθ be compatible and commute with h. Then P n p q det hh1 det h det h1 det h1 det h . p q“ p q p q“ p q p q (iii) If h is self-compatible, then we have

(9.2) det h ε σ h1 1 h . p q“ p q σp q ¨ ¨ ¨ mσpmq σPS ÿm We are now in a position to define Riemannian densities. ` R Definition 9.3. Let g gij GL A be a Riemannian metric. “ p qP n p θ q The Riemannian density is ν g : det g exp 1 Tr log g A ``. ‚ p q “ p q“ 2 p p qq P θ The Riemannian weight ϕg : Aθ C is defined by ‚ Ñ a “ ‰ n ϕg a : 2π τ aν g , a Aθ. p q “ p q 18 p q P “ ‰ n n The Riemannian volume is Volg T : ϕg 1 2π τ ν g . ‚ p θ q “ p q “ p q p q n n n Example 9.7. When g is the standard flat metric gij δij , we have“ Vol‰ g T 2π T . “ p θ q “ p q “ | | 2 A `` Example 9.8. Let gij k δij , k θ , be a conformal deformation of the flat Eucldean metric. Then we have “ P n n n n n n ν g k , ϕg a 2π τ ak , Volg T 2π τ k . p q“ p q “ p q p θ q “ p q Example 9.9. Let g be a product matrix as in“ (9.1),‰ where g1 and g2 are compatible.“ ‰ Then

ν g ν g1 ν g2 . p q“ p q p q The very datum of the Riemannian density gives rise to a non-trivial modular geometry. Namely, unless ν g is a scalar the Riemannian weight ϕg is not a trace. Therefore, the GNS construction p q ˚ ˝ gives rise to non-isometric -representations of the C -algebra Aθ and its opposite algebra Aθ. ˚ H ˝ The latter is represented in the Hilbert space g obtained as the completion with respect to the inner product, ˝ ˚ n ˚ u v ϕg uv 2π τ v ν g u , u,v Aθ. x | yg “ “ p q p q P ˚ 2 The Tomita involution and modular` operator˘ are “Jg a σ‰g a and ∆ a : σν a , where σg is the inner automorphism, p q“ p q p q “ p q 1 1 2 ´ 2 σg a ν g aν g , a Aθ. p q“ p q p q P ` R n 9.3. Laplace-Beltrami operator. Let g gij GL A be a Riemannian metric on T . “ p qP n p θ q θ Definition 9.4 ([33]). The Laplace-Beltrami operator ∆g : Aθ Aθ is defined by Ñ ´1 ij (9.3) ∆gu ν g δi ν g g ν g δj u , u Aθ. “ p q 1ďi,jďn p q p q p q P ÿ `a a ˘ Example 9.10. For the flat metric gij δij the Laplace-Beltrami operator is just the flat Laplacian 2 2 “ ∆ δ1 δ . “ `¨¨¨` n 2 `` Example 9.11. Let g k δij , k A , be a conformally flat metric. In this case, we have “ P θ ´2 ´n n´2 (9.4) ∆g k ∆ k δi k δi. “ ` 1ďiďn p q ÿ ´2 In particular, in dimension n 2 we get ∆g k ∆. This is reminiscent of the conformal invariance of the Laplace-Beltrami“ operator in dimension“ 2. In what follows, we set 1 ij 2 n (9.5) ξ g : g ξiξj , ξ R . | | “ i,j P ` ÿ ˘ `` Note that ξ g A when ξ 0. In fact (see, e.g., [33, Corollary 3.1]) there even are constants | | P θ ‰ 0 d1 d2 such that ă ă 2 2 2 n (9.6) d1 ξ ξ d2 ξ ξ R . | | ď | |g ď | | @ P Proposition 9.12 ([33]). The following holds. ´ 1 1 (1) ∆g is an elliptic 2nd order differential operator with principal symbol ν g 2 ξ gν g 2 . H ˝ p q | | p q (2) ∆g is an essentially selfadjoint operator on g with non-negative discrete spectrum with finite multiplicity. (3) ker ∆g C. “ 10. Connes integration formula for Curved NC Tori In this section, we extend the Connes integration formula of [43] to curved NC tori and present a few consequences. 19 10.1. Spectral Riemannian densities. For every positive-definite matrix g GLn R , we have P p q ´n n´1 Sn´1 (10.1) ξ g d ξ det g , Sn´1 | | “ p q| | ż a Sn´1 2 where ξ g is defined as in (9.5). Indeed, as det g is the volume of the ellipsoid ξ g 1, we get| | p q| | | | “ a 8 2 1 1 (10.2) e´|ξ|g dξ e´t tn´ dt det g Sn´ . Rn “ 0 p q| | ˆ ˙ ż ż a ´|ξ|g Integrating in polar coordinates also shows that Rn e dξ is equal to 8 8 2 2 1 1 ş 2 1 1 ´t |ξ|g n´ n´ ´t n´ ´n n´ (10.3) e t dt d ξ e t dt ξ g d ξ . Sn´1 0 “ 0 Sn´1 | | ż ˆ ż ˙ ˆ ż ˙ˆ ż ˙ Comparing this to (10.2) gives (10.1). Note that combining Connes’ trace theorem with with (10.1) yields the integration formula (2.8). ` A R Tn Definition 10.1. Let g GLn θ be a Riemannian metric on θ . Its spectral Riemannian density is P p q 1 ν g ξ ´ndn´1ξ, ˜ : n´1 g p q “ S Sn´1 | | | | ż where ξ g is defined in (9.5). | | The following statement gathers the main properties of spectral Riemannian densities. Proposition 10.1. Let g GL` A R be a Riemannian metric on Tn. P n p θ q θ A `` (1)ν ˜ g θ , i.e., ν˜ g is a smooth density. (2) Wep qP have p q n ξ 2 ν˜ g π´ 2 e´| |g dξ. p q“ Rn ż A `` 2 n (3) Let k θ be such that k,g 0. Then ν˜ k g k ν˜ g . (4) If g isP self-compatible, thenr s“ p q“ p q (10.4)ν ˜ g ν g det g , p q“ p q“ p q where det g is given by (9.2). a p q ´n A R Proof. As ξ ξ g is a continuous map with values in the closed real subspace θ , it is imme- Ñ | |A R A `` diate thatν ˜ g θ . Therefore, in order to check thatν ˜ g θ we only need to show that ν˜ g has positivep q P spectrum. p q P p q Sn´1 ´n 1 1 If ξ , then (9.6) implies that Sp ξ g d1, d2 , and so Sp ξ g d2, d1 , where Pn p| | qĂr s p| | qĂr s 1 ´ 2 ´n 1 d : d 0. Thus, given any u Hθ, we have ξ u u d2 u u , and so we have i “ i ą P | |g | ě x | y ´n Sn´1 ´1 @ ´n D n´1 1 ξ g u u ξ g u u d ξ d2 u u . | | | “ | | Sn´1 | | | ě x | y ż @ D 1 @ D This implies that Sp ν˜ g d2, 0, , and so this gives the first part. p p qqĂr 8qĂp 8q 2 ´|ξ|g In the same way as in (10.3) we see that Rn e dξ is equal to

