Reconstruction of manifolds in noncommutative geometry Adam Rennie1,2 and Joseph C. Várilly3 1 Institute for Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark 2 Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia 3 Escuela de Matemática, Universidad de Costa Rica, 11501 San José, Costa Rica arXiv:math.OA/0610418v4, 31 enero 2008 Abstract We show that the algebra A of a commutative unital spectral triple (A,H,D) satisfying several additional conditions, slightly stronger than those proposed by Connes, is the algebra of smooth functions on a compact spin manifold. Contents 1 Introduction 2 2 Spectral triples and smooth functional calculus 4 3 Geometric properties of noncommutative manifolds 10 3.1 Axiomatic conditions on commutative spectral triples . . . . . . . . . . . . . . . . 12 3.2 First consequences of the geometric conditions . . . . . . . . . . . . . . . . . . . 15 4 The cotangent bundle 19 5 A Lipschitz functional calculus 25 6 Point-set properties of the local coordinate charts 30 7 Reconstruction of a differential manifold 37 7.1 Local structure of the operator D . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 7.2 Injectivity of the local coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 41 7.3 Riemannian structure of the spectral manifold . . . . . . . . . . . . . . . . . . . . 51 8 Poincaré duality and spinc structures 53 1 9 Conclusion and outlook 59 A Hermitian pairings on finite projective modules 61 B Another look at the geometric conditions 63 References 66 1 Introduction Noncommutative Geometry, as developed over the past several years by Connes and coworkers, has produced a profusion of examples of “noncommutative spaces” [22], many of which partake of the characteristics of smooth Riemannian manifolds, whose metric and differential structure is determined by a generalized Dirac operator. To find a common framework for those examples, Connes proposed in [21] an axiomatic framework for “noncommutative spin manifolds”. The geometry is carried by the notion of spectral triple (A,H,D); the familiar Riemannian spin geometry is recovered when A is a coordinate algebra of smooth functions on a manifold, H a Hilbert space of spinors, and D the Dirac operator determined by the spin structure and Riemannian metric. The question of reconstruction is whether the operator-theoretic framework proposed by Connes, or some variation of it, suffices to determine this spin manifold structure whenever the algebra A is commutative. In [21], Connes held out the hope that it could be so; but the extraction of a manifold from these postulates has proved elusive. In [50], a first attempt at doing so was presented, but was subsequently shown to fall short of the goal [28]. A detailed description of the reconstruction of many geometric features of a spin manifold was presented in [30], but there the starting algebra A was assumed a priori to be the smooth functions on a compact manifold. In this paper, using a slightly stronger set of conditions on a spectral triple, we show that from the further assumption of a commutative coordinate algebra A one can indeed recover a compact boundaryless manifold whose smooth functions coincide with A. One of the key themes of the axioms proposed by Connes was Poincaré duality in 𝐾-theory. Earlier results of Sullivan [58] indicated that in high dimensions, in the absence of 2-torsion and in the simply connected setting, Poincaré duality in 𝐾𝑂-theory characterizes the homotopy type of a compact manifold. While as a guiding principal such an idea is very attractive, we have not found a way to implement this approach to reconstruct a manifold. Instead, we utilize an earlier formulation of Poincaré duality in noncommutative geometry which is phrased at the level of Hochschild chains, and thus is more useful for elaborating a proof. This concrete version of Poincaré duality is described by a “closedness condition” [18, VI.4.𝛾], which historically arose from attempts to fine-tune the Lagrangian of the Standard Model of elementary particles, and conceptually is an analogue of Stokes’ theorem. Poincaré duality in 𝐾-theory plays no role in our reconstruction of a manifold as a compact space 𝑋 with charts and smooth transition functions. However, once that has been achieved, it is needed to show that 𝑋 carries a spinc structure and to identify the class of (A,H,D) as the fundamental class of the spinc manifold. The key to this is Plymen’s characterization of spinc structures [48] as Morita equivalence bimodules for the Clifford action induced by the metric. Indeed, it would be economical to replace Poincaré duality by postulating instead the existence of such bimodules; we 2 touch on this in our final section. Compactness of the manifold, or equivalently the condition that the coordinate algebra have a unit, is an essential technical feature of our proof. However, the reconstruction of noncompact manifolds should also be possible, under some alternative conditions along the lines suggested in [27, 51]. Indeed, many of the crucial arguments used in reconstructing the coordinate charts are completely local. The proof that the Gelfand spectrum 𝑋 = sp(A) is a differential manifold is quite long, but may be conceptually broken into two steps. The first is to construct a vector bundle over 𝑋 which plays the role of the cotangent bundle. Already at this stage we need to deploy all the conditions on our spectral triple (except Poincaré duality in𝐾-theory and a metric condition). In particular, we identify local trivializations and bases of this bundle in terms of the ‘1-forms’ given by the orientability condition. These 1-forms [D, 𝑎 𝑗𝛼], for 𝑗 = 1, . . . , 𝑝, 𝛼 = 1, . . . , 𝑛, generate the sections of this bundle, and the aim now is to show that the maps 𝑎𝛼 = (𝑎1 𝛼, . . . , 𝑎 𝑝 𝛼) : 𝑋 → ℝ𝑝 provide coordinates on suitable open subsets of 𝑋 . This is accomplished by proving that 𝑎𝛼 is locally one-to-one and open. The tools used here are a Lipschitz functional calculus, some measure theoretic results of Voiculescu [61], some basic point set topology and properties of the map 𝑎𝛼, and finally the unique continuation properties for Dirac-type operators [5, 36]. The main tools in the proof are a multivariate 𝐶∞ functional calculus for regular spectral triples [51], which we present here; as well as a Lipschitz functional calculus. The first of these enables us to construct partitions of unity and local inverses within the algebra A. ▶ The plan of the paper is as follows. In Section 2 we give some standard definitions and background results, including the 𝐶∞ functional calculus and its immediate consequences. In Section 3, we introduce the conditions on a spectral triple needed to establish our main result. Section 4 details the construction of the cotangent bundle, while Section 5 develops a Lipschitz functional calculus needed to deal with the topology of our coordinate charts. Sections 6 and 7 contain the detailed proof that we do indeed recover a manifold. We develop the necessary point set topology to establish that the spectrum of our algebra is a manifold: the main issue is the absence of branch points in the chart domains. We show that the algebra generated by A and [D,A] is locally a direct sum of Clifford actions arising from one or several Riemannian metrics, for which D is (again, locally) a direct sum of Dirac-type operators. Then we use the unique continuation properties of Dirac-type operators and the local description of D to banish any branch points and thereby get a manifold. That done, we assemble the Clifford actions globally, and so produce the Clifford action of a single Riemannian metric. In Section 8 we explain in some detail how the (unique) spinc structure arises from Poincaré duality in 𝐾-theory. The Dirac operator is shown to differ from D by at most an endomorphism of the corresponding spinor bundle. Section 9 collects some further remarks on our postulates and their possible variants. Appendix A establishes some basic results about Hermitian pairings on finite projective modules. Appendix B examines additional results about our conditions, in particular the redundancy of the metric condition. 3 Acknowledgments This work has profited from discussions with Alan Carey, Alain Connes, Nigel Higson, Steven Lord, Ryszard Nest, and Iain Raeburn. JCV is grateful to Iain Raeburn for warm hospitality at the University of Newcastle and to Ryszard Nest for a timely visit to Copenhagen. AR thanks JCV for unique hospitality whilst visiting Costa Rica. This work was supported by an ARC grant, DP0211367, by the SNF, Denmark, by a University of Newcastle Visitor Grant, and by the European Commission grant MKTD–CT–2004–509794 at the University of Warsaw. Support from the Universidad de Costa Rica is also acknowledged. Both authors would especially like to thank Piotr Hajac for generous hospitality at IMPAN during the course of this work. 2 Spectral triples and smooth functional calculus The central notion of this paper is that of a spectral triple [19] over a commutative algebra. We begin by recalling several basic definitions, in order to establish a suitable functional calculus for them. Definition 2.1. A spectral triple (A,H,D) is given by: (1) A faithful representation 𝜋 : A → B(H) of a unital ∗-algebra A by bounded operators on a Hilbert space H; and (2) A selfadjoint operator D on H, with dense domain DomD, such that for each 𝑎 ∈ A, [D, 𝜋(𝑎)] extends to a bounded operator on H and 𝜋(𝑎) (1 +D2)−1/2 is a compact operator. The spectral triple is said to be even if there is an operator Γ = Γ∗ ∈ B(H) such that Γ2 = 1 (this determines a ℤ2-grading on H), for which [Γ, 𝜋(𝑎)] = 0 for all 𝑎 ∈ A and ΓD+DΓ = 0 (i.e., 𝜋(A) is even and D is odd with respect to the grading). If no such grading is available, the spectral triple is called odd. Remark 2.1. Since A is faithfully represented on H, we may and shall omit 𝜋, regarding A as a subalgebra of B(H). As such, its norm closure A = 𝐴 is a 𝐶∗-algebra. Remark 2.2. In this paper, we shall always assume that A is unital. Nonunital spectral triples have been studied in [51, 52] under the assumption that A has a dense ideal with local units. Another class of nonunital spectral triples are those arising from Moyal products, analyzed in detail in [27] (and anticipated in [29]). The Moyal example shows that it is important to treat a certain unitization of A as part of the data of a (nonunital) spectral triple, so that it is proper to focus first on the unital case. Definition 2.2. The operator D gives rise to two (commuting) derivations of operators on H; we shall denote them by d𝑥 := [D, 𝑥], 𝛿𝑥 := [|D|, 𝑥], for 𝑥 ∈ B(H). According to Definition 2.1, A lies within Dom d := { 𝑥 ∈ B(H) : [D, 𝑥] ∈ B(H) }. A spectral triple (A,H,D) is called 𝑄𝐶∞ if A ∪ dA ⊆ ∞⋂ 𝑚=1 Dom 𝛿𝑚 . 