8 2 8 2 ´t n´1 ´nş n´1 n´1 ´t Sn´1 e t dt ξ g d ξ t e dt ν˜ g . 0 Sn´1 | | “ 0 | | p q ˆ ż ˙ˆ ż ˙ ˆ ż ˙ 1 n 1 8 1 2 Here Sn´ 2π 2 Γ n 2 ´ , and tn´ e´t dt is equal to | |“ p { q 0 8 8 n´1 dt ş 1 n 1 1 n n 1 1 t 2 e´t t 2 ´ e´tdt Γ π 2 Sn´ ´ . 0 2?t “ 2 0 “ 2 2 “ | | ż ż 2 n ` ˘ ´|ξ|g 2 It then follows that Rn e dξ π ν˜ g . This proves the 2nd part. “ p q 20 ş A `` 2 ´1 Let k θ be such that k,g 0, and setg ˆ k g. As k commutes with g , we have ´1 ´P2 ´1 r ns “ 2 “ ´2 ij ´2 2 ´n n ´n gˆ k g , and so, for all ξ R , we have ξ ˆ k g ξiξj k ξ and ξ ˆ k ξ . “ P | |g “ “ | |g | |g “ | |g Thus, ř ´n n ´n n ν˜ gˆ ξ gˆ dσ0 ξ k ξ g dσ0 ξ k ν˜ g . p q“ Sn´1 | | p q“ Sn´1 | | p q“ p q ż ż This gives the 3rd part. It remains to prove the 4th part. We only have to show that the formula (10.1) continues to hold for self-compatible matrices in GL` A R . Note that this formula immediately extends to n p θ q any g C X, GL` R , where X is any compact Hausdorff space. P p n p qq Suppose that g is self-compatible. The entries gij of g pairwise commute with each other. Thus, if we let B be the unital C˚-algebra that they generate, then we obtain a unital commutative C˚- algebra (cf. [33]). Therefore, under the Gel’fand transform the C˚-algebra B is isomorphic to the C˚-algebra of continuous functions on the Gel’fand spectrum X : Sp B . As (10.1) holds for any ` R “ p q element of C X, GLn we deduce that it holds for g. This proves the 4th part and completes the proof. p p qq  Example 10.2. Let g be of the form, k2I 1 n1 0 A `` g 2 , kj θ , n1 n2 n. “ ˜ 0 k2 In2 ¸ P ` “