4 Remark 2.3. The terminology 𝑄𝐶∞ was introduced in [10], to distinguish “quantum” differentia- bility of operators from “classical” differentiability of smooth functions. One can also define 𝑄𝐶𝑘 , for 𝑘 ∈ ℕ, by requiring only that A ∪ dA ⊆ Dom 𝛿𝑚 for 𝑚 = 1, . . . , 𝑘 . Such spectral triples are more often referred to as regular [21, 30], and have been called smooth in [51]. Definition 2.3. If (A,H,D) is a 𝑄𝐶∞ spectral triple, the family of seminorms 𝑞𝑚 (𝑎) := ∥𝛿𝑚𝑎∥ and 𝑞′𝑚 (𝑎) := ∥𝛿𝑚 ( [D, 𝑎])∥, 𝑚 = 0, 1, 2, . . . (2.1) determine a locally convex topology on A which is finer than the norm topology of 𝐴 (that is given by 𝑞0 alone) and in which the involution 𝑎 ↦→ 𝑎∗ is continuous. Let A𝛿 denote the completion of A in the topology of (2.1). We quote Lemma 16 of [51]. Lemma 2.4. Let (A,H,D) be a𝑄𝐶∞ spectral triple. The Fréchet algebra A𝛿 is a pre-𝐶∗-algebra, and (A𝛿,H,D) is also a 𝑄𝐶∞ spectral triple. □ Recall that a pre-𝐶∗-algebra is a dense subalgebra of a 𝐶∗-algebra which is stable under the holomorphic functional calculus of that 𝐶∗-algebra. There is little loss of generality in assuming that A is complete in the topology given by (2.1), thus is a Fréchet pre-𝐶∗-algebra, and we shall do so. This condition guarantees that the spectrum of an element 𝑎 ∈ A coincides with its spectrum in the 𝐶∗-completion 𝐴, and that any character of the pre-𝐶∗-algebra A extends to a character of 𝐴 as well. We shall denote the character space by 𝑋 := sp(A) = sp(𝐴). Moreover, when A is a Fréchet pre-𝐶∗-algebra, so also is the algebra 𝑀𝑛 (A) of 𝑛 × 𝑛 matrices with entries in 𝐴, whose𝐶∗-completion is 𝑀𝑛 (𝐴); for a proof, see [57]. By a theorem of Bost [6,30], the (topological) 𝐾-theories of A and 𝐴 coincide: 𝐾𝑖 (A) = 𝐾𝑖 (𝐴) for 𝑖 = 0, 1. By replacing any seminorm 𝑞 by 𝑎 ↦→ 𝑞(𝑎) + 𝑞(𝑎∗) if necessary, we may suppose that 𝑞(𝑎) = 𝑞(𝑎∗) for all 𝑎 ∈ A. We note in passing that the multiplication in the Fréchet algebra A is jointly continuous [43]. Lemma 2.5. Let A be a Fréchet pre-𝐶∗-algebra and let 𝐴 be its 𝐶∗-completion. If 𝑞 ∈ 𝐴 is a projector (i.e., a selfadjoint idempotent), and if 0 < 𝜀 < 1, then there is a projector 𝑞 ∈ A such that ∥𝑞 − 𝑞∥ < 𝜀. Proof. Choose 𝛿 with 0 < 𝛿 < 𝜀/32, and let 𝑏 = 𝑏∗ ∈ A be such that ∥𝑏 − 𝑞∥ < 𝛿. Observe that ∥𝑏2 − 𝑏∥ = ∥𝑏2 − 𝑞2 + 𝑞 − 𝑏∥ ⩽ (∥𝑏 + 𝑞∥ + 1) ∥𝑏 − 𝑞∥ ⩽ (3 + 𝛿)∥𝑏 − 𝑞∥ < 𝛿(3 + 𝛿) < 4𝛿. Since A is a Fréchet pre-𝐶∗-algebra, one may, provided 𝛿 is sufficiently small, use holomorphic functional calculus to construct a homotopy within A from 𝑏 to 𝑒 ∈ A such that 𝑒2 = 𝑒 and ∥𝑒 − 𝑏∥ < 2∥𝑏2 − 𝑏∥; see [30, Lemma 3.43], for instance. Let 𝑞 := 𝑒𝑒∗(𝑒𝑒∗ + (1 − 𝑒∗) (1 − 𝑒))−1. Then 𝑞 is a projector in 𝐴, and it lies in A since 𝑒𝑒∗ + (1 − 𝑒∗) (1 − 𝑒) is invertible in the pre-𝐶∗-algebra A. By taking 𝐴 to be faithfully represented on a Hilbert space H, we can write 𝑒, 𝑞 and 𝑏 as operators on H = 𝑒H ⊕ (1 − 𝑒)H, as follows: 𝑒 = ( 1 𝑇 0 0 ) , 𝑞 = ( 1 0 0 0 ) , 𝑏 = ( 𝑅 𝑉 𝑉∗ 𝑆 ) , 5 with 𝑅, 𝑆 selfadjoint and 𝑉,𝑇 : (1 − 𝑒)H → 𝑒H bounded. Then ∥𝑒 − 𝑏∥ < 8𝛿 < 𝜀/4 means ∥(𝑒 − 𝑏)∗(𝑒 − 𝑏)∥ < 𝜀2/16, which entails ∥(𝑅 − 1)2 +𝑉𝑉∗∥ < 𝜀2 16 , ∥(𝑉 − 𝑇)∗(𝑉 − 𝑇) + 𝑆2∥ < 𝜀2 16 , so that ∥𝑉 ∥ < 𝜀/4, ∥𝑉 − 𝑇 ∥ < 𝜀/4, and therefore ∥𝑞 − 𝑒∥ = ∥𝑇 ∥ < 𝜀/2. Hence ∥𝑞 − 𝑞∥ ⩽ ∥𝑞 − 𝑒∥ + ∥𝑒 − 𝑏∥ + ∥𝑏 − 𝑞∥ < 𝜀 2 + 𝜀 4 + 𝛿 < 𝜀. □ A𝑄𝐶∞ spectral triple (A,H,D) for which A is complete has not only a holomorphic functional calculus for A, but also a 𝐶∞ functional calculus for selfadjoint elements: we quote [51, Prop. 22]. Proposition 2.6 (𝐶∞ Functional Calculus). Let (A,H,D) be a 𝑄𝐶∞ spectral triple, and suppose A is complete. Let 𝑓 : ℝ → ℂ be a 𝐶∞ function in a neighbourhood of the spectrum of 𝑎 = 𝑎∗ ∈ A. If we define 𝑓 (𝑎) ∈ 𝐴 using the continuous functional calculus, then in fact 𝑓 (𝑎) lies in A. □ Remark 2.7. For each 𝑎 = 𝑎∗ ∈ A, the𝐶∞-functional calculus defines a continuous homomorphism Ψ : 𝐶∞(𝑈) → A, where 𝑈 ⊂ ℝ is any open set containing the spectrum of 𝑎, and the topology on 𝐶∞(𝑈) is that of uniform convergence of all derivatives on compact subsets. The following proposition extends this result to the case of smooth functions of several variables, yielding a multivariate𝐶∞ functional calculus. Before stating it, we recall the continuous functional calculus for a finite set 𝑎1, . . . , 𝑎𝑛 of commuting selfadjoint elements of a unital𝐶∗-algebra 𝐴. These generate a unital ∗-algebra whose closure in 𝐴 is a 𝐶∗-subalgebra 𝐶∗⟨1, 𝑎1, . . . , 𝑎𝑛⟩; let Δ be its (compact) space of characters. Evaluation of polynomials 𝑝 ↦→ 𝑝(𝑎1, . . . , 𝑎𝑛) yields a surjective morphism from 𝐶 (∏𝑛 𝑗=1 sp 𝑎 𝑗 ) onto 𝐶∗⟨1, 𝑎1, . . . , 𝑎𝑛⟩ ≃ 𝐶 (Δ) which corresponds, via the Gelfand functor, to a continuous injection Δ ↩→ ∏𝑛 𝑗=1 sp 𝑎 𝑗 ; this joint spectrum Δ may thus be regarded as a compact subset of ℝ𝑛. If ℎ ∈ 𝐶 (Δ), we may define ℎ(𝑎1, . . . , 𝑎𝑛) as the image of ℎ |Δ in 𝐶∗⟨1, 𝑎1, . . . , 𝑎𝑛⟩ under the Gelfand isomorphism. Proposition 2.8. Let (A,H,D) be a 𝑄𝐶∞ spectral triple. Let 𝑎1, . . . , 𝑎𝑛 be mutually commuting selfadjoint elements of A, and let Δ ⊂ ℝ𝑛 be their joint spectrum. Let 𝑓 : ℝ𝑛 → ℂ be a𝐶∞ function supported in a bounded open neighbourhood𝑈 of Δ. Then 𝑓 (𝑎1, . . . , 𝑎𝑛) lies in A𝛿. Proof. We first define the operator 𝑓 (𝑎1, . . . , 𝑎𝑛) lying in 𝐴, the 𝐶∗-completion of A, using the continuous functional calculus. Since 𝑓 is a compactly supported smooth function on ℝ𝑛, we may define 𝑓 (𝑎1, . . . , 𝑎𝑛) ∈ 𝐴 alternatively by a Fourier integral: 𝑓 (𝑎1, . . . , 𝑎𝑛) = (2𝜋)−𝑛/2 ∫ ℝ𝑛 𝑓 (𝑠1, . . . , 𝑠𝑛) exp(𝑖 𝑠 · 𝑎) 𝑑𝑛𝑠, (2.2) where 𝑠 · 𝑎 = 𝑠1𝑎1 + · · · + 𝑠𝑛𝑎𝑛. Since 𝛿 (and likewise d = adD) is a norm-closed derivation from A to B(H), we may conclude that 𝑓 (𝑎1, . . . , 𝑎𝑛) ∈ Dom 𝛿 with 𝛿( 𝑓 (𝑎1, . . . , 𝑎𝑛)) = (2𝜋)−𝑛/2 ∫ ℝ𝑛 𝑓 (𝑠1, . . . , 𝑠𝑛) 𝛿(exp(𝑖 𝑠 · 𝑎)) 𝑑𝑛𝑠, (2.3) 6 provided we can establish dominated convergence for the integral on the right hand side [7]. Just as in the one-variable case [51], since each 𝑎 𝑗 ∈ Dom 𝛿, we find that exp(𝑖 𝑠 · 𝑎) = ∏ 𝑗 exp(𝑖𝑠 𝑗𝑎 𝑗 ) lies in Dom 𝛿 also: its factors are given by the expansion 𝛿(exp(𝑖𝑠 𝑗𝑎 𝑗 )) = 𝑖𝑠 𝑗 ∫ 1 0 exp(𝑖𝑡𝑠 𝑗𝑎 𝑗 ) 𝛿(𝑎 𝑗 ) exp(𝑖(1 − 𝑡)𝑠 𝑗𝑎 𝑗 ) 𝑑𝑡, (2.4) and in particular, ∥𝛿(exp(𝑖 𝑠 · 𝑎))∥ ⩽ 𝐶∑ 𝑗 |𝑠 𝑗 |, 𝐶 = max 𝑗 ( ∥𝛿(𝑎 𝑗 )∥ ∏ 𝑖≠ 𝑗 ∥𝑎𝑖∥ ) . A norm bound which dominates the right hand side of (2.3) is thus given by∫ ℝ𝑛 | 𝑓 (𝑠1, . . . , 𝑠𝑛) | ∥𝛿(exp(𝑖 𝑠 · 𝑎))∥ 𝑑𝑛𝑠 ⩽ 𝐶 𝑛∑︁ 𝑗=1 (2𝜋)−𝑛/2 ∫ ℝ𝑛 | 𝑓 (𝑠1, . . . , 𝑠𝑛) | |𝑠 𝑗 | 𝑑𝑛𝑠. Let 𝐴0 be the completion of A for the norm ∥𝑎∥D := ∥𝑎∥ + ∥d𝑎∥; notice that 𝐴0 ⊆ 𝐴. Replacing 𝛿 by d in the previous argument, we find that ∥ 𝑓 (𝑎1, . . . , 𝑎𝑛)∥D ⩽ ∥ 𝑓 ∥1 + ∥d𝑎∥ 𝑛∑︁ 𝑗=1 (2𝜋)−𝑛/2 ∫ ℝ𝑛 | 𝑓 (𝑠1, . . . , 𝑠𝑛) | |𝑠 𝑗 | 𝑑𝑛𝑠. Therefore, 𝑓 (𝑎1, . . . , 𝑎𝑛) can be approximated, in the ∥ · ∥D norm, by Riemann sums for (2.2) belonging to A, and thus 𝑓 (𝑎1, . . . , 𝑎𝑛) ∈ 𝐴0. Since 𝛿 and d are commuting derivations, we obtain that 𝛿( 𝑓 (𝑎1, . . . , 𝑎𝑛)) ∈ Dom d and d( 𝑓 (𝑎1, . . . , 𝑎𝑛)) ∈ Dom 𝛿 for 𝑎 ∈ A, and ∥𝛿(d( 𝑓 (𝑎1, . . . , 𝑎𝑛)))∥ is bounded by a linear combina- tion of expressions ∥𝛿(d𝑎 𝑗 )∥ ∫ | 𝑓 (𝑠1, . . . , 𝑠𝑛) | |𝑠 𝑗 | 𝑑𝑛𝑠 and ∥𝛿𝑎 𝑗 ∥ ∥d𝑎𝑘 ∥ ∫ | 𝑓 (𝑠1, . . . , 𝑠𝑛) | |𝑠 𝑗 𝑠𝑘 | 𝑑𝑛𝑠. In particular, ∥𝛿( 𝑓 (𝑎1, . . . , 𝑎𝑛))∥D also has a bound of this type. For each 𝑚 = 1, 2, 3, . . . , let 𝐴𝑚 be the completion of A for the norm ∑ 𝑘⩽𝑚 ∥𝛿𝑘 (𝑎)∥D. Then 𝛿 extends to a norm-closed derivation from 𝐴𝑚 to B(H), and an ugly but straightforward induction on 𝑚 shows that each 𝛿𝑘 ( 𝑓 (𝑎1, . . . , 𝑎𝑛)) and 𝛿𝑘 (d( 𝑓 (𝑎1, . . . , 𝑎𝑛))) lies in its domain, using the convergence of ∫ | 𝑓 (𝑠1, . . . , 𝑠𝑛) | |𝑝(𝑠1, . . . , 𝑠𝑛) | 𝑑𝑛𝑠 for 𝑝 a polynomial of degree ⩽ 𝑚 + 1. Thus, 𝑓 (𝑎1, . . . , 𝑎𝑛) ∈ 𝐴𝑚. Since A𝛿 = ⋂ 𝑚⩾0 𝐴𝑚, we conclude that 𝑓 (𝑎1, . . . , 𝑎𝑛) ∈ A𝛿. □ Remark 2.9. For most of this paper, the spectral triples we consider will be commutative and will satisfy the first order property [18, VI.4.𝛾], meaning that [[D, 𝑎], 𝑏] = 0 for all 𝑎, 𝑏 ∈ A. In such cases, to define [D, 𝑓 (𝑎1, . . . , 𝑎𝑛)], we may note that for any polynomial 𝑝, the first-order property allows us to write [D, 𝑝(𝑎1, . . . , 𝑎𝑛)] = 𝑛∑︁ 𝑗=1 𝜕𝑗 𝑝(𝑎1, . . . , 𝑎𝑛) [D, 𝑎 𝑗 ], (2.5) with 𝜕𝑗 𝑝 being the 𝑗-th partial derivative of 𝑝. By a 𝐶1-approximation argument – see Proposi- tion 5.1 below – we obtain [D, 𝑓 (𝑎1, . . . , 𝑎𝑛)] = ∑ 𝑗 𝜕𝑗 𝑓 (𝑎1, . . . , 𝑎𝑛) [D, 𝑎 𝑗 ] for any 𝑓 satisfying the hypotheses of Proposition 2.8. Since (A,H,D) is 𝑄𝐶∞, one sees immediately that the right hand side of (2.5) belongs to the smooth domain of 𝛿. 7 We now use the 𝐶∞ functional calculus to prove the existence of certain elements of A, where (A,H,D) is a (unital) commutative 𝑄𝐶∞ spectral triple. The algebra elements we are looking for are smooth partitions of unity and local inverses. Lemma 2.10. Let (A,H,D) be a 𝑄𝐶∞ spectral triple where A is commutative and complete. Let {𝑈𝛼 : 𝛼 = 1, . . . , 𝑛 } be any finite open cover of the compact Hausdorff space 𝑋 = sp(A). Then there exist 𝜙𝛼 ∈ A, for 𝛼 = 1, . . . , 𝑛, such that supp 𝜙𝛼 ⊆ 𝑈𝛼, 0 ⩽ 𝜙𝛼 ⩽ 1, and 𝑛∑︁ 𝛼=1 𝜙𝛼 = 1. (2.6) Proof. SinceA = A𝛿 is a pre-𝐶∗-algebra, its character space is the same as that of its𝐶∗-completion, 𝐴; thus 𝑋 = sp(𝐴) is a compact Hausdorff space. Now, 𝑋 always admits a continuous partition of unity [24] subordinate to the cover {𝑈𝛼}, i.e., we can find 𝜙1, . . . , 𝜙𝑛 ∈ 𝐴 satisfying (2.6). Let 𝑝 ∈ 𝑀𝑛 (𝐴) be the matrix whose (𝛼, 𝛽)-entry is (𝜙𝛼𝜙𝛽)1/2; then 𝑝 is a projector, that is, a selfadjoint idempotent: 𝑝2 = 𝑝 = 𝑝∗. Since𝑀𝑛 (A) is a Fréchet pre-𝐶∗-algebra, it is known [6,30] that the inclusion𝑀𝑛 (A) ↩→ 𝑀𝑛 (𝐴) induces a homotopy equivalence between the respective sets of idempotents in these algebras. Thus, there is a norm-continuous path of idempotents 𝑡 ↦→ 𝑒𝑡 = 𝑒2 𝑡 ∈ 𝑀𝑛 (𝐴) linking 𝑝 = 𝑒0 to an idempotent 𝑒1 ∈ 𝑀𝑛 (A). Moreover, such a path may be chosen so that each ∥𝑝 − 𝑒𝑡 ∥ < 𝜀 for a preassigned 𝜀 > 0 [30, Lemma 3.43]. We choose 𝜀 < 1/3𝑛. Replacing 𝑒𝑡 by 𝑞𝑡 := 𝑒𝑡𝑒 ∗ 𝑡 (𝑒𝑡𝑒∗𝑡 + (1 − 𝑒∗𝑡 ) (1 − 𝑒𝑡))−1, we may link 𝑝 to 𝑞 = 𝑞1 by a path of projectors in 𝑀𝑛 (𝐴); by the proof of Lemma 2.5, we obtain ∥𝑝 − 𝑞𝑡 ∥ < 3𝜀 < 1/𝑛 for 0 ⩽ 𝑡 ⩽ 1. Since the positive element 𝑒1𝑒 ∗ 1 + (1 − 𝑒∗1) (1 − 𝑒1) is invertible in the pre-𝐶∗-algebra 𝑀𝑛 (A), 𝑞 lies in 𝑀𝑛 (A). By the Serre–Swan theorem [59], the projectors 𝑝 and 𝑞 define vector bundles over 𝑋 of the same rank: the rank is given by the matrix trace tr 𝑝 = tr 𝑞, a locally constant integer-valued function in 𝐴 = 𝐶 (𝑋). Write 𝜓𝛼 := 𝑞𝛼𝛼 ∈ A, and notice that 𝜓𝛼 ⩾ 0 in 𝐴; then 𝑛∑︁ 𝛼=1 𝜓𝛼 = tr 𝑞 = tr 𝑝 = 𝑛∑︁ 𝛼=1 𝜙𝛼 = 1. (2.7) We now modify the elements 𝜓𝛼 to obtain a partition of unity subordinate to the cover {𝑈𝛼}. By construction, ∥𝜓𝛼 − 𝜙𝛼∥ ⩽ ∥𝑞 − 𝑝∥ < 1/𝑛 for each 𝛼. Choose a smooth function 𝑔 : ℝ → [0, 1] with support in [𝜀, 1 + 𝜀] such that 0 < 𝑔(𝑡) ⩽ 1 for 𝜀 < 𝑡 ⩽ 1. Then define 𝑉𝛼 := { 𝑥 ∈ 𝑋 : 𝜓𝛼 (𝑥) > 𝜀 } ⊂ 𝑈𝛼. Setting 𝜒𝛼 (𝑥) := 𝑔(𝜓𝛼 (𝑥)) gives 𝜒𝛼 > 0 on 𝑉𝛼 and supp 𝜒𝛼 ⊂ 𝑈𝛼. For all 𝑥 ∈ 𝑋 , there is some 𝜒𝛽 with 𝜒𝛽 (𝑥) > 0: for if not, then 𝜓𝛽 (𝑥) ⩽ 𝜀 for each 𝛽, and∑ 𝛽 𝜓𝛽 (𝑥) ⩽ 𝑛𝜀 < 1, contradicting (2.7). We now define 𝜙𝛼 := 𝜒𝛼 / ∑ 𝛽 𝜒𝛽, which clearly satisfies (2.6). Since 𝜒𝛼 = 𝑔(𝜓𝛼), Proposition 2.6 shows that 𝜒𝛼 ∈ A. Moreover, ∑ 𝛽 𝜒𝛽 is invertible in A, and hence 𝜙𝛼 ∈ A for each 𝛼, as required. □ Corollary 2.11. Given a 𝑄𝐶∞ spectral triple (A,H,D) where A is commutative and complete, let 𝐾 ⊂ 𝑈 ⊂ sp(A) where 𝐾 is compact and 𝑈 is open. Then there is some 𝜓 ∈ A such 0 ⩽ 𝜓 ⩽ 1, 𝜓 ≡ 1 on 𝐾 , and 𝜓 ≡ 0 outside𝑈. Proof. There is a partition of unity {𝜓, 1 − 𝜓} subordinate to the open cover {𝑈, sp(A) \ 𝐾}, with 𝜓 ∈ A. □ 8 Lemma 2.12. Let (A,H,D) again be a 𝑄𝐶∞ spectral triple with A commutative and complete. Let 𝑎 ∈ A have compact support contained in an open subset 𝑈 ⊂ sp(A). Then there exists 𝜙 ∈ A such that 𝜙𝑎 = 𝑎 and supp 𝜙 ⊂ 𝑈. Proof. Choose 𝑏 ∈ 𝐴 = 𝐶 (𝑋), by Urysohn’s lemma, such that 0 ⩽ 𝑏 ⩽ 1, 𝑏(𝑥) = 1 for 𝑥 ∈ supp 𝑎 and 𝑏(𝑥) = 0 for 𝑥 ∉ 𝑈. Pick 𝛿 ∈ (0, 1 2 ) and choose 𝜓 ∈ A satisfying ∥𝑏 − 𝜓∥ < 1 2𝛿. Let 𝑓 : ℝ → [0, 1] be a compactly supported smooth function such that 𝑓 (𝑡) = 0 for 𝑡 ⩽ 𝛿 and 𝑓 (𝑡) = 1 for 1−𝛿 ⩽ 𝑡 ⩽ 2. Then 𝜙 := 𝑓 (𝜓) lies inA by the𝐶∞-functional calculus. Also, for all 𝑥 ∈ supp(𝑎), the estimates 1 − 1 2𝛿 ⩽ 𝜓(𝑥) ⩽ 1 + 1 2𝛿 hold, and so 𝜙(𝑥) = 𝑓 (𝜓(𝑥)) = 1; this shows that 𝜙𝑎 = 𝑎. The continuity of 𝑏 shows that supp( 𝑓 (𝜓)) ⊆ { 𝑥 : 𝜓(𝑥) > 𝛿 } ⊆ { 𝑥 : 𝑏(𝑥) > 1 2𝛿 } ⊆ { 𝑥 : 𝑏(𝑥) ⩾ 1 2𝛿 } ⊂ 𝑈. Thus, supp 𝜙 ⊂ 𝑈, as required. □ Proposition 2.13. Let (A,H,D) be a 𝑄𝐶∞ spectral triple with A commutative and complete. Let 𝑈 ⊂ sp(A) be an open subset, and let ℎ ∈ A satisfy ℎ(𝑥) ≠ 0 for all 𝑥 ∈ 𝑈. Then whenever 𝑎 ∈ A with supp 𝑎 ⊂ 𝑈, A contains the element 𝑎ℎ−1 ∈ 𝐶 (sp(A)) defined by (𝑎ℎ−1) (𝑥) := { 𝑎(𝑥)/ℎ(𝑥) if ℎ(𝑥) ≠ 0, 0 otherwise. (2.8) Proof. The formula (2.8) clearly defines a continuous function on sp(A), so that 𝑎ℎ−1 ∈ 𝐴 = 𝐶 (sp(A)). To check that it lies in A, it is enough to replace ℎ by an invertible element ℎ̃ ∈ A for which ℎ̃(𝑥) = ℎ(𝑥) whenever 𝑎(𝑥) ≠ 0: since A = A𝛿 is a pre-𝐶∗-algebra, ℎ̃−1 will lie in A, and thus 𝑎ℎ−1 = 𝑎ℎ̃−1 ∈ A. Let 𝜀 := inf{ |ℎ(𝑥) | : 𝑥 ∈ supp 𝑎 }; since A is unital, supp 𝑎 is compact and therefore 𝜀 > 0. Let 𝑉 := 𝑈 ∩ { 𝑥 : |ℎ(𝑥) | > 𝜀/2 }. By Corollary 2.11, we can find 𝜙 ∈ A so that 0 ⩽ 𝜙 ⩽ 1, 𝜙 ≡ 0 on supp 𝑎 and 𝜙 ≡ 1 outside 𝑉 . The element ℎ + 1 2𝜀𝜙 ∈ A coincides with ℎ on supp 𝑎 and vanishes only on the compact set { 𝑥 ∉ 𝑉 : ℎ(𝑥) = −1 2𝜀 }. If this set is nonvoid, we can likewise find 𝜓 ∈ A, which is nonzero on this set and vanishes on 𝑉 , so that ℎ̃ := ℎ + 1 2𝜀𝜙 + 𝜓 vanishes nowhere on sp(A). This gives the required ℎ̃ ∈ A such that ℎ̃ ≡ ℎ on supp 𝑎. □ Corollary 2.14. Let (A,H,D) be a 𝑄𝐶∞ spectral triple with A commutative and complete. Let 𝑈 ⊂ sp(A) be an open subset, and let ℎ ∈ 𝑀𝑘 (A) be such that ℎ(𝑥) ∈ 𝑀𝑘 (ℂ) is invertible for all 𝑥 ∈ 𝑈. Let 𝑎 ∈ A with supp 𝑎 ⊂ 𝑈; then the element 𝑎ℎ−1 ∈ 𝑀𝑘 (𝐴) defined by (𝑎ℎ−1) (𝑥) := { (𝑎(𝑥) ⊗ 1𝑘 ) ℎ(𝑥)−1 if ℎ(𝑥) ≠ 0, 0 otherwise, actually lies in the subalgebra 𝑀𝑘 (A). Proof. The proof of Proposition 2.13 goes through with minor modifications; for instance, one may take 𝑉 := 𝑈 ∩ { 𝑥 : | det(ℎ(𝑥)) | > 𝜀/2 }. By adding to ℎ suitable scalar matrices which vanish on supp 𝑎, one constructs an invertible element ℎ̃ ∈ 𝑀𝑘 (A) such that 𝑎ℎ−1 = (𝑎 ⊗ 1𝑘 ) ℎ̃−1, where ℎ̃−1 ∈ 𝑀𝑘 (A) since 𝑀𝑘 (A) is a pre-𝐶∗-algebra. □ 9 3 Geometric properties of noncommutative manifolds The conditions on a spectral triple that we introduce below will control several interdependent features. Before introducing these conditions, we first discuss several such features: summability, metrics and differential structures. We recall the symmetric operator ideals L𝑝,∞(H), for 1 ⩽ 𝑝 < ∞; these are discussed in detail in [18, IV.2.𝛼] and [30, 7.C]; in [16] and [30] they are called L𝑝+(H). The Dixmier ideal L1,∞(H) is the common domain of the Dixmier traces Tr𝜔 : L1,∞(H) → ℂ. These are labelled by an uncountable index set of generalized limits (𝜔-limits [9]), but they are effectively computable only on the subspace of “measurable” operators 𝑇 for which all values Tr𝜔 𝑇 coincide; for instance, trace-class operators satisfy Tr𝜔 𝑇 = 0. Once we have established that a certain operator 𝑇 is indeed measurable, we shall write ⨏ 𝑇 instead of Tr𝜔 𝑇 to denote the common value of its Dixmier traces. If the limit lim 𝑛→∞ 1 log 𝑛 𝑛∑︁ 𝑘=0 𝜇𝑘 (𝑇) exists, where the 𝜇𝑘 (𝑇) are the singular values of𝑇 in nonincreasing order, then𝑇 is measurable and this limit equals ⨏ 𝑇 . A partial converse has been established by Lord, Sedaev and Sukochev [42]: for positive 𝑇 , measurability is equivalent to the existence of this limit. Definition 3.1. A spectral triple (A,H,D) is 𝑝+-summable, with 1 ⩽ 𝑝 < ∞, if (1 + D2)−1/2 ∈ L𝑝,∞(H). For convenience, we abbreviate ⟨D⟩ := (1 + 𝐷2)1/2, recalling that ⟨D⟩ − |D| is bounded, by functional calculus. Remark 3.1. If ⟨D⟩−1 ∈ L𝑝,∞(H), it follows that 𝐴 := ⟨D⟩−𝑝 ∈ L1,∞(H), and hence that Tr𝜔⟨D⟩−𝑝 is finite for any Dixmier trace Tr𝜔. Now if 𝑞 > 𝑝, then ⟨D⟩−𝑞 = 𝐴𝑞/𝑝 lies in the ideal L1(H) of trace-class operators, so Tr𝜔⟨D⟩−𝑞 = 0. It follows that there is at most one value of 𝑝 (independent of 𝜔) for which Tr𝜔⟨D⟩−𝑝 can be both finite and positive. Since Tr𝜔⟨D⟩−𝑝 = 𝜔-lim 𝑛→∞ 1 log 𝑛 𝑛∑︁ 𝑘=0 𝜇𝑘 (⟨D⟩−𝑝) and [9] for any bounded positive sequence {𝑡𝑛}, one can estimate: lim inf 𝑛→∞ 𝑡𝑛 ⩽ 𝜔-lim 𝑛→∞ 𝑡𝑛 ⩽ lim sup 𝑛→∞ 𝑡𝑛, then if lim inf 𝑡𝑛 > 0, every 𝜔-lim 𝑡𝑛 is also positive. If 0 < Tr𝜔⟨D⟩−𝑝 < ∞ for all 𝜔, we shall call 𝑝 the metric dimension of the spectral triple (A,H,D). 10 If 𝜂 : 𝐴 → 𝐴/ℂ 1 is the linear quotient map, then ∥ [D, 𝑎] ∥ depends only on the image 𝜂(𝑎) of 𝑎 ∈ A. Suppose that the set { 𝜂(𝑎) ∈ A/ℂ 1 : ∥ [D, 𝑎] ∥ ⩽ 1 } (3.1) is norm bounded in the Banach space 𝐴/ℂ 1. Then the following formula defines a bounded metric distance on the state space of 𝐴, as follows from [17]: 𝑑 (𝜙, 𝜓) := sup{ |𝜙(𝑎) − 𝜓(𝑎) | : ∥ [D, 𝑎] ∥ ⩽ 1 }. (3.2) (In Appendix B, we show that any irreducible unital spectral triple (A,H,D) determines a possibly unbounded distance function, which actually suffices for the purposes of our proof.) When A is commutative, the distance function (3.2) is determined by its restriction to the subspace of pure states, which may be identified with 𝑋 = sp(𝐴). In the case of A = 𝐶∞(𝑀) where 𝑀 is a compact spinc manifold, and D is a Dirac operator arising from a Riemannian metric 𝑔 on 𝑀 , this 𝑑 coincides [17] with the Riemannian distance function determined by 𝑔. Definition 3.2. If (A,H,D) is a commutative spectral triple for which the set (3.1) is bounded, we define the metric topology on the pure state space of 𝐴 to be the topology defined by the distance function (3.2). Remark 3.2. The equation (3.2) entails the inequality |𝜙(𝑎) − 𝜓(𝑎) | ⩽ ∥ [D, 𝑎] ∥ 𝑑 (𝜙, 𝜓), (3.3) so that all 𝑎 ∈ A are Lipschitz for the metric topology on 𝑋 . When [D, 𝑎] ≠ 0 for 𝑎 ∉ ℂ 1, the two topologies coincide [47, 53] if and only if the set (3.1) is precompact in 𝐴/ℂ 1. Remark 3.3. A priori, the metric topology may be finer than the original weak∗ topology on 𝑋 . In particular, the metric topology need not be compact unless the two topologies coincide. We shall henceforth adopt the convention, when discussing continuous functions on 𝑋 and so forth, that the topology of 𝑋 is by default its weak∗ topology, unless the metric topology is explicitly invoked. To exhibit the differential structure of a spectral triple, we first recall the universal graded differential algebra Ω•A of any associative algebra A [15]. It is generated as an algebra by symbols 𝑎, 𝑑𝑎 for 𝑎 ∈ A subject to the preexisting algebra relations of A, the derivation rule 𝑑 (𝑎𝑏) = 𝑎 𝑑𝑏 + 𝑑𝑎 𝑏, and the relations 𝑑 (𝑎0 𝑑𝑎1 · · · 𝑑𝑎𝑘 ) = 𝑑𝑎0 𝑑𝑎1 · · · 𝑑𝑎𝑘 , 𝑑 (𝑑𝑎1 𝑑𝑎2 · · · 𝑑𝑎𝑘 ) = 0. We may then identify Ω•A with the (normalized) Hochschild complex of A, that is, Ω𝑘A ≃ 𝐶𝑘 (A) := A ⊗ (A/ℂ 1)⊗𝑘 , 𝑎0 𝑑𝑎1 · · · 𝑑𝑎𝑘 ↔ 𝑎0 ⊗ 𝜂(𝑎1) ⊗ · · · ⊗ 𝜂(𝑎𝑘 ). When A is a Fréchet algebra, one generally uses the projective topological tensor product to topologize Ω•A. Definition 3.3. If (A,H,D) is a spectral triple, we shall use the notation CD(A) for the subalgebra of B(H) generated by A and dA = [D,A]. We can define an (algebra) representation 𝜋D : Ω•A → CD(A) by setting 𝜋D(𝑎0 𝑑𝑎1 · · · 𝑑𝑎𝑘 ) := 𝑎0 [D, 𝑎1] · · · [D, 𝑎𝑘 ] . We may regard Ω•A as an involutive algebra by setting (𝑑𝑎)∗ := −𝑑 (𝑎∗); then 𝜋D is a ∗-rep- resentation of the Hochschild chains of A as operators on H. 11 However, this CD(A) is not a graded algebra (although the count of [D, 𝑎] factors does give a filtration), and 𝜋D is not a representation of graded differential algebras. As is well known from physical examples [18,44,56], there may be nontrivial “junk forms” 𝜔 ∈ Ω•A such that 𝜋D(𝜔) = 0 but 𝜋D(𝑑𝜔) ≠ 0. On quotienting out the junk, we obtain a graded differential algebra [18, VI.1]: Λ• DA := CD(A)/𝐽𝜋, where 𝐽𝜋 = 𝜋D(𝑑 (ker 𝜋D)). In particular, 𝐽𝜋 is a differential ideal. The subspaces Λ𝑘 D A = 𝜋D(Ω𝑘A)/𝜋D(𝑑 (Ω𝑘−1A ∩ ker 𝜋D)) give the grading; the differential d on Λ• D A is defined on equivalence classes of operators, modulo junk terms, by d ( 𝑎0 [D, 𝑎1] . . . [D, 𝑎𝑘 ] + junk ) := [D, 𝑎0] [D, 𝑎1] . . . [D, 𝑎𝑘 ] + junk. Here Λ0 D A = A, and Λ1 D A may be identified with the A-bimodule of operators of the form∑ 𝑗 𝑎 𝑗 [D, 𝑏 𝑗 ] (finite sum), with 𝑎 𝑗 , 𝑏 𝑗 ∈ A. Nontrivial junk terms appear in higher degrees. 3.1 Axiomatic conditions on commutative spectral triples From now on, let (A,H,D) be a spectral triple whose algebra A is commutative (and unital). We shall also assume that the 𝐶∗-algebra 𝐴 = A is separable. In [20], Connes introduces several conditions on such an (A,H,D) in order to specify ax- iomatically what a noncommutative spin geometry should be. We now list these conditions (for the commutative case), as well as a few supplementary requirements which we need to establish our main results. Condition 1 (Dimension). The spectral triple (A,H,D) is 𝑝+-summable for a fixed positive integer 𝑝, for which Tr𝜔⟨D⟩−𝑝 > 0 for all 𝜔. By the remark after Definition 3.1, this condition determines 𝑝 uniquely; we then say that the critical summability parameter 𝑝 is the “metric dimension” of (A,H,D). Remark 3.4. A priori, there is no reason why the growth of the eigenvalues of D should be such that 𝑝 is an integer. However, the orientability condition below introduces another dimensionality parameter 𝑝 as the degree of a certain Hochschild cycle, which is necessarily an integer, and we require that these two quantities coincide. This formulation excludes certain interesting “0-dimensional” cases, such as occur when A has finite (linear) dimension. For noncommutative 0-dimensional spectral triples built over matrix algebras, we refer to [32, 37, 46]. Condition 2 (Metric). The set { 𝜂(𝑎) ∈ A/ℂ 1 : ∥ [D, 𝑎] ∥ ⩽ 1 } is norm-bounded in the Banach space 𝐴/ℂ 1. This ensures that the character space 𝑋 = sp(𝐴) is a metric space [17] with the metric distance (3.2). (See Appendix B). Remark 3.5. Since we have assumed that 𝐴 is separable, the space 𝑋 with its weak∗ topology is metrizable. However, the metric on 𝑋 defined by the equation (3.2) does not necessarily give the weak∗ topology. Condition 3 (Regularity). The spectral triple (A,H,D) is 𝑄𝐶∞, as set forth in Definition 2.2. Without loss of generality, we assume that A is complete in the topology given by (2.1) and so is a Fréchet pre-𝐶∗-algebra. 