Assume further that k1, k2 0. This ensures that g is self-compatible, and so by combining (10.4) with Proposition 9.6r we gets“ 2n n ν˜ g det g det k1I1 det k2In k1k2 k1k2 . p q“ p q“ p q p q“ p q “ p q ` A R Tn 10.2. Curved integrationa formula.a Let g GLn θ abe a Riemannian metric on θ . As H ˝ P p q ˚ ˝ the Hilbert space g arises from the GNS construction for the opposite C -algebra Aθ, the A H ˝ right action of θ on itself by right-multiplication extends to a -representation in g of the ˚ ˝ ˚ opposite C -algebra Aθ. The left-action by left-multiplication extends to a representation of Aθ. This not -representation. We obtain a -representation by using the Tomita involution ˚ ´ 1 ˚ 1 ˚ Jg a σ a ν g 2 a ν g 2 . Namely, we get the -representation, p q“ νpgqp q“ p q p q ˚ 1 1 ˝ ˝ ˚ ´ 2 2 a a , where a : Jga Jg ν g aν g . ÝÑ “ “ p q p q ˝ n An intertwining unitary isomorphism from H onto Hθ is the left-multiplication by 2π 2 ν g . g p q p q Let ∆g : Hθ Hθ be the Laplace-Beltrami operator defined by the Riemannian metric g. By Ñ H ˝ a Proposition 9.12 it is essentially selfadjoint on g and has non-negative spectrum. Under the n 1 ˝ unitary isomorphism 2π 2 ν g 2 : H Hθ it corresponds to the operator, p q p q g Ñ 1 1 1 1 1 1 2 ´ 2 ´ 2 2 ij 2 ´ 2 (10.5) ∆g : ν g ∆gν g ν g δiν g g ν g δj ν g . “ p q p q “ 1ďi,jďn p q p q p q p q ÿ This operator isp essentially selfadjoint with non-negative spectrum on Hθ. It is also a 2nd order ij 2 differential operator with principal symbol ξig ξj ξ . All this allows us to define for i,j “ | |g ´s ´s H ˝ H every s 0 the operators ∆g and ∆g by standard Borel functional calculus on g and θ, ą 1 ř 1 ´s ´ 2 ´s 2 respectively. Note that ∆g ν g ∆g ν g . “ p q p p q ´s ´s ´2s Tn Lemma 10.3. For every s 0, the operatorsp ∆g and ∆g are in Ψ θ and have principal 1 ą 1 p q ´2s ´ 2 ´2s 2 symbols ξ g and ν g ξ g ν g , respectively. | | p q | | p q p Proof. As mentioned above, ∆g is a 2nd order elliptic differential operator. It is essentially self- adjoint with non-negative spectrum on Hθ and its principal symbol ξ g is positive and invertible | | ´s for ξ 0. Therefore, it satisfiesp the assumptions of [36, Theorem 7.3], and hence ∆g is an op- ‰ 2 2 1 1 s Tn ´ s ´s ´ 2 ´s 2 erator in Ψ θ with principal symbol ξ g . As ∆g ν g ∆g ν g , it then follows from p q 2 | | “ p q 1 p q 2 1 ´s s Tn ´ 2 ´ s 2 p Corollary 4.11 that ∆g is in Ψ θ and has principal symbol ν g ξ g ν g . The proof is complete. p q pp q | | p q  21 Remark 10.4. The above result actually holds for any s C (see [36, §7]). When s Z´ the result ´s ´s P P is immediate, since ∆g and ∆g then are positive-integer powers of differential operators. When s N this also can be deduced from the case s Z and the results of [32, §11], since ∆´s and P P ´ g ´s p s s ∆g are parametrices of the differential operators ∆g and ∆g, respectively. 2 `` Remark 10.5. In the case of a conformally flat metric gij k δij , k A , by using (9.4) we get p “p P θ n n 2 ´2 ´n n´2 ´ 2 ∆g k k ∆ k δi k δi k “ ` 1ďiďn p q n ` n ÿ n ˘ n 2 ´2 ´ 2 ´ 2 n´2 ´ 2 p k ∆k k δi k δi k . “ ` 1ďiďn p q ÿ ` ˘ In particular, when n 2 we recover the conformally perturbed Laplacian k´1∆k´1 of [15]. “ We are now in a position to prove the main result of this section.