12 Condition 4 (Finiteness). The dense subspace of H which is the smooth domain of D, H∞ := ⋂ 𝑚⩾1 DomD𝑚 is a finitely generated projective A-module. Moreover, there exists a Hermitian pairing (· | ·) : H∞ ×H∞ → A, making ( H∞, (· | ·) ) a full pre-𝐶∗ right A-module; and such that, for some particular Dixmier trace TrΩ, the following relation holds: TrΩ ( (𝜉 | 𝜂) ⟨D⟩−𝑝 ) = ⟨𝜉 | 𝜂⟩, for all 𝜉, 𝜂 ∈ H∞. (3.4) Here ⟨· | ·⟩ denotes the scalar product on H. Remark 3.6. Since A is commutative, we are free to regard H∞ as either a right or left A-module. As a dense subspace of H, it is naturally a left module via the representation 𝜋, but it is algebraically more convenient to treat it as a right module; thus H∞ ≃ 𝑞A𝑚 where 𝑞 ∈ 𝑀𝑚 (A) is (a selfadjoint) idempotent. Remark 3.7. It is proved in Appendix A that (up to positive scalar multiples) the only Hermitian form which can satisfy the conditions listed here is the standard one, expressible as (𝜉 | 𝜂) = ∑ 𝑗 ,𝑘 𝜉 ∗ 𝑗 𝑞 𝑗 𝑘𝜂𝑘 on identifying H∞ with 𝑞A𝑚 with 𝑞 selfadjoint. Condition 5 (Absolute Continuity). For all nonzero 𝑎 ∈ A with 𝑎 ⩾ 0, and for any 𝜔-limit, the following Dixmier trace is positive: Tr𝜔 (𝑎⟨D⟩−𝑝) > 0. Remark 3.8. The absolute continuity condition actually subsumes the dimension condition, and they should properly be regarded as one and the same. This formulation eases the adaptation of the conditions for nonunital algebras [27, 52]. Condition 6 (First Order). The bounded operators in [D,A] commute with A; in other words, [[D, 𝑎], 𝑏] = 0 for all 𝑎, 𝑏 ∈ A. (3.5) This condition says that operators in [D,A] can be regarded as endomorphisms of the A-module H∞; and more generally, that CD(A) ⊆ EndA(H∞). Remark 3.9. For spectral triples over noncommutative algebras, the first-order condition is more elaborate: as well as the representation 𝜋 : A → B(H) we require a commuting representation 𝜋◦ : A◦ → B(H) of the opposite algebra A◦ (or equivalently, an antirepresentation of A that commutes with 𝜋): writing 𝑎 for 𝜋(𝑎) as usual, and 𝑏◦ for 𝜋◦(𝑏), we ask that [𝑎, 𝑏◦] = 0. Now H∞ can be regarded as a right A-module under the action 𝜉 · 𝑏 := 𝜋◦(𝑏) 𝜉. The first-order condition is then expressed as: [[D, 𝑎], 𝑏◦] = 0 for 𝑎, 𝑏 ∈ A; and once again it entails thatCD(A) ⊆ EndA(H∞). Coming back to the commutative case, we take 𝜋◦ = 𝜋 from now on. (But see [37], for instance, for examples of commutative algebras with different left and right actions on H.) 13 Condition 7 (Orientability). Let 𝑝 be the metric dimension of (A,H,D). We require that the spectral triple be even, with ℤ2-grading Γ, if and only if 𝑝 is even. For convenience, we take Γ = 1 when 𝑝 is odd. We say the spectral triple (A,H,D) is orientable if there exists a Hochschild 𝑝-cycle c = 𝑛∑︁ 𝛼=1 𝑎0 𝛼 ⊗ 𝑎1 𝛼 ⊗ · · · ⊗ 𝑎𝑝𝛼 ∈ 𝑍𝑝 (A,A) (3.6a) whose Hochschild class [c] ∈ 𝐻𝐻𝑝 (A) may be called the “orientation” of (A,H,D), such that 𝜋𝐷 (c) ≡ ∑︁ 𝛼 𝑎0 𝛼 [D, 𝑎1 𝛼] · · · [D, 𝑎 𝑝 𝛼] = Γ. (3.6b) Condition 8 (Poincaré duality). The spectral triple (A,H,D) determines a 𝐾-homology class for 𝐴 ⊗ 𝐴. Let 𝜇 = [(A ⊗ A,H,D)] ∈ 𝐾•(A ⊗ A) = 𝐾•(𝐴 ⊗ 𝐴) denote this 𝐾-homology class. We require that 𝜇 be a fundamental class, i.e., that the Kasparov product [31] − ⊗𝐴 𝜇 : 𝐾•(𝐴) → 𝐾•(𝐴) be an isomorphism. (More on this in Section 8). Condition 9 (Reality). There is an antiunitary operator 𝐽 : H → H such that 𝐽𝑎∗𝐽−1 = 𝑎 for all 𝑎 ∈ A; and moreover, 𝐽2 = ±1, 𝐽D𝐽−1 = ±D and also 𝐽Γ𝐽−1 = ±Γ in the even case, according to the following table of signs depending only on 𝑝 mod 8: 𝑝 mod 8 0 2 4 6 𝐽2 = ±1 + − − + 𝐽D𝐽−1 = ±D + + + + 𝐽Γ𝐽−1 = ±Γ + − + − 𝑝 mod 8 1 3 5 7 𝐽2 = ±1 + − − + 𝐽D𝐽−1 = ±D − + − + For the origin of this sign table in 𝐾𝑅-homology, we refer to [30, Sec. 9.5]. Remark 3.10. For a noncommutative algebra A, we would require 𝐽𝑎∗𝐽−1 = 𝑎◦ or, more precisely, 𝐽𝜋(𝑎)∗𝐽−1 = 𝜋◦(𝑎). Thus, 𝐽 implements on H the involution 𝜏 : 𝑎 ⊗ 𝑏◦ ↦→ 𝑏∗ ⊗ 𝑎∗◦ of A ⊗ A◦. Condition 10 (Irreducibility). The spectral triple (A,H,D) is irreducible: that is, the only operators in B(H) (strongly) commuting with D and with all 𝑎 ∈ A are the scalars in ℂ 1. The foregoing conditions are an elaboration of those set forth in [21] for the reconstruction of a spin manifold. We add a final condition, which provides a cohomological version of Poincaré duality: see the discussion in [18, VI.4.𝛾]. Condition 11 (Closedness). The 𝑝+-summable spectral triple (A,H,D) satisfies the following closedness condition: for any 𝑎1, . . . , 𝑎𝑝 ∈ A, the operator Γ [D, 𝑎1] · · · [D, 𝑎𝑝]⟨D⟩−𝑝 has vanish- ing Dixmier trace; thus, for any Tr𝜔, Tr𝜔 ( Γ [D, 𝑎1] · · · [D, 𝑎𝑝] ⟨D⟩−𝑝 ) = 0. (3.7) 14 Remark 3.11. By setting Φ(𝑎0, . . . , 𝑎𝑝) := Tr𝜔 ( Γ 𝑎0 [D, 𝑎1] · · · [D, 𝑎𝑝] ⟨D⟩−𝑝 ) , the equation (3.7) may be rewritten [18, VI.2] as 𝐵0Φ = 0, where 𝐵0 is defined on (𝑘 + 1)-linear functionals by (𝐵0𝜙) (𝑎1, . . . , 𝑎𝑘 ) := 𝜙(1, 𝑎1, . . . , 𝑎𝑘 ) + (−1)𝑘𝜙(𝑎1, . . . , 𝑎𝑘 , 1). We quote Lemma 3 of [18, VI.4.𝛾], adapted to the present case where A is commutative and (A,H,D) is 𝑝+-summable. Lemma 3.12 (Connes). Let (A,H,D) be 𝑝+-summable and satisfy Condition 6 (first order). Then for each 𝑘 = 0, 1, . . . , 𝑝 and 𝜂 ∈ Ω𝑘A, a Hochschild cocycle 𝐶𝜂 ∈ 𝑍 𝑝−𝑘 (A,A∗) is defined by 𝐶𝜂 (𝑎0, . . . , 𝑎𝑝−𝑘 ) := Tr𝜔 ( Γ 𝜋D(𝜂) 𝑎0 [D, 𝑎1] · · · [D, 𝑎𝑝−𝑘 ] ⟨D⟩−𝑝 ) . Moreover, if Condition 11 (closedness) also holds, then 𝐶𝜂 depends only on the class of 𝜋D(𝜂) in Λ𝑘 D A, and 𝐵0𝐶𝜂 = (−1)𝑘 𝐶𝑑𝜂. □ 3.2 First consequences of the geometric conditions We now describe some immediate consequences of these conditions, which already give a reasonable picture of the spaces and bundles we shall employ. In this subsection, (A,H,D) will always denote a 𝑄𝐶∞ spectral triple whose algebra A is commutative and complete (and unital, too). In other words, Condition 3 (regularity) is taken for granted. We shall write, as before, 𝑋 = sp(A) = sp(𝐴) where 𝐴 is the separable 𝐶∗-completion of A; it is a metrizable compact Hausdorff space under its weak∗ topology. Lemma 3.13. Under Conditions 6 and 10 (first order, irreducibility), the algebra A contains no nontrivial projector. Proof. Let 𝑞 ∈ A be a projector. Then [D, 𝑞] = [D, 𝑞2] = 𝑞 [D, 𝑞] + [D, 𝑞] 𝑞 = 2𝑞 [D, 𝑞], where the first order condition gives the last equality. Hence (2𝑞−1) [D, 𝑞] = 0, implying [D, 𝑞] = 0 since 2𝑞 − 1 is invertible. Thus, 𝑞 commutes with D, and with all 𝑎 ∈ A since A is commutative: by irreducibility, 𝑞 must be a scalar, either 𝑞 = 0 or 𝑞 = 1. □ Corollary 3.14. Under the same Conditions 6 and 10, the space 𝑋 is connected. Proof. It is enough to show that there are no nontrivial projectors in the 𝐶∗-algebra 𝐴. Let 𝑞 ∈ 𝐴 be a projector; by Lemma 2.5, we can find a projector 𝑞 ∈ A such that ∥𝑞 − 𝑞∥ < 1 2 . Since 𝑞 is either 0 or 1 by Lemma 3.13, the same must be true of 𝑞. Therefore, 𝐶 (𝑋) contains no nontrivial projectors, and so 𝑋 is connected. □ Lemma 3.15. Under Condition 4 (finiteness), the dense subspace H∞ ⊂ H consists of continuous sections of a complex vector bundle 𝑆 → 𝑋 . If (A,H,D) is also irreducible, then 𝑆 has constant rank. Proof. Condition 4 says that there is an integer 𝑚 > 0 and a projector 𝑞 ∈ 𝑀𝑚 (A) such that H∞ ≃ 𝑞A𝑚 as a (right) A-module. We may regard 𝑞 as an element of 𝑀𝑚 (𝐴); the 𝐴-module 𝑞𝐴𝑚 is well-defined as a finitely generated projective module over 𝐴 = 𝐶 (𝑋). By the Serre–Swan 15 theorem [59], 𝑞𝐴𝑚 ≃ Γ(𝑋, 𝑆), the space of continuous sections of a complex vector bundle 𝑆 → 𝑋 . From the finiteness axiom, the Hermitian pairing on 𝑞A𝑚 gives 𝑆 → 𝑋 the structure of a Hermitian vector bundle. If (A,H,D) is irreducible, then 𝑋 is connected by Corollary 3.14, and the rank of 𝑆 must be constant. We shall denote this rank by 𝑁 . □ Lemma 3.16. Under Conditions 4 to 7 (finiteness, absolute continuity, first order, orientability), the algebra CD(A) is a unital selfadjoint subalgebra of Γ(𝑋,End 𝑆). The operator norm of each 𝑇 ∈ CD(A) coincides with its norm as an endomorphism of 𝑆. Proof. Since (A,H,D) is 𝑄𝐶∞, each operator 𝑇 ∈ CD(A) maps H∞ into itself; and the first order condition ensures that 𝑇 is an A-linear map on H∞. If 𝑇 = ∑ 𝑗 𝑎 𝑗 [D, 𝑏 𝑗 ] ∈ Λ1 D A = 𝜋D(Ω1A), the adjoint operator 𝑇∗ = −∑ 𝑗 [D, 𝑏∗𝑗 ] 𝑎∗𝑗 = ∑ 𝑗 𝑏 ∗ 𝑗 [D, 𝑎∗𝑗 ] − [D, 𝑏∗𝑗𝑎∗𝑗 ] lies in Λ1 D A also, so Λ1 D A is a selfadjoint linear subspace of B(H). Thus, the algebra CD(A) generated by A and Λ1 D A is a ∗-subalgebra of B(H). Moreover, since the pairing on H∞ is determined by the scalar product on H via (3.4), we conclude that (𝜉 | 𝑇𝜂) = (𝑇∗𝜉 | 𝜂) for each 𝑇 ∈ CD(A). Consequently, 𝑇 yields an adjointable 𝐴-module map of the 𝐶∗-module Γ(𝑋, 𝑆); that is, CD(A) ⊂ End𝐴 (Γ(𝑋, 𝑆)) = Γ(𝑋,End 𝑆). The algebra CD(A) contains Γ2 = 1. We use the inequality [49, Cor. 2.22] between positive elements of the 𝐶∗-algebra 𝐴: (𝑇𝜉 | 𝑇𝜉) ⩽ ∥𝑇 ∥2 End 𝑆 (𝜉 | 𝜉), where ∥𝑇 ∥End 𝑆 denotes the norm of 𝑇 in the 𝐶∗-algebra Γ(𝑋,End 𝑆). Therefore, when 𝜉 ∈ H∞, ⟨𝑇𝜉 | 𝑇𝜉⟩ = TrΩ ( (𝑇𝜉 | 𝑇𝜉) ⟨D⟩−𝑝 ) ⩽ ∥𝑇 ∥2 End 𝑆 TrΩ ( (𝜉 | 𝜉) ⟨D⟩−𝑝 ) = ∥𝑇 ∥2 End 𝑆 ⟨𝜉 | 𝜉⟩. Majorizing this inequality over { 𝜉 ∈ H∞ : ⟨𝜉 | 𝜉⟩ ⩽ 1 }, we obtain ∥𝑇 ∥ ⩽ ∥𝑇 ∥End 𝑆. To see that these norms are indeed equal, suppose that 0 ⩽ 𝑀 < ∥𝑇 ∥2 End 𝑆, so that 𝑀 − 𝑇∗𝑇 is selfadjoint and not positive in Γ(𝑋,End 𝑆). Then we can find a nonzero 𝜉 ∈ Γ(𝑋, 𝑆) such that (𝑇𝜉 | 𝑇𝜉) − 𝑀 (𝜉 | 𝜉) is positive and nonzero. In view of Condition 5, this implies that 𝑀 ⟨𝜉 | 𝜉⟩ = TrΩ ( 𝑀 (𝜉 | 𝜉) ⟨D⟩−𝑝 ) < TrΩ ( (𝑇𝜉 | 𝑇𝜉) ⟨D⟩−𝑝 ) = ⟨𝑇𝜉 | 𝑇𝜉⟩ = ∥𝑇𝜉∥2 ⩽ ∥𝑇 ∥2 ⟨𝜉 | 𝜉⟩, so that 𝑀 < ∥𝑇 ∥2 since 𝜉 ≠ 0. This is true for all 𝑀 < ∥𝑇 ∥2 End 𝑆, thus ∥𝑇 ∥ = ∥𝑇 ∥End 𝑆. □ Corollary 3.17. Under the same Conditions 4 to 7, the algebra of sections CD(A) is pointwise a direct sum of matrix algebras: (CD(A))𝑥 ≃ 𝑀𝑘1 (ℂ) ⊕ · · · ⊕ 𝑀𝑘𝑟 (ℂ) for 𝑥 ∈ 𝑋, where 𝑘1 + · · · + 𝑘𝑟 = 𝑁 . 16 Proof. Lemma 3.16 shows that (CD(A))𝑥 is a selfadjoint subalgebra of the finite-dimensional algebra End 𝑆𝑥; hence it is a direct sum of full matrix algebras. This subalgebra has full rank, since Γ = 𝜋D(c) lies in CD(A), so that Γ2 𝑥 is the identity element in End 𝑆𝑥 . □ The following result allows us to sidestep several questions of domains. Proposition 3.18. Let 𝑇 : H∞ → H∞ be A-linear. Then 𝑇 extends to a bounded operator on H. Proof. Let 𝜉1, . . . , 𝜉𝑚 ∈ H∞ ≃ 𝑞A𝑚 be defined by 𝜉 𝑗 := 𝑞𝜀 𝑗 = ∑ 𝑘 𝑞𝑘 𝑗𝜀𝑘 where 𝜀 𝑗 ∈ A𝑚 is the column-vector with 1 in the 𝑗 th slot and zeroes elsewhere. Every 𝜉 ∈ H∞ can be written in the form 𝜉 = ∑𝑚 𝑗=1 𝜉 𝑗𝑎 𝑗 , for some 𝑎 𝑗 ∈ A. By the properties of our chosen Hermitian pairing, we get (𝜉 𝑗 | 𝜉𝑘 )H∞ = (∑ 𝑟 𝑞𝑟 𝑗𝜀𝑟 �� ∑ 𝑠 𝑞𝑠𝑘𝜀𝑠 ) A𝑚 = ∑︁ 𝑟,𝑠 𝑞 𝑗𝑟𝑞𝑠𝑘𝛿𝑟𝑠 = 𝑞 𝑗 𝑘 . Thus, as already noted, (𝜉 | 𝜉) = ∑ 𝑗 ,𝑘 𝑎 ∗ 𝑗 𝑞 𝑗 𝑘𝑎𝑘 . The A-linearity of 𝑇 gives 𝑇𝜉 = ∑𝑚 𝑗=1(𝑇𝜉 𝑗 )𝑎 𝑗 , and by hypothesis, 𝑇𝜉 𝑗 ∈ H∞. Therefore, (𝑇𝜉 𝑗 | 𝑇𝜉𝑘 ) = ( 𝑇 ∑ 𝑛 𝑞𝑛 𝑗𝜀𝑛 �� 𝑇 ∑ 𝑙 𝑞𝑙𝑘𝜀𝑙 ) = ( 𝑇 ∑ 𝑛,𝑟 𝑞𝑛𝑟𝑞𝑟 𝑗𝜀𝑛 �� 𝑇 ∑ 𝑙,𝑠 𝑞𝑙𝑠𝑞𝑠𝑘𝜀𝑙 ) = ( 𝑇 ∑ 𝑛,𝑟 𝑞𝑛𝑟𝜀𝑛𝑞𝑟 𝑗 �� 𝑇 ∑ 𝑙,𝑠 𝑞𝑙𝑠𝜀𝑙𝑞𝑠𝑘 ) = ∑︁ 𝑟,𝑠 𝑞 𝑗𝑟 (𝑇𝜉𝑟 | 𝑇𝜉𝑠)𝑞𝑠𝑘 . Denote by Θ𝜉,𝜂, for 𝜉, 𝜂 ∈ H∞, the “ketbra” operator 𝜌 ↦→ 𝜉 (𝜂 | 𝜌). The pairing (𝑇𝜉 | 𝑇𝜉) may be expanded as follows: (𝑇𝜉 | 𝑇𝜉) = ∑︁ 𝑗 ,𝑘 𝑎∗𝑗 (𝑇𝜉 𝑗 | 𝑇𝜉𝑘 )𝑎𝑘 = ∑︁ 𝑗 ,𝑘,𝑟,𝑠 𝑎∗𝑗 (𝜉 𝑗 | 𝜉𝑟) (𝑇𝜉𝑟 | 𝑇𝜉𝑠) (𝜉𝑠 | 𝜉𝑘 )𝑎𝑘 = ∑︁ 𝑟,𝑠 (𝜉 | 𝜉𝑟) (𝑇𝜉𝑟 | 𝑇𝜉𝑠) (𝜉𝑠 | 𝜉) = ∑︁ 𝑟,𝑠 ( 𝑇𝜉𝑟 (𝜉𝑟 | 𝜉) �� 𝑇𝜉𝑠 (𝜉𝑠 | 𝜉)) = ∑︁ 𝑟,𝑠 ( Θ𝑇𝜉𝑟 ,𝜉𝑟 𝜉 �� Θ𝑇𝜉𝑠 ,𝜉𝑠𝜉) = ∑︁ 𝑟,𝑠 ( Θ𝜉𝑠 (𝑇𝜉𝑠 |𝑇𝜉𝑟 ),𝜉𝑟 𝜉 �� 𝜉) ⩽ ∑︁ 𝑟,𝑠 Θ𝜉𝑠 (𝑇𝜉𝑠 |𝑇𝜉𝑟 ),𝜉𝑟 (𝜉 | 𝜉). In the last line here the norm is both the operator norm and the endomorphism norm, which coincide by Lemma 3.16. The norm of each ketbra is finite, since each 𝑇𝜉𝑟 ∈ H∞ by hypothesis. Now we can estimate the operator norm of 𝑇 ; for 𝜉 ∈ H∞ we get the bound ⟨𝑇𝜉 | 𝑇𝜉⟩ = TrΩ ( (𝑇𝜉 | 𝑇𝜉) ⟨D⟩−𝑝 ) ⩽ ∑︁ 𝑟,𝑠 Θ𝜉𝑠 (𝑇𝜉𝑠 |𝑇𝜉𝑟 ),𝜉𝑟 TrΩ ( (𝜉 | 𝜉) ⟨D⟩−𝑝 ) = ∑︁ 𝑟,𝑠 Θ𝜉𝑠 (𝑇𝜉𝑠 |𝑇𝜉𝑟 ),𝜉𝑟 ⟨𝜉 | 𝜉⟩. Since the 𝜉𝑟 are a fixed finite set of vectors, the operator norm of 𝑇 is finite, with ∥𝑇 ∥2 ⩽ ∑︁ 𝑟,𝑠 (𝑇𝜉𝑠 | 𝑇𝜉𝑟) (𝜉𝑟 | 𝜉𝑟) 1/2 (𝜉𝑠 | 𝜉𝑠) 1/2 . This last expression for the norm follows from [49, Lemma 2.30] or [30, Lemma 4.21]. □ 17 Lemma 3.19. Under Conditions 1, 4, 5 and 7 (𝑝+-summability, finiteness, absolute continuity, orientability), the A-valued Hermitian pairing on H∞ given by (3.4) is independent of the choice of Dixmier trace. Proof. Connes’ character theorem [18, Thm. IV.2.8] – we refer to [30] and [10] for its detailed proof – shows that any operator of the form 𝑇 = Γ ∑ 𝛼 𝑎 0 𝛼 [D, 𝑎1 𝛼] · · · [D, 𝑎 𝑝 𝛼] ⟨D⟩−𝑝, (3.8) where c = ∑ 𝛼 𝑎 0 𝛼 ⊗ 𝑎1 𝛼 ⊗ · · · ⊗ 𝑎𝑝𝛼 is a Hochschild cycle, is a measurable operator. Condition 7 provides us with such a Hochschild 𝑝-cycle c for which 𝜋(c) = Γ. Using Γ2 = 1, we can rewrite (3.4) as ⟨𝜉 | 𝜂⟩ = TrΩ ( (𝜉 | 𝜂) ⟨D⟩−𝑝 ) = TrΩ ( Γ(𝜉 | 𝜂)Γ ⟨D⟩−𝑝 ) . (3.9) If 𝑎 = (𝜉 | 𝜂), then 𝑎c is also a Hochschild cycle for A – as an easy consequence of the cycle property of c and the commutativity of A – so the right hand side of (3.9) is TrΩ(𝑇), where 𝑇 = Γ 𝜋𝐷 (𝑎c) ⟨D⟩−𝑝 is indeed of the form (3.8). Thus TrΩ may be replaced, in the formula (3.4), by any other Dixmier trace Tr𝜔. □ It was noted in [19] that the orientability condition yields the following expression for D in terms of commutators d𝑎 = [D, 𝑎] and [D2, 𝑎]. Lemma 3.20. Under Condition 7 (orientability), the operator D verifies the following formula (as an operator on H∞): D = 1 2 (−1)𝑝−1Γ 𝑛∑︁ 𝛼=1 𝑝∑︁ 𝑗=1 (−1) 𝑗−1𝑎0 𝛼 d𝑎1 𝛼 · · · d𝑎 𝑗−1 𝛼 [D2, 𝑎 𝑗 𝛼] d𝑎 𝑗+1 𝛼 · · · d𝑎𝑝𝛼 + 1 2 (−1)𝑝−1Γ dΓ, (3.10) where Γ = ∑𝑛 𝛼=1 𝑎 0 𝛼 d𝑎1 𝛼 · · · d𝑎 𝑝 𝛼 and we write dΓ := ∑𝑛 𝛼=1 d𝑎0 𝛼 d𝑎1 𝛼 · · · d𝑎𝑛𝛼. Proof. First note that on the domain H∞, the derivation ad𝐷2 may be written as [D2, 𝑎] = D d𝑎 + d𝑎D for all 𝑎 ∈ A. (3.11) Thus, the summation over 𝑗 in (3.10) telescopes, to give 𝑛∑︁ 𝛼=1 𝑝∑︁ 𝑗=1 (−1) 𝑗−1𝑎0 𝛼 d𝑎1 𝛼 · · · d𝑎 𝑗−1 𝛼 [D2, 𝑎 𝑗 𝛼] d𝑎 𝑗+1 𝛼 · · · d𝑎𝑝𝛼 = 𝑛∑︁ 𝛼=1 (𝑎0 𝛼D d𝑎1 𝛼 · · · d𝑎 𝑝 𝛼 + (−1)𝑝−1𝑎0 𝛼 d𝑎1 𝛼 · · · d𝑎 𝑝 𝛼D) = − 𝑛∑︁ 𝛼=1 d𝑎0 𝛼 d𝑎1 𝛼 · · · d𝑎 𝑝 𝛼 +DΓ + (−1)𝑝−1ΓD = −dΓ + 2 (−1)𝑝−1ΓD, and (3.10) follows on multiplying both sides by 1 2 (−1)𝑝−1Γ. □ 18 Corollary 3.21. Under Conditions 6 and 7 (first order, orientability), the commutator [D, 𝑎], for 𝑎 ∈ A, has the expansion [D, 𝑎] = 1 2 (−1)𝑝−1Γ 𝑛∑︁ 𝛼=1 𝑝∑︁ 𝑗=1 (−1) 𝑗−1𝑎0 𝛼 d𝑎1 𝛼 · · · (d𝑎 𝑗 𝛼 d𝑎 + d𝑎 d𝑎 𝑗𝛼) · · · d𝑎𝑝𝛼 . Proof. The first order condition entails that 𝑎 commutes with all operator factors in the expan- sion (3.10), except the [D2, 𝑎 𝑗 𝛼] factors. For those, (3.11) and [[D, 𝑎 𝑗𝛼], 𝑎] = 0 imply [[D2, 𝑎 𝑗 𝛼], 𝑎] = [D[D, 𝑎 𝑗𝛼], 𝑎] + [[D, 𝑎 𝑗𝛼]D, 𝑎] = [D, 𝑎] [D, 𝑎 𝑗𝛼] + [D, 𝑎 𝑗𝛼] [D, 𝑎] . □ 4 The cotangent bundle Throughout this section, (A,H,D) will be a spectral triple whose algebra A is (unital and) com- mutative and complete; and 𝑋 = sp(A) will be its metrizable compact Hausdorff character space. Moreover, we shall assume that Conditions 1, 3–7 and 11 hold, namely that (A,H,D) is 𝑝+-sum- mable, 𝑄𝐶∞ and has the properties of finiteness, absolute continuity, first order, orientability and closedness. Lemma 4.1. The operator [D, 𝑎] [D, 𝑏] + [D, 𝑏] [D, 𝑎] is a junk term, for any 𝑎, 𝑏 ∈ A. Proof. We must show that 𝑑𝑎 𝑑𝑏 + 𝑑𝑏 𝑑𝑎 belongs to 𝑑 (ker 𝜋D) in the universal graded differential algebra Ω•A. Since 𝑑𝑎 𝑑𝑏 + 𝑑𝑏 𝑑𝑎 = 𝑑 (𝑎 𝑑𝑏 − 𝑑 (𝑏𝑎) + 𝑏 𝑑𝑎) = 𝑑 (𝑎 𝑑𝑏 − 𝑑𝑏 𝑎), it is enough to notice that the first-order condition gives 𝜋D(𝑎 𝑑𝑏 − 𝑑𝑏 𝑎) = 𝑎 [D, 𝑏] − [D, 𝑏] 𝑎 = 0. □ Lemma 4.2. The image of Γ = 𝜋D(c) in Λ 𝑝 D A is nonzero. Proof. The Hochschild cycle c ∈ 𝑍𝑝 (A,A) defines a Hochschild 0-cocycle (a trace) 𝐶c on A by Lemma 3.12. Taking into account Lemma 3.19, it is given by 𝐶c(𝑎) = ⨏ Γ𝜋D(c) 𝑎⟨D⟩−𝑝 = ⨏ 𝑎⟨D⟩−𝑝 . Since 𝐶c(1) = ⨏ ⟨D⟩−𝑝 > 0, (4.1) this cocycle does not vanish. Moreover, Condition 11 entails that 𝐶c depends only on the class of 𝜋D(c) in Λ 𝑝 D (A). If this class were zero, so that 𝜋D(c) ∈ 𝜋D(𝑑 (ker 𝜋D)), then we could write it as a finite sum of the form 𝜋D(c) = ∑ 𝛽 d𝑏1 𝛽 · · · d𝑏𝑝 𝛽 . But the closedness condition would then apply to show that 𝐶c(1) = ⨏ Γ𝜋D(c)⟨D⟩−𝑝 = ∑︁ 𝛽 ⨏ Γ d𝑏1 𝛽 d𝑏2 𝛽 · · · d𝑏 𝑝 𝛽 ⟨D⟩−𝑝 = 0, contradicting (4.1). Hence, the class of Γ has a nonzero image in Λ 𝑝 D (A). □ 19 Corollary 4.3. Let Γ′ ∈ CD(A) be defined by Γ′ := 1 𝑝! ∑︁ 𝜎∈𝑆𝑝 (−1)𝜎 ∑︁ 𝛼 𝑎0 𝛼 d𝑎𝜎(1)𝛼 d𝑎𝜎(2)𝛼 . . . d𝑎𝜎(𝑝)𝛼 , (4.2) on skewsymmetrizing the expression for Γ obtained from (3.6). If 𝑎 ∈ A is positive and nonzero, then 𝑎Γ′ ≠ 0. Proof. Let 𝑎 ∈ A be positive, 𝑎 ≠ 0. Since A is commutative and c ∈ 𝑍𝑝 (A,A), the product 𝑎c is also a Hochschild 𝑝-cycle. Now the absolute continuity condition implies that 𝐶𝑎c(1) = 𝐶c(𝑎) = ⨏ 𝑎⟨D⟩−𝑝 > 0. Since 𝜋D(𝑎c) = 𝑎Γ, the proof of Lemma 4.2 shows that the class [𝑎Γ] in Λ 𝑝 D A is nonzero. Now, Lemma 4.1 shows that [𝑎Γ′] = [𝑎Γ]. In particular, 𝑎Γ′ ≠ 0 as an element of B(H). □ Thus, the skewsymmetrization Γ′ of Γ given by (4.2) is nonzero as an operator on H, and a fortiori as a section in Γ(𝑋,End 𝑆). In fact, this section vanishes nowhere on 𝑋 , as the proof of the following Proposition shows. ▶ In what follows, whenever 𝑇 is a continuous (local) section of End 𝑆 → 𝑋 , we write either 𝑇 (𝑥) or 𝑇 |𝑥 to denote its value in the fibre End 𝑆𝑥 . The support of 𝑇 will mean its support as a section, namely, the complement of the largest open subset𝑉 ⊆ 𝑋 such that 𝑇 (𝑥) = 0 in End 𝑆𝑥 for all 𝑥 ∈ 𝑉 . Proposition 4.4. There is an open cover {𝑈1, . . . ,𝑈𝑛} of 𝑋 such that, for each 𝛼 = 1, . . . , 𝑛, the operators [D, 𝑎1 𝛼], . . . , [D, 𝑎 𝑝 𝛼] are pointwise linearly independent sections of Γ(𝑈𝛼,End 𝑆). Proof. Let 𝑍 := { 𝑥 ∈ 𝑋 : Γ′(𝑥) = 0 } be the zero set of Γ′. If 𝑉 were a nonvoid open subset of 𝑍 , then, using Lemma 2.10, we could find a nonzero positive 𝑏 ∈ A such that supp 𝑏 ⊂ 𝑉 ; but this would imply 𝑏Γ′ = 0, contradicting Corollary 4.3. Therefore, 𝑍 has empty interior. The pairing on H∞, or rather, on the completed 𝐴-module Γ(𝑋, 𝑆), induces a 𝐶∗-norm on Γ(𝑋,End 𝑆) = End𝐴 (Γ(𝑋, 𝑆)) which in turn determines a norm on each fibre End 𝑆𝑥 so that ∥𝑇 ∥End 𝑆 = sup𝑥∈𝑋 ∥𝑇 (𝑥)∥End 𝑆𝑥 for 𝑇 ∈ Γ(𝑋,End 𝑆). Choose 𝜀 > 0; unless 𝑍 = ∅, there is an open set 𝑊 ⊃ 𝑍 such that sup𝑦∈𝑊 ∥Γ′(𝑦)∥End 𝑆𝑦 < 𝜀. Next choose 𝑎 ∈ A, positive and nonzero, with supp 𝑎 ⊂ 𝑊 . By Lemma 2.12, there exists 𝜓 ∈ A such that 0 ⩽ 𝜓 ⩽ 1, 𝜓𝑎 = 𝑎 and supp𝜓 ⊂ 𝑊 . Hence, by Lemma 3.16, ∥𝜓Γ′∥ = ∥𝜓Γ′∥End 𝑆 < 𝜀. By Corollary 4.3, the Hochschild 0-cocycles 𝐶𝑎Γ and 𝐶𝑎Γ′ are equal, and they define positive functionals on A. For any Dixmier trace Tr𝜔, we know that 𝐶𝑎Γ (1) = 𝐶Γ (𝑎) = Tr𝜔 (𝑎⟨D⟩−𝑝), 𝐶𝑎Γ′ (1) = Tr𝜔 (ΓΓ′ 𝑎⟨D⟩−𝑝) = Tr𝜔 (ΓΓ′ 𝜓𝑎⟨D⟩−𝑝). (4.3) This yields the following estimate: |𝐶𝑎Γ′ (1) | = 𝐶𝑎Γ′ (1) = Tr𝜔 (ΓΓ′𝜓𝑎⟨D⟩−𝑝) ⩽ ∥ΓΓ′ 𝜓∥ Tr𝜔 (𝑎⟨D⟩−𝑝) < 𝜀 Tr𝜔 (𝑎⟨D⟩−𝑝). Since Tr𝜔 (𝑎⟨D⟩−𝑝) > 0, this is inconsistent with (4.3) when 0 < 𝜀 < 1. We conclude that the set 𝑍 must necessarily be empty, so that Γ′(𝑥) is nonvanishing on 𝑋 . 20 Thus, at each 𝑥 ∈ 𝑋 , there is some 𝛼 such that 𝑎0 𝛼 (𝑥) ≠ 0 and d𝑎1 𝛼 (𝑥), . . . , d𝑎 𝑝 𝛼 (𝑥) in End 𝑆𝑥 have a nonzero skewsymmetrized product, and therefore are linearly independent. We may now define𝑈𝛼 to be the open set of all 𝑥 for which this linear independence holds; and𝑈1 ∪ · · · ∪𝑈𝑛 = 𝑋 from the nonvanishing of Γ′. □ Lemma 4.5. Fix 𝛼 ∈ {1, . . . , 𝑛} and let 𝑎 ∈ A, writing 𝑎 =: 𝑎𝑝+1 𝛼 for notational convenience. Then (−1)𝑝−1 2(𝑝 + 1)! ∑︁ 𝜎∈𝑆𝑝+1 (−1)𝜎𝑎0 𝛼 [D, 𝑎 𝜎(1) 𝛼 ] · · · [D, 𝑎𝜎(𝑝)𝛼 ] [D, 𝑎𝜎(𝑝+1) 𝛼 ] = 0. (4.4) Proof. By Corollary 3.21, we may write Γ [D, 𝑎] = 1 2 (−1)𝑝−1 𝑛∑︁ 𝛼=1 𝑝∑︁ 𝑗=1 (−1) 𝑗−1𝑎0 𝛼 d𝑎1 𝛼 . . . (d𝑎 𝑗 𝛼 d𝑎𝑝+1 𝛼 + d𝑎𝑝+1 𝛼 d𝑎 𝑗𝛼) . . . d𝑎𝑝𝛼 . For each 𝛼, every term in the sum over 𝑗 contains a symmetric product of one-forms, so its skewsymmetrization vanishes. □ Remark 4.6. For brevity, we shall denote by Γ′ 𝛼 the 𝛼th summand of Γ′ in (4.2), and by Γ′ 𝛼 ∧ d𝑎 the operator on the left hand side of (4.4). Now, d𝑎 and each d𝑎 𝑗𝛼, and therefore each Γ′ 𝛼, is an endomorphism of H∞; Lemma 4.5 shows that Γ′ 𝛼 (𝑥) ∧ d𝑎(𝑥) = 0 in End 𝑆𝑥 , for each 𝑥 ∈ 𝑈𝛼, where the notation ∧ now denotes skewsymmetrization with respect to the several d𝑎 𝑗𝛼 (𝑥). Remark 4.7. From now on we shall assume, without any loss of generality, that each 𝑎 𝑗 𝛼, for 𝑗 = 1, . . . , 𝑝, 𝛼 = 1, . . . , 𝑛, is selfadjoint. (Otherwise, we just take selfadjoint and skewadjoint parts, allowing some repetition of the sets𝑈𝛼.) Consequently, each [D, 𝑎 𝑗𝛼] is skewadjoint. Proposition 4.8. The operators [D, 𝑎 𝑗𝛼], for 𝛼 = 1, . . . , 𝑛 and 𝑗 = 1, . . . , 𝑝, generate Λ1 D A as a finitely generated projective A-module. Proof. Let 𝑎 ∈ A; choose (and fix) 𝛼 such that 𝑎Γ′ 𝛼 ≠ 0. Then by Lemma 4.5, Γ′ 𝛼 ∧ d𝑎 = 0 in CD(A), and thus Γ′ 𝛼 (𝑥) ∧ d𝑎(𝑥) = 0 in End 𝑆𝑥 , for each 𝑥 ∈ 𝑈𝛼. Let 𝐸𝑥 be the (complex) vector subspace of End 𝑆𝑥 spanned by the endomorphisms d𝑎1 𝛼 (𝑥), . . . , d𝑎 𝑝 𝛼 (𝑥). The exterior algebra Λ•𝐸𝑥 is represented on 𝑆𝑥 by (𝑣1 ∧ · · · ∧ 𝑣𝑘 ) · 𝜉 := 1 𝑘! ∑︁ 𝜎∈𝑆𝑘 (−1)𝜎𝑣𝜎(1) · · · 𝑣𝜎(𝑘) 𝜉, and this representation is faithful, on account of Proposition 4.4. Similarly, we can represent on 𝑆𝑥 the exterior algebra of the vector subspace 𝐸′ 𝑥 := span{d𝑎1 𝛼 (𝑥), . . . , d𝑎 𝑝 𝛼 (𝑥), d𝑎(𝑥)} ⊇ 𝐸𝑥 . Now Lemma 4.5 implies that Λ•𝐸′ 𝑥 = Λ•𝐸𝑥 , and thus 𝐸′ 𝑥 = 𝐸𝑥; therefore, d𝑎(𝑥) lies in 𝐸𝑥 , for all 𝑥 ∈ 𝑈𝛼. 21 Choose a partition of unity {𝜙𝛼}𝑛𝛼=1 ⊂ A subordinate to the open cover {𝑈𝛼}, as in Lemma 2.10. Then for any 𝑎 ∈ A we may write [D, 𝑎] = ∑ 𝛼 𝜙𝛼 [D, 𝑎]. Then for each 𝑥 ∈ 𝑈𝛼, the linear independence of the d𝑎 𝑗𝛼 (𝑥) yields unique constants 𝑐 𝑗𝛼 (𝑥) such that 𝜙𝛼 (𝑥) d𝑎(𝑥) = 𝑝∑︁ 𝑗=1 𝑐 𝑗𝛼 (𝑥) d𝑎 𝑗𝛼 (𝑥). (4.5) Since supp 𝜙𝛼 ⊂ 𝑈𝛼, (4.5) defines a continuous local section in Γ(𝑈𝛼,End 𝑆). By uniqueness, 𝑐 𝑗𝛼 (𝑥) = 0 outside supp 𝜙𝛼, so we may extend 𝑐 𝑗𝛼 by zero to a function on all of 𝑋 , and thus we may regard these local sections as elements of Γ(𝑋,End 𝑆). We claim that each 𝑐 𝑗𝛼 lies in A (and in particular is continuous). Define an A-valued Hermitian pairing on Λ1 D A by setting (d𝑎 | d𝑏) := 𝐶𝑝 tr((d𝑎)∗ d𝑏), (4.6) where𝐶𝑝 is a suitable positive normalization constant, and tr denotes the matrix trace in EndAH∞ = 𝑞𝑀𝑚 (A)𝑞, where Γ(𝑋, 𝑆) = 𝑞𝐴𝑚. To see that this pairing takes values in A, we use the following localization argument. Choose a finite open cover {𝑉𝜌} of 𝑋 such that 𝑆 is trivial over each 𝑉𝜌. For each 𝜌, choose 𝑁 = rank 𝑆 elements 𝜉𝜌1 , . . . , 𝜉 𝜌 𝑁 ∈ H∞ which, when regarded as sections of 𝑆, are linearly independent over 𝑉𝜌. Moreover, these sections can be chosen so that {𝜉𝜌1 (𝑥), . . . , 𝜉 𝜌 𝑁 (𝑥)} is an orthonormal basis in each fibre 𝑆𝑥 , for 𝑥 ∈ 𝑉𝜌; this means that for all 𝑏 ∈ A with supp 𝑏 ⊂ 𝑉𝜌 the orthogonality relations 𝑏(𝜉𝜌 𝑖 | 𝜉𝜌 𝑗 ) = 𝑏𝛿𝑖 𝑗 hold. That may be achieved by Gram–Schmidt orthogonalization, on invoking Proposition 2.13 to see that local inverses of elements in A also lie in A. Next, choose a partition of unity {𝜓𝜌} subordinate to the cover {𝑉𝜌}, with 𝜓𝜌 ∈ A, as in Lemma 2.10. Then tr((d𝑎)∗ d𝑏) = ∑︁ 𝜌 𝜓𝜌 tr((d𝑎)∗ d𝑏) = ∑︁ 𝜌 𝑁∑︁ 𝑗=1 𝜓𝜌 (d𝑏 𝜉𝜌𝑗 | d𝑎 𝜉𝜌 𝑗 ). The right hand side is a finite sum of elements of A, and so belongs to A. If 𝑏 = (d𝑎 | d𝑎), then 𝑏(𝑥) is the trace of a positive element of End 𝑆𝑥 , so 𝑏(𝑥) = 0 if and only if d𝑎(𝑥) = 0; thus the pairing is positive definite. Consider the matrix 𝑔𝛼 = [𝑔 𝑗 𝑘𝛼 ] ∈ 𝑀𝑝 (A) given by 𝑔 𝑗 𝑘 𝛼 := (d𝑎 𝑗𝛼 | d𝑎𝑘𝛼) = −𝐶𝑝 tr(d𝑎 𝑗𝛼 d𝑎𝑘𝛼). (4.7) The matrix 𝑔𝛼 (𝑥) ∈ 𝑀𝑝 (ℂ) has the form 𝐶𝑝 tr(𝑚𝛼 (𝑥)∗𝑚𝛼 (𝑥)) where 𝑚𝛼 (𝑥) ∈ (End 𝑆𝑥)𝑝 is the 𝑝-column with linearly independent entries d𝑎 𝑗𝛼 (𝑥). Thus, for 𝑥 ∈ 𝑈𝛼, each 𝑔𝛼 (𝑥) is a positive definite Gram matrix, hence invertible, when 𝑥 ∈ 𝑈𝛼. Let 𝑔−1 𝛼 (𝑥) := [𝑔𝛼,𝑖 𝑗 ] (𝑥) denote the inverse matrix. We may now invoke Corollary 2.14 – recall that A is complete – to conclude that 𝜙𝛼𝑔𝛼,𝑖 𝑗 is an element of A for 𝑖, 𝑗 = 1, . . . , 𝑝. Now if 𝜙𝛼 d𝑎 = ∑ 𝑖 𝑐𝑖𝛼 d𝑎𝑖𝛼 with supp 𝑐𝑖𝛼 ⊂ 𝑈𝛼 as in (4.5), we find that 𝑐 𝑗𝛼 = ∑︁ 𝑖,𝑘 𝑐𝑖𝛼 𝑔𝛼, 𝑗 𝑘 𝑔 𝑘𝑖 𝛼 = −𝐶𝑝 ∑︁ 𝑖,𝑘 𝑔𝛼, 𝑗 𝑘 tr(d𝑎𝑘𝛼 𝑐𝑖𝛼 d𝑎𝑖𝛼) = −𝐶𝑝 ∑︁ 𝑘 𝑔𝛼, 𝑗 𝑘 tr(d𝑎𝑘𝛼 𝜙𝛼 d𝑎) = ∑︁ 𝑘 𝜙𝛼 𝑔𝛼, 𝑗 𝑘 (d𝑎𝑘𝛼 | d𝑎), where each 𝜙𝛼 𝑔𝛼, 𝑗 𝑘 ∈ A and (d𝑎𝑘𝛼 | d𝑎) ∈ A by previous arguments; we conclude that 𝑐 𝑗𝛼 ∈ A. 22 Finally, for any 𝑎 ∈ A, we may now write d𝑎 = 𝑛∑︁ 𝛼=1 𝜙𝛼 d𝑎 = 𝑛∑︁ 𝛼=1 𝑝∑︁ 𝑗=1 𝑐 𝑗𝛼 d𝑎 𝑗𝛼 ∈ Λ1 DA. (4.8) Since the coefficients 𝑐 𝑗𝛼 in this finite sum lie in A, the d𝑎 𝑗𝛼 = [D, 𝑎 𝑗𝛼] generate the A-module Λ1 D A. To see that Λ1 D A is a projective A-module, we rewrite the coefficients in (4.8) as 𝑐 𝑗𝛼 =∑𝑝 𝑘=1 𝜓 𝑗 𝑘𝛼 (d𝑎 𝑘 𝛼 | d𝑎), and we get, for 𝑏 ∈ A, 𝑏 d𝑎 = 𝑛∑︁ 𝛼=1 𝑝∑︁ 𝑗 ,𝑘=1 𝜓 𝑗 𝑘𝛼 (d𝑎𝑘𝛼 | 𝑏 d𝑎) d𝑎 𝑗𝛼, and therefore Λ1 D A ≃ 𝑄A𝑛𝑝 via standard isomorphisms [30, Prop. 3.9], where 𝑄 ∈ 𝑀𝑛𝑝 (A) is the projector with entries 𝑄𝛼𝑘,𝛽𝑙 := ∑𝑝 𝑚=1 𝜓𝑙𝑚𝛽 (d𝑎 𝑘 𝛼 | d𝑎𝑙 𝛽 ). □ We shall frequently need to replace the expansion (4.8) by a “localized” version for a single 𝛼, as follows. Corollary 4.9. If 𝑎 ∈ A is such that supp d𝑎 ⊂ 𝑈𝛼, then there exist 𝑐1𝛼, . . . , 𝑐𝑝𝛼 ∈ A, compactly supported in𝑈𝛼, such that d𝑎 = 𝑝∑︁ 𝑗=1 𝑐 𝑗𝛼 d𝑎 𝑗𝛼 . (4.9) More generally, if 𝑏 ∈ A, then for any open 𝑉 ⊂ 𝑈𝛼 there are continuous functions 𝑏 𝑗𝛼 : 𝑉 → ℂ for 𝑗 = 1, . . . , 𝑝, such that d𝑏(𝑥) = 𝑝∑︁ 𝑗=1 𝑏 𝑗𝛼 (𝑥) d𝑎 𝑗𝛼 (𝑥) for all 𝑥 ∈ 𝑉, (4.10) and such that each 𝑐 𝑏 𝑗𝛼 ∈ A whenever 𝑐 ∈ A with supp 𝑐 ⊂ 𝑉 . Proof. If supp d𝑎 ⊂ 𝑈𝛼, then we may choose the partition of unity of the previous proof such that 𝜙𝛼 (𝑥) = 1 on supp d𝑎, by Corollary 2.11. Thus 𝜙𝛼 d𝑎 = d𝑎 and 𝜙𝛽 d𝑎 = 0 for 𝛽 ≠ 𝛼. Thus both (4.5) and (4.8) reduce to (4.9). By construction, supp 𝑐 𝑗𝛼 ⊆ supp 𝜙𝛼. In the same way, if 𝑐 ∈ A with supp 𝑐 ⊂ 𝑉 , we may expand 𝑐 d𝑏 =: ∑𝑝 𝑗=1 𝑐 ′ 𝑗𝛼 d𝑎 𝑗𝛼 with 𝑐′ 𝑗𝛼 ∈ A and supp 𝑐′ 𝑗𝛼 ⊆ supp 𝑐. Uniqueness of the coefficients at each 𝑥 ∈ 𝑉 shows that 𝑐′ 𝑗𝛼 (𝑥) = 𝑐(𝑥)𝑏 𝑗𝛼 (𝑥), where each function 𝑏 𝑗𝛼 does not depend on 𝑐; also, 𝑏 𝑗𝛼 is continuous because its restriction to each compact subset of 𝑉 is continuous. □ ▶ With the local linear independence and spanning provided by Propositions 4.4 and 4.8, we now obtain a (complex) vector subbundle 𝐸 of End 𝑆, such that Λ1 D A ⊆ Γ(𝑋, 𝐸). This vector bundle will eventually play the role of the complexified cotangent bundle 𝑇∗ ℂ (𝑋), although at this stage we have not yet identified a suitable differential structure on 𝑋 . 