n ˝ ´ 2 Theorem 10.6 (Curved Integration Formula). For every a Aθ, the operator a ∆g is strongly measurable, and we have P

n ˝ ´ 2 1 n´1 (10.6) a ∆g cˆnτ aν˜ g , cˆn : S . ´ “ p q “ n| | ż n“ ‰ n ´ 2 ´ 2 ´n Tn Proof. We know by Lemma 10.3 that ∆g and ∆g are operators in Ψ θ . Let a Aθ. For every positive trace ϕ on L 1,8, we have p q P

n 1 1 n 1 n 1 n ˝ ´ 2 ´ ´ 2 p ´ 2 ´ ´ 2 (10.7) ϕ a ∆g ϕ ν g 2 aν g 2 ∆g ϕ a ν g 2 ∆g ν g 2 ϕ a∆g . “ p q p q “ p q p q “ n ` ˘ ”` ˘ ı ˝ ”´ `2 ˘ ˘ı ` ˘ Combining this with Corollary 8.3 we deduce that a ∆g is strongly measurable, andp we have

n n ˝ ´ 2 ´ 2 1 a a τ ac ´ n . (10.8) ∆g ∆g ∆ 2 ´ “´ “ n g ż żn ´ 2 ´n “ p ‰ By Lemma 10.3 the principal symbol of ∆g isp ξ , and so we have | |g ´n n´1 n´1 c ´ n ξ d ξ S ν g nc ν g . ∆ 2 gp ˜ ˆn ˜ g “ Sn´1 | | “ | | p q“ p q ż Combining this with (10.8)p completes the proof.  Remark 10.7. In the definition (9.3) of the Laplace-Beltrami operator we may replace ν g by any A `` p q smooth density ν θ . The resulting operator, denoted by ∆g,ν in [33], is essentially selfadjoint P H ˝ ˝ on the Hilbert space ν of the GNS representation of Aθ with respect to the weight ϕν a τ aν , 1 1 p q“ r s ˝ H ˝ ˝ ´ 2 2 a Aθ (see [33]). The -representation a a of Aθ in ν is given by a ν aν . With these modificationsP we have ˚ Ñ “ n ˝ ´ 2 a ∆g,ν cˆnτ aν˜ g a Aθ. ´ “ p q @ P ż By Proposition 10.1 the spectral Riemannian“ density‰ ν ˜ g agrees with the Riemannian density ν g when g is self-compatible. Therefore, in this case wep obtainq the following result. p q Corollary 10.8. Assume that g is a self-compatible Riemannian metric. Then, for all a Aθ, we have P n ˝ ´ 2 (10.9) a ∆g cˆnτ aν g cnϕg a , ´ “ p q “ p q ż ´n “ ‰ where cn 2π cˆn is the same constant as in (2.8). In particular, for a 1 we get “ p q “ n ´ 2 n (10.10) ∆g cn Volg T . ´ “ p θ q ż Remark 10.9. When θ 0 every Riemannian metric is self-compatible. We then recover the integration formula (2.8)“ for arbitrary Riemannian metrics on the ordinary torus Tn. 22 Remark 10.10. The formula (10.9) shows that, in the self-compatible case, the NC integral allows us to recover the Riemannian density ν g and the Riemannian weight ϕg. As mentioned in the introduction this provides us with a nicep q link between the notions of integrals in NCG and the noncommutative measure theory in operator algebras, where the role of Radon measures is played by weights on C˚-algebras. Remark 10.11. In order to recover the Riemannian density ν g it is enough to have (10.9) for p q elements a of the smooth algebra Aθ. Indeed, by using the Fourier series expansion ν g k p q “ νk g U in Aθ, we get p q k k ˚ ´1 k ´1 ˝ ´n ř νk g ν g U τ U ν g cˆ U ∆ . p q“ p q| “ p q p q “ n ´ p q s g ż This confirms our claim. @ D “ ‰ “ 10.3. Lower dimensional volumes. In the same way as in the case of ordinary manifolds, the Z Tn trace formula (8.1) allows us to extend the NC integral to all operators in Ψ θ , including those operators that are not weak trace class or even bounded. p q m Definition 10.2. For every P Ψ Aθ , m Z, its NC integral is defined by P p q P 1 P : Res P . ´ “ n p q ż When the metric g is self-compatible the integration formula (10.9) shows that the NC integral recaptures the Riemannian density. For general Riemannian metrics, it recaptures the density Tn ν˜ g , and so it is natural to call the latter the spectral Riemannian density of θ ,g . n p q p q ´1 ´ 2 Along the same lines of thought as that of