23 Proposition 4.10. For each 𝑥 ∈ 𝑈𝛼, define a 𝑝-dimensional complex vector space by 𝐸𝑥 := span{d𝑎1 𝛼 (𝑥), . . . , d𝑎 𝑝 𝛼 (𝑥)} ⊆ End 𝑆𝑥 . (4.11) Then these spaces form the fibres of a complex vector bundle 𝐸 → 𝑋 . Proof. We prove that 𝐸 is a vector bundle by providing transition functions satisfying the usual Čech cocycle condition. For each pair of indices 𝛼, 𝛽, Corollary 4.9 provides continuous functions 𝑐𝑘 𝑗𝛼𝛽 : 𝑈𝛼 ∩𝑈𝛽 → ℂ such that d𝑎𝑘𝛼 (𝑥) = 𝑝∑︁ 𝑗=1 𝑐𝑘𝑗𝛼𝛽 (𝑥) d𝑎 𝑗 𝛽 (𝑥) for all 𝑥 ∈ 𝑈𝛼 ∩𝑈𝛽. (4.12) Whenever 𝑥 ∈ 𝑈𝛼 ∩𝑈𝛽 ∩𝑈𝛾, this entails the additional relation d𝑎𝑘𝛼 (𝑥) = 𝑝∑︁ 𝑗=1 𝑐𝑘𝑗𝛼𝛽 (𝑥) d𝑎 𝑗 𝛽 (𝑥) = 𝑝∑︁ 𝑗 ,𝑙=1 𝑐𝑘𝑗𝛼𝛽 (𝑥) 𝑐 𝑗 𝑙𝛽𝛾 (𝑥) d𝑎𝑙𝛾 (𝑥), and the linear independence of the d𝑎𝑙𝛾 (𝑥) shows that 𝑐𝑘𝑙𝛼𝛾 (𝑥) = 𝑝∑︁ 𝑗=1 𝑐𝑘𝑗𝛼𝛽 (𝑥) 𝑐 𝑗 𝑙𝛽𝛾 (𝑥) for all 𝑥 ∈ 𝑈𝛼 ∩𝑈𝛽 ∩𝑈𝛾 . In particular, the matrix 𝑐𝛼𝛽 (𝑥) = [𝑐𝑘 𝑗𝛼𝛽 (𝑥)] is invertible with 𝑐−1 𝛼𝛽 (𝑥) = 𝑐𝛽𝛼 (𝑥) for 𝑥 ∈ 𝑈𝛼 ∩ 𝑈𝛽. The relation (4.12) and its analogue with 𝛼 and 𝛽 exchanged show that the vector space 𝐸𝑥 of (4.11) is well defined, independently of 𝛼. Moreover, the cocycle conditions 𝑐𝛼𝛽𝑐𝛽𝛾 = 𝑐𝛼𝛾 hold over every nonvoid𝑈𝛼 ∩𝑈𝛽 ∩𝑈𝛾, so these are continuous transition matrices for a vector bundle 𝐸 → 𝑋 , whose total space is the disjoint union 𝐸 := ⊎ 𝑥∈𝑋 𝐸𝑥 . □ Corollary 4.11. If for each 𝑥 ∈ 𝑈𝛼 ⊂ 𝑋 , we define the real vector space 𝐸ℝ,𝑥 = ℝ- span{d𝑎1 𝛼 (𝑥), . . . , d𝑎 𝑝 𝛼 (𝑥)}, then 𝐸ℝ := ⊎ 𝑥∈𝑋 𝐸ℝ,𝑥 is the total space of a real vector bundle over 𝑋 . Proof. We need only show that the transition functions are actually real matrices. Since each d𝑎 𝑗 𝛽 is skewadjoint, taking the adjoint (in End 𝑆𝑥) of (4.12) yields d𝑎𝑘𝛼 (𝑥) = 𝑝∑︁ 𝑗=1 𝑐𝑘𝑗𝛼𝛽 (𝑥) d𝑎 𝑗 𝛽 (𝑥). By uniqueness of the coefficients, we conclude that 𝑐𝑘 𝑗𝛼𝛽 = 𝑐𝑘 𝑗𝛼𝛽 for each 𝑗 , 𝑘, 𝛼, 𝛽. □ We conclude this Section by indicating that the functions 𝑎 𝑗𝛼 are not constant on sets with nonempty interior, and more importantly, that the operator D is actually local. 24 Lemma 4.12. Let 𝑌 ⊂ 𝑈𝛼 be a level set of the function 𝑎 𝑗𝛼, for some 𝑗 = 1, . . . , 𝑝. Then 𝑌 is closed and its interior Int𝑌 is empty. Proof. Clearly 𝑌 is closed, since 𝑎 𝑗𝛼 ∈ 𝐶 (𝑋). Suppose that Int𝑌 were nonempty; then there would be a nonzero element 𝑓 ∈ A such that supp 𝑓 ⊂ Int𝑌 . Let 𝜆 ∈ ℝ be the value of 𝑎 𝑗𝛼 on the level set 𝑌 , so that 𝜆 𝑓 = 𝑎 𝑗𝛼 𝑓 . Taking commutators with D gives 𝜆 [D, 𝑓 ] = 𝑎 𝑗𝛼 [D, 𝑓 ] + [D, 𝑎 𝑗𝛼] 𝑓 = 𝑎 𝑗𝛼 [D, 𝑓 ] + 𝑓 [D, 𝑎 𝑗𝛼], using the first order condition. Therefore, 𝑓 (𝑦) d𝑎 𝑗𝛼 (𝑦) = 0 for all 𝑦 ∈ 𝑌 . This contradicts supp 𝑓 ⊂ 𝑈𝛼, since by definition𝑈𝛼 ⊆ { 𝑥 ∈ 𝑋 : d𝑎 𝑗𝛼 (𝑥) ≠ 0 } for each 𝑗 . □ Corollary 4.13. For each 𝛼 = 1, . . . , 𝑛, let 𝑎𝛼 : 𝑈𝛼 → ℝ𝑝 be the mapping with components 𝑎 𝑗𝛼, 𝑗 = 1, . . . , 𝑝. Then any level set of the mapping 𝑎𝛼 is a closed set with empty interior. Proof. Any such level set for 𝑎𝛼 is the intersection of level sets for the several 𝑎 𝑗𝛼. □ Corollary 4.14. If 𝑎 ∈ A, then supp(d𝑎) ⊆ supp 𝑎. Proof. Suppose that 𝑌 := (𝑋 \ supp 𝑎) is nonempty, otherwise there is nothing to prove. Then 𝑌 is a nonvoid open subset of 𝑋 , and the function 𝑎 vanishes on its closure 𝑌 . Arguing as in the proof of Lemma 4.12, with 𝜆 = 0 and 𝑎 𝑗𝛼 replaced by 𝑎, we see that d𝑎(𝑦) = 0 for all 𝑦 ∈ 𝑌 . □ Lemma 4.15. If 𝑉 ⊆ 𝑈𝛼 is open, then 𝑎𝛼 (𝑉) has nonempty interior in ℝ𝑝. Proof. The level sets of each 𝑎 𝑗𝛼 on𝑈𝛼 are closed with no interior, so no 𝑎 𝑗𝛼 is constant on 𝑉 , or on any open subset of 𝑉 . Then the range of 𝑎1 𝛼 contains a nontrivial interval (i.e., not a point), and so it contains an open interval (𝑠, 𝑡) ⊂ ℝ. Let 𝑉1 = 𝑉 ∩ (𝑎1 𝛼)−1((𝑠, 𝑡)), an open subset of 𝑉 . Likewise, since 𝑎2 𝛼 is not constant on𝑉1, we can find an open𝑉2 ⊆ 𝑉1 which 𝑎 𝑗𝛼 maps onto an open subinterval of 𝑎 𝑗𝛼 (𝑉1) for 𝑗 = 1, 2; and so on. After 𝑝 steps, we obtain an open subset 𝑉𝑝 ⊆ 𝑉 that 𝑎𝛼 maps onto an open rectangle in ℝ𝑝. □ Corollary 4.16. Let 𝑥 ∈ 𝑈𝛼 be such that 𝑥 is neither a local maximum nor minimum of any of the functions 𝑎 𝑗𝛼, 𝑗 = 1, . . . , 𝑝. Then there is an open neighbourhood of 𝑥 on which 𝑎𝛼 is an open map. Proof. The hypothesis says that 𝑎𝛼 (𝑥) is not an endpoint of any interval in 𝑎 𝑗𝛼 (𝑈𝛼) for any 𝑗 . Thus we can find an open rectangle 𝑎𝛼 (𝑥) ⊂ 𝑅 ⊆ 𝑎𝛼 (𝑈𝛼), whereby 𝑎−1 𝛼 (𝑅) is an open neighbourhood of 𝑥, such that every point 𝑦 ∈ 𝑎−1 𝛼 (𝑅) also satisfies the hypothesis of the corollary. □ Corollary 4.17. If 𝐵 ⊆ 𝑎𝛼 (𝑈𝛼) ⊆ ℝ𝑝 has empty interior, then 𝑎−1 𝛼 (𝐵) ∩𝑈𝛼 has empty interior also. Proof. If 𝑎−1 𝛼 (𝐵) ∩𝑈𝛼 had an interior point, then so too would 𝑎𝛼 (𝑎−1 𝛼 (𝐵) ∩𝑈𝛼) = 𝐵. □ 5 A Lipschitz functional calculus Definition 5.1. Let (A,H,D) again be a spectral triple whose algebraA is (unital and) commutative and complete. From now on, we shall say that (A,H,D) is a spectral manifold of dimension 𝑝 if it is 𝑄𝐶∞, 𝑝+-summable, and satisfies the metric, first order, finiteness, absolute continuity, orientability, irreducibility and closedness conditions, that is, all postulates of subsection 3.1 except perhaps Conditions 8 and 9. 25 Throughout this section, we shall assume that (A,H,D) is a spectral manifold of dimension 𝑝. We shall use without comment the open cover {𝑈𝛼} of 𝑋 = sp(A) provided by Proposition 4.4, and the vector bundle 𝐸 → 𝑋 afforded by Proposition 4.10. As indicated earlier, we shall also assume that the 𝑎 𝑗𝛼 appearing in (3.6), for 𝑗 ≠ 0, are selfadjoint. Our next task is to develop a Lipschitz version of the functional calculus. For each 𝛼 = 1, . . . , 𝑛, we shall denote the joint spectrum of 𝑎1 𝛼, . . . , 𝑎 𝑝 𝛼 by Δ𝛼, and shall write 𝑎𝛼 := (𝑎1 𝛼, . . . , 𝑎 𝑝 𝛼) as a continuous mapping from𝑈𝛼 to ℝ𝑝. We recall Nachbin’s extension [45] of the Stone–Weierstrass approximation theorem to subal- gebras of differentiable functions. Proposition 5.1 (Nachbin [45]). Let 𝑈 be an open subset of ℝ𝑝, and let B ⊂ 𝐶𝑟 (𝑈,ℝ) for 𝑟 ∈ {1, 2, . . . }. A necessary and sufficient condition for the algebra generated by B to be dense in 𝐶𝑟 (𝑈,ℝ), in the 𝐶𝑟 topology, is that the following conditions hold: 1. For each 𝑥 ∈ 𝑈, there exists 𝑓 ∈ B such that 𝑓 (𝑥) ≠ 0. 2. Whenever 𝑥, 𝑦 ∈ 𝑈 with 𝑥 ≠ 𝑦, there exists 𝑓 ∈ B such that 𝑓 (𝑥) ≠ 𝑓 (𝑦). 3. For each 𝑥 ∈ 𝑈 and each nonzero tangent vector 𝜉𝑥 ∈ 𝑇𝑥𝑈, there exists 𝑓 ∈ B such that 𝜉𝑥 ( 𝑓 ) ≠ 0. In particular, the real polynomials on ℝ𝑝, restricted to 𝑈, are 𝐶𝑟-dense in 𝐶𝑟 (𝑈,ℝ); and thus also, the complex-valued polynomials are 𝐶𝑟-dense in 𝐶𝑟 (𝑈) = 𝐶𝑟 (𝑈,ℂ). Lemma 5.2. Let (A,H,D) be a spectral manifold of dimension 𝑝, and let 𝑌 be a compact subset of 𝑈𝛼 with nonempty interior. Let 𝑎 ∈ A with supp 𝑎 ⊂ 𝑌 , and suppose there is a bounded 𝐶1 function 𝑓 : 𝐿 → ℂ such that 𝑎 = 𝑓 (𝑎1 𝛼, . . . , 𝑎 𝑝 𝛼), where 𝐿 is a bounded open subset of ℝ𝑝 with Δ𝛼 ⊂ 𝐿. Then there are positive constants 𝐶′ 𝛼 and 𝐶𝑌 , independent of 𝑎 and 𝑓 , such that 𝐶′ 𝛼 ∥ [D, 𝑎] ∥ ⩽ ∥𝑑𝑓 ∥∞ ⩽ 𝐶𝑌 ∥ [D, 𝑎] ∥ (5.1) where ∥𝑑𝑓 ∥2 ∞ := sup𝑡∈Δ𝛼 ∑ 𝑗 |𝜕𝑗 𝑓 (𝑡) |2. Proof. By Proposition 5.1, we may approximate 𝑓 by polynomials 𝑝𝑘 such that 𝑝𝑘 → 𝑓 and 𝜕𝑗 𝑝𝑘 → 𝜕𝑗 𝑓 for each 𝑗 , uniformly on Δ𝛼. The first order condition shows that (2.5) holds for each 𝑝𝑘 . In the limit, the proof of Proposition 2.8 yields [D, 𝑎] = [D, 𝑓 (𝑎1 𝛼, . . . , 𝑎 𝑝 𝛼)] = 𝑝∑︁ 𝑗=1 𝜕𝑗 𝑓 (𝑎1 𝛼, . . . , 𝑎 𝑝 𝛼) [D, 𝑎 𝑗𝛼] . (5.2) Since supp 𝑎 ⊂ 𝑌 , Corollary 2.11 provides an element 𝜙 ∈ A such that 0 ⩽ 𝜙 ⩽ 1 and supp 𝜙 ⊂ 𝑈𝛼 while 𝜙(𝑥) = 1 for 𝑥 ∈ 𝑌 ; and consequently, 𝜙𝑎 = 𝑎. Moreover, 𝜙[D, 𝑎] = [D, 𝑎], on account of Corollary 4.14. The elements 𝑔 𝑗 𝑘𝛼 ∈ A defined by (4.7) form a positive definite matrix of functions on𝑈𝛼; again let 𝑔𝛼,𝑖 𝑗 denote the entries of its inverse matrix. The proof of Proposition 4.8 shows that 𝜙 𝑔𝛼,𝑖 𝑗 lies in A, for all 𝑖, 𝑗 . 26 For 𝑥 ∈ 𝑈𝛼, we compute that tr (∑︁ 𝑗 𝑔𝛼,𝑖 𝑗 d𝑎 𝑗𝛼 d𝑎 ) (𝑥) = 𝑝∑︁ 𝑗 ,𝑘=1 𝜕𝑘 𝑓 (𝑎1 𝛼, . . . , 𝑎 𝑝 𝛼) (𝑥)𝑔𝛼,𝑖 𝑗 (𝑥) tr ( d𝑎 𝑗𝛼 (𝑥) d𝑎𝑘𝛼 (𝑥) ) = −𝐶−1 𝑝 𝑝∑︁ 𝑗 ,𝑘=1 𝜕𝑘 𝑓 (𝑎1 𝛼, . . . , 𝑎 𝑝 𝛼) (𝑥)𝑔𝛼,𝑖 𝑗 (𝑥)𝑔 𝑗 𝑘𝛼 (𝑥) = −𝐶−1 𝑝 𝜕𝑖 𝑓 (𝑎1 𝛼, . . . , 𝑎 𝑝 𝛼) (𝑥). The trace is defined pointwise in the endomorphism algebra Γ(𝑈𝛼,End 𝑆). Let ∥ · ∥𝑥 be the operator norm in End 𝑆𝑥 induced by the hermitian pairing (4.6) on this algebra. Then | (𝜕𝑖 𝑓 ) (𝑎1 𝛼, . . . , 𝑎 𝑝 𝛼) (𝑥) | = 𝐶𝑝 ����tr(∑︁ 𝑗 𝑔𝛼,𝑖 𝑗 d𝑎 𝑗𝛼 d𝑎 ) (𝑥) ���� ⩽ ����∑︁ 𝑗 ( 𝑔𝛼,𝑖 𝑗 d𝑎 𝑗𝛼 �� 𝜙 d𝑎 ) (𝑥) ���� ⩽ ∑︁ 𝑗 ( 𝜙 𝑔𝛼,𝑖 𝑗 d𝑎 𝑗𝛼 �� 𝜙 𝑔𝛼,𝑖 𝑗 d𝑎 𝑗𝛼 )1/2(𝑥) ( d𝑎 �� d𝑎 )1/2(𝑥) ⩽ ∑︁ 𝑗 𝑝 ∥𝜙(𝑥)𝑔𝛼,𝑖 𝑗 (𝑥) d𝑎 𝑗𝛼 (𝑥)∥𝑥 ∥d𝑎(𝑥)∥𝑥 ⩽ ∑︁ 𝑗 𝑝 ∥𝑔𝛼,𝑖 𝑗 (𝑥) d𝑎 𝑗𝛼 (𝑥)∥𝑥 ∥d𝑎(𝑥)∥𝑥 ⩽ 𝐵𝑖 (𝑥) ∥d𝑎(𝑥)∥𝑥 , (5.3) where 𝐵𝑖 (𝑥) is independent of 𝑎 and 𝑓 and is bounded on 𝑌 . Since ∥d𝑎(𝑥)∥𝑥 ⩽ ∥ [D, 𝑎] ∥ by Lemma 3.16, taking 𝐶𝑌 := max𝑖 sup{ 𝐵𝑖 (𝑥) : 𝑥 ∈ 𝑌 } yields the second inequality in (5.1). On the other hand, for any 𝑥 ∈ 𝑈𝛼, the estimate ∥d𝑎(𝑥)∥𝑥 ⩽ ∑︁ 𝑗 | (𝜕𝑗 𝑓 ) (𝑎1 𝛼, . . . , 𝑎 𝑝 𝛼) (𝑥) | ∥d𝑎 𝑗𝛼 (𝑥)∥𝑥 ⩽ ∑︁ 𝑗 ∥d𝑎 𝑗𝛼∥End 𝑆 | (𝜕𝑗 𝑓 ) (𝑎1 𝛼, . . . , 𝑎 𝑝 𝛼) (𝑥) | ⩽ ∑︁ 𝑗 ∥ [D, 𝑎 𝑗𝛼] ∥ ∥𝑑𝑓 ∥∞, again by Lemma 3.16, shows that ∥ [D, 𝑎] ∥ ⩽ (𝐶′ 𝛼)−1∥𝑑𝑓 ∥∞ if we take (𝐶′ 𝛼)−1 := ∑ 𝑗 [D, 𝑎 𝑗𝛼] ; this gives the first inequality in (5.1). □ We need to remove the hypothesis that 𝑎 ∈ A has compact support in𝑌 . This is possible because the proof of Lemma 5.2 relies on pointwise estimates. Corollary 5.3. Let 𝑎 ∈ A and let𝑌 ⊂ 𝑈𝛼 be any compact subset such that 𝑎 |𝑌 = ( 𝑓 ◦𝑎𝛼) |𝑌 for some 𝐶1 function 𝑓 defined and bounded in a neighbourhood of 𝑎𝛼 (𝑌 ) ⊂ ℝ𝑝. Then there are positive constants 𝐶′ 𝛼 and 𝐶𝑌 , independent of 𝑎 and 𝑓 , such that 𝐶′ 𝛼 ∥ [D, 𝑎] ∥𝑌 ⩽ ∥𝑑𝑓 ∥𝑌,∞ ⩽ 𝐶𝑌 ∥ [D, 𝑎] ∥ (5.4) where ∥𝑑𝑓 ∥2 𝑌,∞ := sup𝑡∈𝑎𝛼 (𝑌 ) ∑ 𝑗 |𝜕𝑗 𝑓 (𝑡) |2, and ∥ [D, 𝑎] ∥𝑌 := sup𝑥∈𝑌 ∥ [D, 𝑎] (𝑥)∥𝑥 . 