Section 2, we can regard the operator cn ∆g as the quantum volume element, and then we define the quantum length element as the operator, n 1 1 1 ´1 ´ 2 n ´ n ´ 2 ds : c ∆g cn ∆g . “ n “ ´1 Tn Note that Lemma 10.3 ensures us that ds` is an operator˘ in Ψ θ , and hence is an infinitesimal 1 p q operator of order n by Proposition 4.13. In particular, we can define the NC integrals of the powers dsk, k 1,...,n. This leads us to the following definition of the lower dimensional spectral volumes“ of curved NC tori. Definition 10.3. For k 1,...,n, the k-dimensional spectral volume of Tn,g is “ p θ q pkq Vol Tn : dsk. g p θ q “´ ż Remark 10.12. When the RiemannianĄ metric g is self-compatible, by using (10.10) we get pnq n n n ´1 ´ 2 n Vol T ds c ∆g Volg T . g p θ q“´ “ n ´ “ p θ q ż ż pnq Ą Tn Thus, in this case the spectral volume Volg θ agrees with the Riemannian volume in the sense of Definition 9.3. p q Ą 11. Scalar Curvature of Curved NC Tori In this section, we explain how the results of the previous section enables us to setup a natural notion of scalar curvature for general curved NC tori. As mentioned in Section 2, for any (compact) Riemannian manifold M n,g , the functional 1 2 8 ´ n´ p q C M a c1 ads recaptures the (dressed) scalar curvature κ g ν g . For curved NC p q Q Ñ ´ n p q p q tori Tn,g this leads us to the following definition. p θ q ş A R Tn Definition 11.1. Assume n 3, and let g GLn θ be a Riemannian metric on θ . The ě n P p q scalar curvature functional of T ,g is the functional Rg : Aθ C given by p θ q Ñ 1 R a a˝dsn´2, a A . g 1 θ p q“´cn ´ P ż 23 Remark 11.1. This definition is consistent with the definition of the scalar curvature functional of spectral triples [12, Definition 1.147] (see also [14, Remark 5.4]) and with the intrinsic definition of the modular curvature of conformally flat NC 2-tori [14, §5.2] (see also [22]). R n In what follows we assume n 3 and let g GLn A be a Riemannian metric on T . ě P p θ q θ Proposition 11.2. There is a unique κ g Aθ such that p qP (11.1) Rg a τ aκ g a Aθ. p q“ p q @ P Namely, we have “ ‰ n n 2 κ g π c ´ n `1 . (11.2) 3 4 Γ 1 ∆ˆ 2 p q“´ p q 2 ´ g ` ˘ Proof. Let a Aθ. In view of the definitions of ds and ´, we have P 1 1 n´2 n 1 1 n´2 n 1 ˝ n´2 ´ n ˝ ´ 2 `ş ´ n ˝ ´ 2 ` Rg a a ds cn a ∆g cn Res a ∆g . p q“´c1 ´ “´c1 ´ “´nc1 n ż n ż n By arguing as in (10.7) and using (5.3) we get ` ˘ n 1 n 1 ˝ ´ 2 ` ´ 2 ` a a τ ac ´ n `1 . Res ∆g Res ∆g ∆ˆ 2 “ “ g n´2 ` ˘1 ´ ` ˘n “ ‰ n p 2 n Moreover, it follows from (2.11) that nc1 cn 12 4π Γ 2 1 . Thus, n “ p q p ´ q n n 2 R a π τ ac ´ n `1 . g 3 4 Γ 1 ∆ˆ 2 p q“´ p q 2 ´ g That is, (11.1) holds with κ g given by (11.2). ` ˘ “ ‰ In the same way as in Remarkp q 10.11, the formula (11.1) uniquely determines the Fourier coef- ficients of ν g , and so ν g is unique. The proof is complete.  p q p q Definition 11.2. ν g is the scalar curvature of Tn,g . p q p θ q Remark 11.3. When θ 0 we recover the usual notion of scalar curvature for Riemannian metrics on the ordinary torus T“n. Remark 11.4. In dimension n 4 and for conformally flat metrics we recover the definition of the scalar curvature of conformally“ flat metrics in [20, 22] with a different sign convention and n 2 n 2 n up to the factor 6 4π Γ 2 1 6 4π . Note that the factor Γ 2 1 in (11) is essential to relate κ g to thep symbolq p of´ theq resolvent“ p q and perform the analyticp continuat´ q ion at n 2 (see Propositionp q 11.7 and Remark 11.10). “ The scalar curvature of conformally flat tori is often expressed in terms of the subleading coefficient in the short time heat trace asymptotics (see, e.g., [14, 15]). This heat coefficient is ´1 given by multi-integrals involving the 2nd subleading symbol σ´4 ξ; λ of the resolvent ∆g λ . We shall now explain how to derive a similar expression from thep formulaq (11.2). p ´ q n n ´ 2 `1 ´ 2 `1 We need to determine the symbol of degree n of ∆g . Note that as ∆g phas order n 2 its symbol of degree n agrees with its 2nd´ subleading symbol. Let s 0. We have ´ ` ´ ą p p ´s 1 ´s ´1 ∆g λ ∆g λ dλ, “ 2iπ Γ p ´ q ż L H where the integral converges in p θ and Γ is ap downward oriented contour of the form Γ ˚ p q “ Λ r , with Λ r : λ C ; ℜλ 0 or λ r and 0 r inf Sp ∆g (see [36]). It is shown B p q p q “´ t 1 P ď | |ď u ă ă p qq n in [36] that ∆g λ belongs to a suitable class of ΨDOs with parameter on T . In particular, p ´ q θ it has a symbol σ ξ; λ 0 σ 2 j ξ; λ , where p p q„ jě ´ ´ p q p 2 ´1 ř σ 2 ξ; λ ξ λ , ´ p q“ | |g ´ 1 2 ´1 α α σ 2 j ξ; λ ξ λ` p2˘ k ξ δ σ 2 l ξ; λ , j 0. ´ ´ p q“´ α! | |g ´ Bξ ´ p q ´ ´ p q ě k`l`|α|“j lÿăj ` ˘ 24 2 Here p ξ p2 ξ p1 ξ p0 is the symbol of ∆g with p2 ξ ξ g. We have the following refinementp q “ of Lemmap q` 10.3.p q` p q “ | | p ´s Lemma 11.5 ([36, Theorem 7.3]). Let s 0. The operator ∆ has symbol ρ s; ξ 0 ρ 2s j s; ξ , ă g p q„ jě ´ ´ p q where ř 1 ´s p (11.3) ρ 2s j s; ξ λ σ 2 j ξ; λ dλ, ξ 0. ´ ´ p q“ 2iπ ´ ´ p q ‰ żγξ 2 Here γξ is any clockwise-oriented Jordan curve in C , 0 that encloses Sp ξ . zp´8 s p| |gq Remark 11.6. The above result actually holds for any s C (see [36]). P n n For s 2 1 we obtain the symbol ρ´n 2 1; ξ by setting j 2 in (11.3). Moreover, it “ ´ 2 p ´ q “ Sn´1 follows from (9.6) that Sp ξ g d1, d2 with 0 d1 d2. In particular, if ξ , then 2 p| | qĂr s ă ă P Sp ξ d1, d2 , and so may take γξ γ, where γ is any given clockwise-oriented Jordan curve p| |gqĂr s “ in C , 0 that encloses d1, d2 . We then get zp´8 s r s n 1 n n´1 ´ 2 `1 n´1 c ´ n `1 ρ ξ d ξ λ σ 4 ξ λ d ξdλ. ∆ 2 ´n 1; ´ ; g “ Sn´1 p 2 ´ q “ 2iπ Sn´1 p q ż ż żγ Combining thisp with (11.2) we then arrive at the following result. A R Tn Proposition 11.7. Assume n 3, and let g GLn θ be a Riemannian metric on θ . Then we have ě P p q 3 n n n 2 ´ 2 `1 n´1 (11.4) κ g 4π Γ 1 λ σ´4 ξ; λ d ξdλ, p q“´2iπ p q 2 ´ Sn´1 p q ż żγ ` ˘ where σ 4 ξ; λ and γ are as above. ´ p q n 1 8 n 2 8 3 2 Remark 11.8. For ℜλ 0 we have Γ n 1 λ´ 2 ` t 2 ´ e´tλdt 2 tn´ e´t λdt. Inserting ą p 2 ´ q “ 0 “ 0 this into the integral on the r.h.s. of (11.4) and using the homogeneity of σ 4 ξ; λ we find that ş ş ´ p q 6 n 2 ´λ (11.5) κ g 4π e σ´4 ξ; λ dλ dξ. p q“´2iπ p q Rn Γ1 p q ż ˆ ż ˙ 1 π where Γ is any upward-oriented contour arg λ φ , 0 φ 2 . This kind of formula for the scalar curvature is not new; it is used byt| various| authors“ u ă (see,ă e.g., [14, 15, 18]). The advantage of using the integral in (11.4) with respect to that given in (11.5) is two-fold. First, we get an integration domain that is one-dimension smaller. Second, this domain is bounded, and so we avoid the convergence issues with integrating over unbounded domains in (11.5). ´1 ´λ Remark 11.9. We observe from [37, 54] that 2iπ 1 e σ 4 ξ; λ dλ dξ agrees with the p q Rn Γ ´ p q coefficient a2 ∆g in the short time asymptotics, p q ş ` ş ˘ n ˝ ´t∆g ´ 2 j (11.6) Tr b e t t τ ba2j ∆g , b Aθ. „ jě0 p q P “ ‰ ÿ “ ‰ Therefore, from (11.5) we also can infer that