27 Proof. Choose 𝜙 ∈ A with 𝜙(𝑥) = 1 for all 𝑥 ∈ 𝑌 and supp 𝜙 ⊂ 𝑈𝛼. Then for all 𝑥 ∈ 𝑌 , the proof of Lemma 5.2 gives us | (𝜕𝑖 𝑓 ◦ 𝑎𝛼) (𝑥) | ⩽ ∑︁ 𝑗 𝑝 ∥𝑔𝛼,𝑖 𝑗 (𝑥) d𝑎 𝑗𝛼 (𝑥)∥𝑥 ∥d𝑎(𝑥)∥𝑥 = ∑︁ 𝑗 𝑝 ∥𝜙(𝑥)𝑔𝛼,𝑖 𝑗 (𝑥) d𝑎 𝑗𝛼 (𝑥)∥𝑥 ∥d𝑎(𝑥)∥𝑥 . Taking suprema over 𝑥 ∈ 𝑌 yields the second inequality (note that ∥ [D, 𝑎] ∥𝑌 ⩽ ∥ [D, 𝑎] ∥, as a result of Lemma 3.16). □ Denote by 𝐴D the completion of A in the norm ∥𝑎∥D := ∥𝑎∥ + ∥[D, 𝑎] ∥. Lemma 5.4. Let 𝑌 ⊂ 𝑈𝛼 be a compact set on which 𝑎𝛼 : 𝑌 → ℝ𝑝 is one-to-one. Then for all 𝑏 ∈ A𝐷 there exists a unique bounded Lipschitz function 𝑔 : 𝑎𝛼 (𝑌 ) → ℂ such that 𝑏 |𝑌 = 𝑔 ◦ 𝑎𝛼 |𝑌 . Proof. Since 𝑎𝛼 is one-to-one on 𝑌 , the functions 𝑎1 𝛼, . . . , 𝑎 𝑝 𝛼 separate the points of 𝑌 , so there is a unique bounded continuous function 𝑔 ∈ 𝐶 (𝑎𝛼 (𝑌 )) such that 𝑏 |𝑌 = 𝑔(𝑎1 𝛼, . . . , 𝑎 𝑝 𝛼). Choose 𝑏𝑘 = 𝑔𝑘 (𝑎1 𝛼, . . . , 𝑎 𝑝 𝛼) ∈ Awhere the 𝑔𝑘 are smooth functions defined on a neighbourhood of 𝑎𝛼 (𝑌 ), such that 𝑏𝑘 |𝑌 → 𝑏 |𝑌 and [D, 𝑏𝑘 ] |𝑌 → [D, 𝑏] |𝑌 . This is possible since 𝐴D is the completion of A and functions of the form 𝑓 ◦ 𝑎𝛼, with 𝑓 smooth, lie in A and separate the points of 𝑌 . Therefore, ∥ [D, 𝑏 𝑗 ] − [D, 𝑏𝑘 ] ∥𝑌 = sup 𝑥∈𝑌 ∥ [D, 𝑏 𝑗 ] (𝑥) − [D, 𝑏𝑘 ] (𝑥)∥𝑥 → 0 as 𝑗 , 𝑘 → ∞, and {𝑔𝑘 } is a Cauchy sequence in the norm 𝑓 ↦→ sup𝑦∈𝑌 | 𝑓 (𝑦) | + ∥𝑑𝑓 ∥𝑌,∞, by Corollary 5.3. Hence there is a bounded Lipschitz function ℎ : 𝑎𝛼 (𝑌 ) → ℂ such that ℎ := lim𝑘 𝑔𝑘 in this norm. Thus, (ℎ ◦ 𝑎𝛼) |𝑌 = lim 𝑘 (𝑔𝑘 ◦ 𝑎𝛼) |𝑌 = lim 𝑘 𝑏𝑘 |𝑌 = 𝑏 |𝑌 . Since 𝑏 |𝑌 = 𝑔 ◦ 𝑎𝛼, it follows that (𝑔 − ℎ) ◦ 𝑎𝛼 = 0 and we have established that 𝑔 is Lipschitz on 𝑎𝛼 (𝑌 ), with Lipschitz constant bounded above by 𝐶𝑌 ∥ [D, 𝑎] ∥. □ Proposition 5.5. Suppose that 𝐵 ⊆ 𝑈𝛼 is such that 𝑎𝛼 : 𝐵 → ℝ𝑝 is one-to-one. Then the map 𝑎−1 𝛼 : 𝑎𝛼 (𝐵) → 𝐵 is continuous for the metric topology of 𝐵 (and thus also for its weak∗ topology). Proof. Choose 𝑥, 𝑦 ∈ 𝐵 and let 𝑌 be any weak∗-compact subset of 𝐵 with 𝑥, 𝑦 ∈ 𝑌 . If 𝑎 ∈ A, let 𝑓𝑎 be the unique Lipschitz function on 𝑎𝛼 (𝑌 ) with 𝑎 |𝑌 = 𝑓𝑎 (𝑎1 𝛼, . . . , 𝑎 𝑝 𝛼). Write 𝑡 := 𝑎𝛼 (𝑥), 𝑠 := 𝑎𝛼 (𝑦) in ℝ𝑝. We now find, using Corollary 5.3 and Lemma 5.4, that 𝑑 (𝑥, 𝑦) = sup{ |𝑎(𝑥) − 𝑎(𝑦) | : 𝑎 ∈ A, ∥ [D, 𝑎] ∥ ⩽ 1 } = sup{ |( 𝑓𝑎 ◦ 𝑎𝛼) (𝑥) − ( 𝑓𝑎 ◦ 𝑎𝛼) (𝑦) | : 𝑎 ∈ A, ∥ [D, 𝑎] ∥ ⩽ 1 } = sup{ | 𝑓𝑎 (𝑡) − 𝑓𝑎 (𝑠) | : 𝑎 ∈ A, ∥ [D, 𝑎] ∥ ⩽ 1 } ⩽ sup{𝐶𝑌 ∥ [D, 𝑎] ∥ |𝑡 − 𝑠 | : 𝑎 ∈ A, ∥ [D, 𝑎] ∥ ⩽ 1 } = 𝐶𝑌 |𝑡 − 𝑠 |, (5.5) where |𝑡 − 𝑠 | is the Euclidean distance between 𝑡 and 𝑠 in ℝ𝑝. □ 28 Corollary 5.6. If 𝑌 ⊂ 𝑈𝛼 is compact and if 𝑥, 𝑦 ∈ 𝑌 satisfy 𝑎𝛼 (𝑥) ≠ 𝑎𝛼 (𝑦), then 𝑑 (𝑥, 𝑦) ⩽ 𝐶𝑌 |𝑎𝛼 (𝑥) − 𝑎𝛼 (𝑦) |. □ Note that if 𝑌,𝑌 ′ are compact subsets of 𝑈𝛼 with 𝑌 ⊆ 𝑌 ′, then 𝐶𝑌 ⩽ 𝐶𝑌 ′ by the proof of Lemma 5.2. Thus, in the previous Corollary, the minimal value of 𝐶𝑌 is 𝐶{𝑥,𝑦}. Corollary 5.7. If 𝐵 ⊆ 𝑈𝛼 is such that 𝑎𝛼 : 𝐵 → ℝ𝑝 is one-to-one, then the map 𝑎𝛼 |𝐵 is a homeomorphism onto its image (for either the metric or the weak∗ topology of 𝐵). Proof. The map 𝑎𝛼 : 𝐵 → 𝑎𝛼 (𝐵) ⊂ ℝ𝑝 is continuous for the weak∗ topology on 𝐵 since each 𝑎 𝑗𝛼 lies in A, and thus it is also continuous for the metric topology on 𝐵. The estimate (5.5) shows that the inverse map 𝑎−1 𝛼 : 𝑎𝛼 (𝐵) → 𝐵 is continuous for the metric topology on 𝐵, and a posteriori for the weak∗ topology. □ Corollary 5.8. If𝑉 ⊆ 𝑈𝛼 is an weak∗-open subset such that 𝑎𝛼 : 𝑉 → ℝ𝑝 is one-to-one, then 𝑎𝛼 (𝑉) is an open subset of ℝ𝑝. Proof. By Lemma 4.15, the set 𝑎𝛼 (𝑉) has nonempty interior in ℝ𝑝. Now 𝑎𝛼 (𝑉) \ Int 𝑎𝛼 (𝑉) is the boundary of Int 𝑎𝛼 (𝑉) in the relative topology of 𝑎𝛼 (𝑉). Since 𝑎𝛼 is a homeomorphism from 𝑉 onto 𝑎𝛼 (𝑉), it cannot map interior points to this boundary. Thus, since 𝑉 is open, this boundary is empty, and hence 𝑎𝛼 (𝑉) = Int 𝑎𝛼 (𝑉) is open in ℝ𝑝. □ Lemma 5.9. On any subset 𝑉 ⊂ 𝑈𝛼 on which 𝑎𝛼 is one-to-one, the weak∗ and metric topologies coincide. Proof. By Lemma 5.4, the restriction of any function 𝑎 ∈ A to a compact subset 𝑌 ⊂ 𝑉 may be written as 𝑎 |𝑌 = ( 𝑓𝑎 ◦ 𝑎𝛼) |𝑌 , where 𝑓𝑎 is a bounded Lipschitz function on 𝑎𝛼 (𝑌 ). We need only show that convergence of a sequence 𝑉 ∋ 𝑥𝑚 → 𝑦 for the weak∗ topology implies the convergence 𝑥𝑚 → 𝑦 in the metric topology. Choose such a weak∗-convergent sequence {𝑥𝑚}; without loss of generality, we may suppose that it is contained in a compact subset 𝑌 ⊂ 𝑉 such that each 𝑥𝑚 ∈ 𝑌 and hence also 𝑦 ∈ 𝑌 . Weak∗ convergence of the sequence 𝑥𝑚 means that |𝑎(𝑥𝑚) − 𝑎(𝑦) | → 0 for all 𝑎 ∈ A. This implies that |𝑎(𝑥𝑚) − 𝑎(𝑦) | ∥ [D, 𝑎] ∥ → 0 as 𝑚 → ∞ for all 𝑎 ∈ Dom d with d𝑎 = [D, 𝑎] ≠ 0. By Lemma 5.4, this is equivalent to | ( 𝑓𝑎 ◦ 𝑎𝛼) (𝑥𝑚) − ( 𝑓𝑎 ◦ 𝑎𝛼) (𝑦) | ∥ [D, 𝑎] ∥ → 0 as 𝑚 → ∞ for all such 𝑎. Using Corollary 5.3, this is equivalent to |𝑎𝛼 (𝑥𝑚) − 𝑎𝛼 (𝑦) | → 0 as 𝑚 → ∞. The metric convergence follows: 𝑑 (𝑥𝑚, 𝑦) = sup{ |𝑎(𝑥𝑚) − 𝑎(𝑦) | : ∥ [D, 𝑎] ∥ ⩽ 1 } = sup{ |( 𝑓 ◦ 𝑎𝛼) (𝑥𝑚) − ( 𝑓 ◦ 𝑎𝛼) (𝑦) | : ∥𝑑𝑓 ∥∞ ⩽ 𝐶𝑌 } ⩽ 𝐶𝑌 |𝑎𝛼 (𝑥𝑚) − 𝑎𝛼 (𝑦) | → 0 as 𝑚 → ∞, on invoking Corollary 5.3 once more. □ 29 6 Point-set properties of the local coordinate charts We must analyze the possible failure of 𝑎𝛼 to be one-to-one, by using some point-set topology. Some of this extra effort is due to the arbitrariness of the Hochschild cycle c. For example, consider the manifold 𝕊2, with volume form 𝑥 𝑑𝑦 ∧ 𝑑𝑧 + 𝑦 𝑑𝑧∧ 𝑑𝑥 + 𝑧 𝑑𝑥 ∧ 𝑑𝑦. Each𝑈𝛼 (the subset where 𝑎0 𝛼 is nonzero, and 𝑑𝑎1 𝛼, 𝑑𝑎2 𝛼 are linearly independent) consists of two open hemispheres, missing only an equator. For instance, there is a chart domain consisting of 𝕊2 \ {𝑧 = 0} with local coordinates (𝑥, 𝑦); but these “coordinates” are actually two-to-one on this domain. This rather simple problem can be addressed by restricting the coordinate maps to the connected components of the𝑈𝛼, thereby increasing the number of charts. Ultimately, in the next section, we shall show that this is the worst that can happen. We begin by considering some set-theoretic properties of the map 𝑎𝛼 : 𝑈𝛼 → ℝ𝑝 for a fixed 𝛼, with a view to showing that 𝑎𝛼 is at worst finitely-many-to-one on an open dense subset of 𝑈𝛼. (Eventually, this behaviour will be improved to local injectivity.) The linear functional 𝜇D(𝑎) := ⨏ 𝑎 ⟨D⟩−𝑝 on A extends to a continuous linear functional on 𝐴, and by the Riesz representation theorem it is given by a regular Borel measure that we also denote by 𝜇D. Lemma 6.1. The measure 𝜇D has no atoms. Proof. By [55, Lemma 14, p. 408], we can decompose 𝑋 as a disjoint union 𝑋 = 𝑋′ ⊎ 𝐶 where 𝐶 is countable, its closure 𝐶 is the support of the atomic part of 𝜇D, and 𝑋′ has no isolated points. Recalling that ⟨𝜉 | 𝜂⟩ = 𝜇D((𝜉 | 𝜂)) for 𝜉, 𝜂 ∈ H∞, we get a corresponding Hilbert-space decomposition H = 𝐿2(𝑋, 𝑆) = 𝐿2(𝑋′, 𝑆 |𝑋 ′) ⊕ 𝐿2(𝐶, 𝑆 |𝐶). If supp 𝜉 = {𝑥} ⊆ 𝐶, then 𝑎𝜉 = 𝑎(𝑥)𝜉 for any 𝑎 ∈ A, so that ℂ𝜉 would be a 1-dimensional subrepresentation of 𝜋(𝐴) and thus 𝜋(𝐴) would contain a nontrivial projector, contradicting Corol- lary 3.14. □ We can write 𝜇D as a sum of finite measures 𝜇D,𝛼 concentrated on each𝑈𝛼 by⨏ 𝑎 ⟨D⟩−𝑝 = ⨏ Γ2 𝑎 ⟨D⟩−𝑝 = 𝑛∑︁ 𝛼=1 ⨏ Γ 𝑎𝑎0 𝛼 d𝑎1 𝛼 . . . d𝑎 𝑝 𝛼 ⟨D⟩−𝑝 =: 𝑛∑︁ 𝛼=1 𝜇D,𝛼 (𝑎) since the third expression depends only on the skew-symmetrization of the d𝑎 𝑗𝛼, by the proof of Proposition 4.4, and each skew-symmetrized Γ′ 𝛼 vanishes off the respective𝑈𝛼. We can transfer these measures to the several 𝑎𝛼 (𝑈𝛼) by setting 𝜆D,𝛼 ( 𝑓 ) := 𝜇D,𝛼 ( 𝑓 (𝑎1 𝛼, . . . , 𝑎 𝑝 𝛼)) for 𝑓 ∈ 𝐶𝑐 (𝑎𝛼 (𝑈𝛼)). Each 𝜆D,𝛼 is a nonatomic regular Borel measure on 𝑎𝛼 (𝑈𝛼) ⊂ ℝ𝑝. Its Lebesgue decomposition 𝜆D,𝛼 = 𝜆𝑠D,𝛼 + 𝜆 ac D,𝛼 provides measures 𝜆𝑠 D,𝛼 and 𝜆ac D,𝛼 that are respectively singular and absolutely continuous with respect to the Lebesgue measure on 𝑎𝛼 (𝑈𝛼). The singular part 𝜆𝑠 D,𝛼 is concentrated on a set of Lebesgue measure zero, whereas the absolutely continuous part 𝜆ac D,𝛼 has support 𝑎𝛼 (𝑈𝛼). 30 Definition 6.1. Define 𝑛𝛼 : 𝑈𝛼 → {1, 2, . . . ,∞} by 𝑛𝛼 (𝑦) := # ( 𝑎−1 𝛼 (𝑎𝛼 (𝑦)) ∩𝑈𝛼 ) , where # denotes cardinality (all infinite cardinals being treated simply as ∞). Lemma 6.2. The set 𝑛−1 𝛼 (∞) of infinite-multiplicity points has empty interior in𝑈𝛼. Proof. Let spac(𝑎𝛼) ⊆ Δ𝛼 be the absolutely continuous joint spectrum of 𝑎𝛼 = (𝑎1 𝛼, . . . , 𝑎 𝑝 𝛼), regarded as 𝑝 commuting selfadjoint operators on H. Over spac(𝑎𝛼), the multiplicity function 𝑚𝛼 of (the representation of) 𝑎𝛼 is 𝐿1 with respect to Lebesgue measure. Without changing the measure class, we may take 𝑚𝛼 to be precisely the multiplicity of 𝑎𝛼, namely, 𝑚𝛼 (𝑡) = 𝑁 𝑛𝛼 (𝑎−1 𝛼 (𝑡) ∩𝑈𝛼) for 𝑡 ∈ spac(𝑎𝛼), where 𝑁 is the rank of the bundle 𝑆. This is well defined since 𝑛𝛼 is constant on 𝑎−1 𝛼 (𝑡) ∩𝑈𝛼. Suppose that𝑉 ⊂ 𝑎𝛼 (𝑛−1 𝛼 (∞)) is a Borel subset of 𝑎𝛼 (𝑈𝛼). If𝑉 had positive Lebesgue measure, it would follow that ∫ spac (𝑎𝛼) 𝑚𝛼 (𝑡) 𝑑𝑝𝑡 ⩾ ∫ 𝑉 𝑚𝛼 (𝑡) 𝑑𝑝𝑡 = ∞. (6.1) However, by [61], discussed also in [18, Sec. IV.2.𝛿], there is an equality∫ spac (𝑎𝛼) 𝑚𝛼 (𝑡) 𝑑𝑝𝑡 = 𝐶′ 𝑝 (𝑘−𝑝 (𝑎𝛼))𝑝, where𝐶′ 𝑝 is a constant depending only on 𝑝, and Voiculescu’s modulus 𝑘−𝑝 (𝑎𝛼) is a positive number which measures the size of the joint absolutely continuous spectrum. This number is finite, since by [18, Prop. IV.2.14], see also [10, 30] for similar estimates: 𝑘−𝑝 (𝑎𝛼) ⩽ 𝐶𝑝 max 𝑗 ∥ [D, 𝑎 𝑗𝛼] ∥ (⨏ ⟨D⟩−𝑝 )1/𝑝 < ∞. This contradicts (6.1); consequently, 𝑉 is a Lebesgue nullset. Since any subset of 𝑎𝛼 (𝑈𝛼) on which 𝜆𝑠 D,𝛼 is concentrated also has Lebesgue measure zero, and elsewhere 𝑚𝛼 is finite, we conclude that 𝑎𝛼 (𝑛−1 𝛼 (∞)) is a Lebesgue nullset, and in particular it has empty interior. Now Lemma 4.15 entails that 𝑛−1 𝛼 (∞) has empty interior in𝑈𝛼. □ ▶ The remainder of this section proves that the topological structure of 𝑈𝛼 is sufficiently nice for us to deploy our geometric tools in the following section. These tools will prove the local injectivity of 𝑎𝛼 : 𝑈𝛼 → ℝ𝑝. From now until Theorem 7.20, we fix 𝛼 and work solely within 𝑈𝛼 using the relative weak∗ topology. Thus, all closures, interiors and boundaries are taken in this relative weak∗ topology, unless specifically noted. Also, for 𝐸 ⊆ Δ𝛼 ⊂ ℝ𝑝, “𝑎−1 𝛼 (𝐸)” will mean 𝑎−1 𝛼 (𝐸) ∩ 𝑈𝛼. Thus, up until Theorem 7.20, we restrict the universe to𝑈𝛼. Definition 6.2. Consider the following subsets of𝑈𝛼, for 𝑘 = 1, 2, . . . : 𝐷𝑘 := 𝑛−1 𝛼 ({1, . . . , 𝑘}), 𝐸𝑘 := Int𝐷𝑘 , 𝑁𝑘+1 := 𝑈𝛼 \ 𝐷𝑘 = 𝑛 −1 𝛼 ({𝑘 + 1, . . . ,∞}), 𝑊𝑘 := Int(𝑛−1 𝛼 (𝑘)). (6.2) 31 Remark 6.3. If 𝑍 ⊂ 𝑈𝛼, its closure in the relative (weak∗) topology of𝑈𝛼 is 𝑍𝑋 ∩𝑈𝛼. Elements of this closure are limits of sequences in 𝑍 , since 𝑋 is metrizable because the𝐶∗-algebra 𝐴 is assumed to be separable. Lemma 6.4. The set 𝑁𝑘+1 is open and the set 𝐷𝑘 is closed in𝑈𝛼. Proof. If 𝐷𝑘 is finite, there is nothing to prove; otherwise, choose any convergent sequence {𝑥𝑖} ⊂ 𝐷𝑘 with 𝑥𝑖 → 𝑥 ∈ 𝑈𝛼. Each element 𝑥𝑖 has multiplicity ⩽ 𝑘 , and so 𝑎−1 𝛼 (𝑎𝛼 ({𝑥𝑖})) consists of at most 𝑘 sequences with at most 𝑘 limit points 𝑦1, . . . , 𝑦𝑚, 𝑚 ⩽ 𝑘 , since 𝑈𝛼 is Hausdorff. Thus 𝑥 is one of the 𝑦 𝑗 , say 𝑥 = 𝑦1, and 𝑎𝛼 (𝑦) = 𝑎𝛼 (𝑥) for 𝑦 ∈ 𝐷𝑘 if and only if 𝑦 ∈ {𝑦1, . . . , 𝑦𝑚}. (Notice that𝑈𝛼 can contain no isolated points, by Corollary 3.14.) Hence the limit point 𝑥 has multiplicity 𝑚 ⩽ 𝑘 , and so 𝑥 ∈ 𝐷𝑘 . □ Notice that 𝐷𝑘 is nonempty for some finite 𝑘 , since 𝑛−1 𝛼 (∞) ≠ 𝑈𝛼. The next Proposition shows that the awkward possibility of 𝑛−1 𝛼 (∞) being dense cannot occur. We require a preparatory lemma. Lemma 6.5. If 𝑛−1 𝛼 (∞) were dense in 𝑈𝛼, then each 𝑁𝑘+1 would be an open dense subset of 𝑈𝛼, every neighbourhood of an infinite-multiplicity point in 𝑛−1 𝛼 (∞) would contain elements of 𝐷𝑘 for arbitrarily large 𝑘 , and every neighbourhood of a finite-multiplicity point in some 𝐷𝑘 would contain infinitely many points in 𝑛−1 𝛼 (∞). Proof. The first statement is clear, since 𝑛−1 𝛼 (∞) ⊆ 𝑁𝑘+1 for each finite 𝑘 . By Lemma 6.2, the union ⋃∞ 𝑘=1 𝐷𝑘 is dense in 𝑈𝛼 and, by the proof of Lemma 6.4, no infinite- multiplicity point can be the limit of a sequence contained in any 𝐷𝑘 . The last statement is just the assumed density of 𝑛−1 𝛼 (∞). □ Proposition 6.6. The subset 𝑛−1 𝛼 (∞) is nowhere dense in𝑈𝛼. Proof. Let 𝑌 be a compact subset of 𝑈𝛼 with nonempty interior. Suppose, arguendo, that the set 𝑛−1 𝛼 (∞) ∩ Int𝑌 is dense in Int𝑌 . Since 𝑎𝛼 cannot then be one-to-one on 𝑌 , we can choose a finite-multiplicity value 𝑡 ∈ 𝑎𝛼 (Int𝑌 ) and two distinct points 𝑦, 𝑦′ ∈ 𝑌 such that 𝑎𝛼 (𝑦) = 𝑎𝛼 (𝑦′) = 𝑡. If necessary, we can add a small compact neighbourhood of 𝑦′ to 𝑌 . Using Lemma 6.5, we can find a sequence {𝑡𝑚} ⊂ 𝑎𝛼 (𝑌 ∩ 𝑛−1 𝛼 (∞)) such that |𝑡 − 𝑡𝑚 | < 𝜀𝑚 for all 𝑚, where 𝜀𝑚 → 0 with 0 < 𝜀𝑚 < (2𝐶𝑌 )−1𝑑 (𝑦, 𝑦′) for all 𝑚, where 𝐶𝑌 is the constant appearing on the right hand side of the estimate (5.4). Suppose that for all 𝑚, all points 𝑧 ∈ 𝑎−1 𝛼 (𝑡𝑚) ∩ 𝑌 satisfy 𝑑 (𝑦, 𝑧) ⩽ 𝐶𝑌𝜀𝑚. Then 𝑑 (𝑦′, 𝑧) ⩾ ��𝑑 (𝑦, 𝑦′) − 𝑑 (𝑦, 𝑧)�� ⩾ 𝑑 (𝑦, 𝑦′) − 𝐶𝑌𝜀𝑚 > 𝐶𝑌𝜀𝑚 . Consequently, on applying Corollary 5.6 to the pair of points 𝑦′, 𝑧 ∈ 𝑌 , we obtain 𝐶𝑌𝜀𝑚 < 𝑑 (𝑦′, 𝑧) ⩽ 𝐶𝑌 ��𝑎𝛼 (𝑦′) − 𝑎𝛼 (𝑧)�� = 𝐶𝑌 |𝑡 − 𝑡𝑚 | < 𝐶𝑌𝜀𝑚, (6.3) a contradiction. On the other hand, if there is some 𝑚 and some 𝑧 ∈ 𝑎−1 𝛼 (𝑡𝑚) ∩ 𝑌 such that 𝑑 (𝑦, 𝑧) > 𝐶𝑌𝜀𝑚, then