1 n ´ 2 a2 ∆g 4π κ g . p q“´6p q p q In particular, when θ 0 we recover the well-known formula for a2 ∆g in terms of κ g (see [26]). This shows the consistency“ of our normalization constants. p q p q Remark 11.10. We obtain from (11.4) the scalar curvature of NC 2-tori by analytic continuation n 1 of Γ n 1 λ´ 2 ` at n 2. Namely, p 2 ´ q “ 12 κ g σ´4 ξ; λ dξdλ. p q“´2iπ S1 p q ż żγ This is consistent with the definition of the definition of the scalar curvature functional in dimen- sion 2 (see [14, Eq. (5.23)]). In particular, in the same way as in Remark 11.8, for conformally flat NC 2-tori we recover the formulas for the scalar curvature in terms of σ´4 ξ; λ of [14, 15]. 25 p q Remark 11.11. Rosenberg [48] introduced the analogue the Levi-Civita connection and Riemann curvature tensor for Riemannian metrics on NC tori. This leads us to an alternative notion of scalar curvature for curved NC tori.

Appendix A. Proof of Equality (2.10) In this appendix we briefly sketch a proof of Eq. (2.10). Suppose that M n,g is a closed p q 1 1 8 ´ n ´ 2 Riemannian manifold, and let f C M . In view of (2.9) and of the fact that ds cn ∆g we have P p q “ n´2 n 1 n´2 n´2 1 ´ n ´ 2 ` 2 ´ n ´s (A.1) fds cn Res f∆g cn Ress n 1 Tr f∆ . ´ “ n “ n “ 2 ´ g ż ` ˘ “´s ‰ As is well-known (see, e.g., [29]), the residues of the zeta function Tr f∆g are related to the coefficients of the short time heat trace asymptotics, “ ‰ n ´t∆g ´ 2 `j Tr fe 0` t fa2j ∆g ν g . „tÑ p q p q jě0 żM “ ‰ ÿ In particular, we have 1 ´s n ´ (A.2) Ress n 1 Tr f∆ Γ 1 fa2 ∆g ν g . “ 2 ´ g “ 2 ´ p q p q żM “ ‰ ` ˘ The first few coefficients a2j ∆g are known (see [26]). For j 1, we have p q “ 1 n ´ 2 a2 ∆g 4π κ g . p q“´6p q p q 2 ´ n´ 1 1 Combining this with (A.1) and (A.2) shows that fds cn M fκ g ν g with cn given by (2.11). The proof is complete. “ ´ p q p q ş ş Appendix B. L2-Integration Formulas In this appendix we compare the integration formulas of this paper with the curved integration formula of [42]. The latter is an L2-version of the curved integration formula (10.6), in the sense H 2 Tn that it holds for any element of θ L θ . The approach of [42] is based on the trace formula for noncommutative tori of [43, Theorem“ p 6.5]q and a “curved” L 1,8-Cwikel estimate which is proved in [42]. The approach to the trace formula in [43] relies on a C˚-algebraic approach to the principal symbol [43, 50]. In [42] the “curved” L 1,8-Cwikel estimate is deduced from a flat L 1,8-Cwikel estimate via various operator theoretic considerations. The point we would like to make in this appendix is that in order to get L2-integration formulas we only need the flat L 1,8-Cwikel estimate of [42]. We shall actually get an L2-integration formulas for ΨDOs (see Theorem B.4 below). The main result of [42] then follows from this as a mere special case. In particular, this allows us to bypass the C˚-algebraic considerations of [43, 50] and the operator theoretic considerations of [42] to get an even more general result. To wit note that if x Hθ, then the left multiplication by x maps Aθ to Hθ, but in general P this operator need not be bounded. In what follows we denote by 2 the norm of Hθ. The flat L 1,8-Cwikel estimate of [42] can be stated as follows. }¨}

1,8 ´ n Proposition B.1 (Flat L -Cwickel Estimate [42]). For every x Hθ the operator x 1 ∆ 2 is bounded and weak-trace class. Moreover, there is C 0 independentP of x such that p ` q ą ´ n (B.1) x 1 ∆ 2 C x 2. p ` q 1,8 ď } } › › We observe we actually have a pseudodifferential› › version of the above Cwikel estimate. Proposition B.2 (Pseudodifferential L 1,8-Cwikel Estimate). Let P Ψ´n Tn . For every P p θ q x Hθ the operator xP is bounded and weak trace class. Moreover, there is CP 0 independent ofPx such that ą

(B.2) xP CP x 2. 1,8 ď } } 26 › › › › n Proof. As P has order n, the operator 1 ∆ 2 P is a zeroth order ΨDO by Corollary 4.11, and ´ p ` q so it is bounded by Proposition 4.12. Let x Hθ. The first part of Proposition B.1 ensures us that n P n n x 1 ∆ ´ 2 is bounded and weak-trace class, and so the operator xP x 1 ∆ ´ 2 1 ∆ 2 P isp in` theq ideal L 1,8. Moreover, we have “ p ` q ¨ p ` q n n n n ´ 2 2 ´ 2 2 xP 1, x 1 ∆ 1 ∆ P 1, x 1 ∆ 1, 1 ∆ P . } } 8 ď} p ` q ¨ p ` q } 8 ď} p ` q } 8}p ` q } Combining this with (B.1) gives the result. 

´ n Remark B.3. Applying the above result to P 1 ∆g 2 allows us to recover the curved L 1,8-Cwikel estimate of [42]. “ p ` q We are now in a position to get an L2-version of Corollary 8.3. ´n Tn H Theorem B.4. Let P Ψ θ . For every x θ the operator xP is strongly measurable, and we have P p q P 1 (B.3) xP τ xcP , ´ “ n s ż “ where cP is defined as in (5.2). Proof. In the same way as in the proof of Corollary 8.3, in order to show that xP is strongly measurable and satisfies the trace formula (B.3), we only have to show that, for every positive trace ϕ on L1,8, we have 1 ϕ xP τ xcP . p q“ n We know by Corollary 8.3 that the equality holds for“ all‰ x Aθ. It follows from the continuity P of ϕ and the estimate (B.2) that the l.h.s. is a continuous linear form on Hθ. As cP Aθ the P r.h.s. is continuous as well. Combining this with the density of Aθ in Hθ we then deduce that the equality holds for any x Hθ. This completes the proof.  P By using Theorem B.4 and arguing along the same lines as that of the proof of Theorem 10.6 we recover the main result of [42] in the following form.

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School of Mathematics, Sichuan University, Chengdu, China E-mail address: [email protected]

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