we reach the same impasse on replacing 𝑦′ by 𝑦 in (6.3). We conclude that no such sequence 𝑡𝑚 → 𝑡 can exist, so that in particular 𝑦 has a neighbourhood excluding 𝑛−1 𝛼 (∞). That is to say, 𝑛−1 𝛼 (∞) ∩ Int𝑌 is not dense in Int𝑌 . By the arbitrariness of 𝑌 , 𝑛−1 𝛼 (∞) is nowhere dense in𝑈𝛼. □ 32 Corollary 6.7. The map 𝑎𝛼 is finitely-many-to-one on an open dense subset of𝑈𝛼. □ Lemma 6.8. Within𝑈𝛼, the following relations hold: 𝑛−1 𝛼 (𝑘) \𝑊𝑘 ⊂ 𝜕𝑊𝑘 ∪ 𝜕𝑁𝑘+1, 𝑛−1 𝛼 (𝑘) \𝑊 𝑘 ⊂ 𝜕𝑁𝑘+1, 𝜕𝐷𝑘 = 𝜕𝑁𝑘+1, and thus𝑈𝛼 = Int𝐷𝑘 ∪ 𝑁 𝑘+1. Moreover, 𝜕𝑊𝑘 ⊆ 𝐷𝑘 and 𝑛−1 𝛼 (∞) ∩ ∞⋃ 𝑘=1 𝑊 𝑘 = ∅. Proof. Consider the set 𝐻𝑘 := 𝑈𝛼 \ ( 𝐷𝑘−1 ∪𝑊 𝑘 ∪ 𝑁 𝑘+1 ) . It is open in 𝑈𝛼, and 𝐻𝑘 ⊂ 𝑛−1 𝛼 (𝑘). However, 𝐻𝑘 ∩ Int(𝑛−1 𝛼 (𝑘)) = 𝐻𝑘 ∩𝑊𝑘 = ∅, entailing 𝐻𝑘 = ∅. Therefore𝑈𝛼 = 𝐷𝑘−1 ∪𝑊 𝑘 ∪ 𝑁 𝑘+1, and so 𝑛−1 𝛼 (𝑘) \𝑊𝑘 ⊂ 𝐷𝑘−1 ∪ 𝜕𝑊𝑘 ∪ 𝑁 𝑘+1. However, both 𝐷𝑘−1 and 𝑁𝑘+1 are disjoint from 𝑛−1 𝛼 (𝑘) by definition, so 𝑛−1 𝛼 (𝑘) \𝑊𝑘 ⊂ 𝜕𝑊𝑘 ∪ 𝜕𝑁𝑘+1. Taking the closure of 𝑊𝑘 in 𝑈𝛼 gives the second relation. The third relation follows on recalling that𝑈𝛼 \ 𝑁𝑘+1 = 𝐷𝑘 . It is clear that 𝜕𝑊𝑘 ⊆ 𝐷𝑘 , since 𝑊𝑘 ⊂ 𝐷𝑘 and 𝐷𝑘 is closed. The last relation follows from 𝑊 𝑘 ⊂ 𝐷𝑘 , since 𝑛−1 𝛼 (∞) is disjoint from each 𝐷𝑘 . □ Lemma 6.9. The interiors of 𝐷𝑘 and of ⋃𝑘 𝑗=1𝑊 𝑗 coincide: 𝐸𝑘 = Int (⋃𝑘 𝑗=1𝑊 𝑗 ) . Proof. Take 𝑥 ∈ 𝐸𝑘 . Then every open neighbourhood of 𝑥 in 𝐷𝑘 must meet some 𝑊 𝑗 with 𝑗 ⩽ 𝑘 , so that 𝑥 ∈ ⋃𝑘 𝑗=1𝑊 𝑗 . Hence 𝐸𝑘 ⊆ ⋃𝑘 𝑗=1𝑊 𝑗 . On the other hand, since each 𝐷 𝑗 is closed in 𝑈𝛼, we get ⋃𝑘 𝑗=1𝑊 𝑗 ⊆ ⋃𝑘 𝑗=1 𝐷 𝑗 = 𝐷𝑘 . The inclusion Int (⋃𝑘 𝑗=1𝑊 𝑗 ) ⊆ 𝐸𝑘 follows at once. □ These topological preliminaries now allow us to prove three basic results about the mapping 𝑎𝛼 and the set𝑈𝛼. Proposition 6.10. There exists a weak∗-open cover {𝑍 𝑗 } 𝑗⩾1 of ⋃∞ 𝑘=1𝑊𝑘 ⊆ 𝑈𝛼 such that 𝑎𝛼 : 𝑍 𝑗 → 𝑎𝛼 (𝑍 𝑗 ) ⊂ ℝ𝑝 is a homeomorphism and an open map, for each 𝑗 ⩾ 1. Proof. First observe that 𝑊1 is an open subset of 𝑈𝛼 such that 𝑎𝛼 : 𝑊1 → 𝑎𝛼 (𝑊1) is an open homeomorphism. This follows from Corollaries 5.7 and 5.8 and the definition of𝑊1. Choose 𝑥 ∈ 𝑊𝑘 , 𝑘 > 1. Since 𝑎−1 𝛼 (𝑎𝛼 (𝑥)) consists of 𝑘 points of 𝑊𝑘 and 𝑊𝑘 is Hausdorff, we may choose 𝑘 disjoint open subsets of𝑊𝑘 , each containing precisely one of the preimages 𝑥1, . . . , 𝑥𝑘 of 𝑎𝛼 (𝑥). Call these sets𝑉1, . . . , 𝑉𝑘 , and suppose that for some 𝑗 = 1, . . . , 𝑘 , the map 𝑎𝛼 : 𝑉 𝑗 → ℝ𝑝 is not one-to-one. Then there exist 𝑧1, 𝑧2 ∈ 𝑉 𝑗 with 𝑧1 ≠ 𝑧2, such that 𝑎𝛼 (𝑧1) = 𝑎𝛼 (𝑧2). Again we can separate these two points, and thereby suppose that only one of 𝑧1, 𝑧2 lies in 𝑉 𝑗 . If for every 𝑥 ∈ 𝑊𝑘 we can repeat this process finitely often to obtain neighbourhoods of each of the 𝑥 𝑗 on which 𝑎𝛼 is one-to-one, we are done. 33 On the other hand, if in this manner we always find pairs of distinct points 𝑧1𝑚 ≠ 𝑧2𝑚 from successively smaller neighbourhoods of 𝑥 𝑗 , we thereby obtain two sequences, both converging to 𝑥 𝑗 . We may alternate the labelling of each pair, if necessary, so that both sequences are infinite. Then for 𝑖 = 1, 2, 𝑎𝛼 (𝑧𝑖𝑚) converges to 𝑎𝛼 (𝑥 𝑗 ) for 𝑗 = 1, . . . , 𝑘 , and so in any open neighbourhood 𝑁 of 𝑎𝛼 (𝑥 𝑗 ) there are infinitely many such 𝑎𝛼 (𝑧𝑖𝑚). Now take a neighbourhood 𝑉𝑙 of each 𝑥𝑙 , for 𝑙 ≠ 𝑗 , such that 𝑎𝛼 (𝑉𝑙) maps onto 𝑁 (this is possible; we started with disjoint neighbourhoods of the 𝑥𝑙 which map onto a neighbourhood of 𝑎𝛼 (𝑥 𝑗 )). Then there is a sequence {𝑦𝑙𝑚} ⊂ 𝑉𝑙 mapping onto 𝑎𝛼 (𝑧𝑖𝑚). A quick count now shows that each element of the sequence 𝑎𝛼 (𝑧𝑖𝑚) has (at least) 𝑘 + 1 preimages in𝑊𝑘 : contradiction. Hence, for all 𝑥 ∈ 𝑊𝑘 we may find a neighbourhood 𝑉 of 𝑥 in 𝑊𝑘 such that 𝑎𝛼 : 𝑉 → 𝑎𝛼 (𝑉) is one-to-one, and thus is an open homeomorphism for the weak∗ topology on 𝑉 , by Corollary 5.7. In this way we obtain a open cover of the (locally compact, metrizable) set ⋃∞ 𝑘=1𝑊𝑘 of the desired form; now let {𝑍 𝑗 } 𝑗⩾1 be an enumeration of a countable subcover. Notice that we have also shown that each 𝑍 𝑗 is included in some𝑊𝑘 , with 𝑘 = 𝑘 ( 𝑗). □ The next Proposition is the most crucial consequence of the Lipschitz functional calculus. It allows us to extend the homeomorphism 𝑎𝛼 : 𝑍 𝑗 → 𝑎𝛼 (𝑍 𝑗 ) to the closure 𝑍 𝑗 as a homeomorphism. It is essential in the next Proposition that we use the relative topology of𝑈𝛼. Proposition 6.11. For each open set 𝑍 𝑗 as in Proposition 6.10, the function 𝑎𝛼 extends to a homeomorphism (for both the metric and weak∗ topologies) 𝑎𝛼 : 𝑍 𝑗 → 𝑎𝛼 (𝑍 𝑗 ). The same is true when 𝑍 𝑗 is replaced by any open set 𝑉 ⊂ 𝑈𝛼 on which 𝑎𝛼 is one-to-one. In particular, the weak∗ and metric topologies agree on 𝑉 . Proof. Take any 𝑏 ∈ A. Then by Lemma 5.4, for any compact 𝑌 ⊂ 𝑍 𝑗 there is a unique bounded Lipschitz function 𝑔 : 𝑎𝛼 (𝑌 ) → ℂ such that 𝑏 |𝑌 = 𝑔 ◦ 𝑎𝛼 |𝑌 . Now we can cover 𝑍 𝑗 by open sets which are interiors of compact subsets, and so obtain many function representations of 𝑏 on these sets. By uniqueness, they agree on overlaps, and 𝑏 = 𝑔 ◦ 𝑎𝛼 for each of these local representations. Hence we arrive at a single function 𝑔 : 𝑎𝛼 (𝑍 𝑗 ) → ℂ with 𝑏 |𝑍 𝑗 = 𝑔 ◦ 𝑎𝛼 |𝑍 𝑗 . The function 𝑔 is bounded since 𝑏 is bounded, but might be only locally Lipschitz (since 𝑎−1 𝛼 might only be locally Lipschitz). To proceed, suppose first that 𝑔 is a 𝐶1-function on 𝑎𝛼 (𝑍 𝑗 ). The proof of Lemma 5.2, in particular (5.3), and Corollary 5.3 guarantee that for any subset 𝑌 of 𝑍 𝑗 with compact closure contained in 𝑍 𝑗 , sup 𝑥∈𝑌 |𝜕𝑗𝑔(𝑎𝛼 (𝑥)) | ⩽ sup 𝑥∈𝑌 𝐵 𝑗 (𝑥) ∥ [D, 𝑏] (𝑥)∥𝑥 < ∞. The finiteness of the right hand side follows because ∥ [D, 𝑏] (𝑥)∥𝑥 ⩽ ∥ [D, 𝑏] ∥, and 𝐵 𝑗 is continuous on all of𝑈𝛼, so that 𝐵 𝑗 is bounded on 𝑌 . To see that 𝑔 extends to a locally Lipschitz function on 𝑎𝛼 (𝑍 𝑗 ), we argue as follows. If 𝑡 ∈ 𝑎𝛼 (𝑍 𝑗 ), and if {𝑡𝑛} and {𝑡′𝑛} are two sequences in 𝑎𝛼 (𝑍 𝑗 ) such that 𝑡𝑛 → 𝑡 and 𝑡′𝑛 → 𝑡 in 𝑎𝛼 (𝑈𝛼), then |𝑔(𝑡𝑛) − 𝑔(𝑡′𝑛) | ⩽ 𝐶 |𝑡𝑛 − 𝑡′𝑛 |, where 𝐶 = sup 𝑥∈𝑌 𝐵 𝑗 (𝑥) ∥ [D, 𝑏] ∥. For the estimate, since 𝑍 𝑗 ⊆ 𝑊𝑘 for a suitable 𝑘 , it is enough take𝑌 to be the union of the 𝑘 preimages of the sequences {𝑡𝑛} and {𝑡′𝑛} and their limits, which is compact in𝑈𝛼. Thus �̃�(𝑡) := lim𝑛→∞ 𝑔(𝑡𝑛) 34 is well defined, and coincides with 𝑔(𝑡) whenever 𝑡 ∈ 𝑎𝛼 (𝑍 𝑗 ) already. The upshot is a bounded continuous function �̃� : 𝑎𝛼 (𝑍 𝑗 ) → ℝ. Its Lipschitz norm on any compact subset 𝑌 ⊂ 𝑍 𝑗 satisfies ∥𝑑�̃�∥𝑌 ⩽ 𝐶𝑌 ∥ [D, 𝑏] ∥, and so �̃� is locally Lipschitz. The continuity of �̃� and 𝑏 yields 𝑏 = �̃� ◦ 𝑎𝛼 over the set 𝑍 𝑗 . For an arbitrary 𝑏 ∈ A, we may remove the assumption that 𝑔 be𝐶1 on 𝑎𝛼 (𝑍 𝑗 ) by approximating 𝑏 by a sequence {𝑏𝑟} in the norm 𝑎 → ∥𝑎∥ + ∥[D, 𝑎] ∥, where 𝑏𝑟 |𝑍 𝑗 = 𝑔𝑟 ◦ 𝑎𝛼 |𝑍 𝑗 with each 𝑔𝑟 being 𝐶1. For any subset 𝑌 ⊂ 𝑍 𝑗 with compact closure, we get an estimate sup 𝑥∈𝑌 |𝜕𝑗𝑔𝑟 (𝑎𝛼 (𝑥)) − 𝜕𝑗𝑔𝑠 (𝑎𝛼 (𝑥)) | ⩽ sup 𝑥∈𝑌 𝐵 𝑗 (𝑥) ∥ [D, 𝑏𝑟 − 𝑏𝑠] ∥, and thus {𝑔𝑟} is a Cauchy sequence in the Lipschitz norm of 𝑎𝛼 (𝑌 ). Each 𝑔𝑟 extends to a locally Lipschitz function �̃�𝑟 on 𝑎𝛼 (𝑍 𝑗 ), and these converge uniformly on compact subsets to a locally Lipschitz function �̃� satisfying 𝑏 = �̃� ◦ 𝑎𝛼 over 𝑍 𝑗 . Since we can thereby extend the function representation for all functions 𝑏 ∈ A, we conclude that 𝑎𝛼 separates points of 𝑍 𝑗 , and thus it is one-to-one on this set. By Corollaries 5.7 and 5.8, 𝑎𝛼 : 𝑍 𝑗 → ℝ𝑝 is an open map, which is a homeomorphism onto its image. By Lemma 5.9, the weak∗ and metric topologies agree. □ Here is a first and critical consequence of Proposition 6.11. Lemma 6.12. For each 𝑘 ⩾ 1, the set 𝑊𝑘 is a disjoint union of 𝑘 open subsets 𝑊1𝑘 , . . . ,𝑊𝑘𝑘 , on each of which 𝑎𝛼 is injective, such that 𝑎𝛼 (𝑊 𝑗 𝑘 ) = 𝑎𝛼 (𝑊 𝑗 ′𝑘 ) for 𝑗 , 𝑗 ′ = 1, . . . , 𝑘 . Proof. Choose any point 𝑥 ∈ 𝑊𝑘 , and choose disjoint open sets 𝑉 𝑗 ⊂ 𝑊𝑘 , for 𝑗 = 1, . . . , 𝑘 , each containing precisely one of the preimages of 𝑥 in 𝑎−1 𝛼 (𝑎𝛼 (𝑥)). We may suppose by Proposition 6.10 that 𝑎𝛼 is one-to-one on each 𝑉 𝑗 , and on replacing the 𝑉 𝑗 by the components of 𝑎−1 𝛼 (⋂ 𝑗 𝑎𝛼 (𝑉 𝑗 ) ) , we may suppose also that the various 𝑉 𝑗 have the same image 𝑎𝛼 (𝑉 𝑗 ) in ℝ𝑝. By Proposition 6.11, 𝑎𝛼 extends to a homeomorphism on each 𝑉 𝑗 . Suppose first that there exists 𝑧 ∈ 𝑉1 ∩ 𝑉 𝑙 ∩𝑊𝑘 for some 𝑙 ⩾ 2. Then, since 𝑎𝛼 is a homeo- morphism on each 𝑉 𝑗 , the set 𝑎−1 𝛼 (𝑎𝛼 (𝑧)) ∩ ⋃𝑘 𝑗=1𝑉 𝑗 consists of fewer than 𝑘 points. So there must be some 𝑧′ ∈ 𝑎−1 𝛼 (𝑎𝛼 (𝑧)) ∩𝑊𝑘 with 𝑧′ ∉ 𝑉 𝑗 for 𝑗 = 1, . . . , 𝑘 . The open set 𝑊𝑘 \ ⋃𝑘 𝑗=1𝑉 𝑗 includes an open neighbourhood 𝑂 of 𝑧′. Now 𝑎−1 𝛼 (𝑎𝛼 (𝑂)) is an open set in 𝑊𝑘 meeting each 𝑉 𝑗 , since 𝑧 ∈ 𝜕𝑉1 and 𝑎𝛼 (𝑉1) = 𝑎𝛼 (𝑉 𝑗 ) for each 𝑗 . Consider a sequence {𝑡𝑚} ⊂ 𝑎𝛼 (𝑂) ∩ 𝑎𝛼 (𝑉 𝑗 ) with 𝑡𝑚 → 𝑎𝛼 (𝑧) = 𝑎𝛼 (𝑧′). Then 𝑎−1 𝛼 ({𝑡𝑚}) meets each 𝑉 𝑗 and 𝑂. Hence for 𝑚 sufficiently large, 𝑛𝛼 (𝑎−1 𝛼 (𝑡𝑚)) > 𝑘 , which is impossible within𝑊𝑘 . We conclude that the relatively closed subsets𝑉 𝑗∩𝑊𝑘 of𝑊𝑘 are disjoint. Since𝑊𝑘 is metrizable (by Remark 3.5) and is thus a normal topological space, there are disjoint open sets𝑈1, . . . ,𝑈𝑘 ⊂ 𝑊𝑘 with 𝑉 𝑗 ∩𝑊𝑘 ⊂ 𝑈 𝑗 . Now choose a point 𝑧 on the boundary of 𝑉1 lying in 𝑊𝑘 (if there is no such point, then 𝑉1 is a union of connected components of 𝑊𝑘 , and so too are the other 𝑉 𝑗 , and we are done). Choose an open neighbourhood𝑈 ⊂ 𝑊𝑘 of 𝑧 on which 𝑎𝛼 is one-to-one. Note that 𝑎𝛼 is one-to-one on𝑈 ∩𝑉1, on𝑈, and on𝑉1. Let𝑈′ := 𝑈1 ∩𝑈; we claim that 𝑎𝛼 : 𝑈′∪𝑉1 → ℝ𝑝 is one-to-one. For if not, there would exist 𝑦 ∈ 𝑈′\𝑉1 and 𝑦′ ∈ 𝑉1 \𝑈′ such that 𝑎𝛼 (𝑦) = 𝑎𝛼 (𝑦′). However, 𝑦 ∉ 𝑉1 and 𝑎−1 𝛼 (𝑎𝛼 (𝑦′)) consists of 𝑘 points already, so this forces 𝑦 ∈ 𝑉 𝑗 ∩ 𝑎−1 𝛼 (𝑎𝛼 (𝑦′)) for some 𝑗 > 1; otherwise 𝑦 would have multiplicity at least 𝑘 + 1, contradicting 𝑦 ∈ 𝑊𝑘 . However, 𝑦 ∈ 𝑈′ \ 𝑉1 ⊂ 𝑈1, which forbids 35 𝑦 ∈ 𝑉 𝑗 for 𝑗 > 1. The upshot is that 𝑎−1 𝛼 (𝑎𝛼 (𝑈′ ∪ 𝑉1)) is a union of 𝑘 disjoint open sets on each of which 𝑎𝛼 is one-to-one. It is clear that𝑈 may be chosen such that their images under 𝑎𝛼 coincide. By this argument, we may continue this process by taking a boundary point of 𝑈′ ∪ 𝑉1 within 𝑊𝑘 , finding a neighbourhood on which 𝑎𝛼 is one-to-one, and deducing that 𝑎𝛼 is one-to-one on the union. In this way we cover the entire connected component of 𝑊𝑘 in which 𝑥 lies. Thus 𝑊𝑘 is a disjoint union of 𝑘 open subsets𝑊1𝑘 , . . . ,𝑊𝑘𝑘 , and by construction we see that 𝑎𝛼 (𝑊 𝑗 𝑘 ) = 𝑎𝛼 (𝑊 𝑗 ′𝑘 ) for all 𝑗 , 𝑗 ′. □ Corollary 6.13. Each set𝑊 𝑗 𝑘 is a smooth manifold with the coordinate map 𝑎𝛼 |𝑊 𝑗𝑘 . □ Remark 6.14. If there is only a single nonempty𝑊𝑘 , and𝑈𝛼 = 𝑊𝑘 , Lemma 6.12 shows that we are in the situation of the 2-sphere described at the beginning of this section, where 𝑘 = 2. Consider now 𝐷𝑘 , the subset of multiplicity at most 𝑘 . Clearly, ⋃𝑘 𝑗=1𝑊 𝑗 ⊆ 𝐸𝑘 = Int𝐷𝑘 . If it were true that for all 𝑘 one could find some 𝑚 > 𝑘 with 𝐷𝑘 ⊆ Int𝐷𝑚, then one could deduce from Lemmas 6.8 and 6.9 that 𝑈𝛼 = Int (⋃ 𝑘𝑊 𝑘 ) ∪ 𝑛−1 𝛼 (∞). For lack of such a guarantee (at present), we name the following subsets where the multiplicities may be troublesome. We shall eventually show that these subsets are empty. Definition 6.3. Consider the following subsets of𝑈𝛼: 𝐵𝑘 := { 𝑥 ∈ 𝐷𝑘 \ 𝐸𝑘 : 𝑥 ∉ 𝐸𝑚 for all 𝑚 > 𝑘 }, 𝐵(𝛼) := ⋃ 𝑘⩾1 𝐵𝑘 , 𝐶𝑘 := 𝐸𝑘 \ 𝑘⊎ 𝑗=1 𝑊 𝑗 . The set 𝐵𝑘 consists of (some) boundary points of 𝐷𝑘 , while 𝐶𝑘 collects interior points 𝑥 of 𝐷𝑘 , if any, that lie on the boundary of some 𝑊 𝑗 with 𝑗 ⩽ 𝑘 . The points 𝐶𝑘 will be branch points of the “branched manifold” ⋃ 𝑘⩾1 𝐸𝑘 ⊆ 𝑈𝛼. Lemma 6.15. If the multiplicity is bounded on 𝑈𝛼, then 𝑛−1 𝛼 (∞) = 𝐵(𝛼) = ∅. If the multiplicity is unbounded, then 𝑥 ∈ 𝐷𝑘 \ 𝐸𝑘 lies in 𝐵𝑘 if and only if 𝑥 = lim𝑛→∞ 𝑥𝑛 for some sequence {𝑥𝑛} satisfying 𝑛𝛼 (𝑥𝑛) → ∞. Proof. The first statement is obvious: if the multiplicity is bounded, by 𝑚 say, then 𝐷𝑚 = 𝑈𝛼 and every 𝐷𝑘 is included in 𝐷𝑚 = 𝐸𝑚