Serre-Swan Theorem for Non Commutative C -Algebras 1

Home , Projective module

Serre-Swan Theorem for non commutative C∗-algebras Katsunori Kawamura Research Institute for Mathematical Sciences Kyoto University, Kyoto 606-8502, Japan

For a Hilbert C∗-module X over a C∗-algebra , we construct a A vector bundle X associated with X. We represent X as a vector space E ΓX of some class of holomorphic sections on X with the following properties: E

(i) ΓX is a Hilbert -module and ΓX = X, A ∼ (ii) X has a ﬂat connection which deﬁnes the action of on ΓX , E A (iii) X has a hermitian metric H which induces the C∗-inner product E of ΓX . This section representation of modules is a kind of generalization of the

Serre-Swan theorem to non commutative C∗-algebras. We show that X is isomorphic to an associated bundle of an inﬁnite dimensional E Hopf bundle with the structure group U(1).

Keywords: non commutative geometry, Serre-Swan theorem, Hilbert C∗-module 1991 MSC: 46L87

1 Introduction

The Serre-Swan theorem is described as follows:

Theorem 1.1 (Serre-Swan) Let Ω be a connected compact Hausdorff space and C(Ω) the algebra of all complex valued continuous functions on Ω. As- sume X is a module over C(Ω). Then X is finitely generated projective iff there is a complex vector bundle E on Ω such that X is isomorphic onto the module of all continuous sections of E.

Proof. See [9].

1 By this theorem, finitely generated projective modules over C(Ω) and complex vector bundles on Ω are in one-to-one correspondence up to isomorphisms. In non commutative geometry [5, 13], some class of module over non commutative C∗-algebra is treated as a vector bundle of “non commutative space” by generalizingA Serre-Swan theorem for commutative C - A ∗ algebra, for example, 2 2 matrix algebra M2(C) and its free (right)module m × M2(C) M2(C) M2(C) of rank m. Specially, it is believed that a finitely≡ generated⊕ projective · · · ⊕ module is regarded as a non commutative vector bundle because the condition of modules is used in Theorem 1.1. On the other hand, for a unital general non commutative C∗-algebra , there is a uniform Kählerbundle ( , p, B) [3] unique up to equivalence classA P of , such that is ∗ isomorphic onto the uniform Kählerfunction algebra withA -productA on ( , p, B) which is a natural generalization of Gel’fand representation.∗ P Example 1.1 We show the example of a functional representation of the simplest, nontrivial and non commutative C∗-algebra = M2(C). The A 1 uniform Kählerbundle ( , p, B) of M2(C) is given as follows: Let = CP , B = b , where B is a one-pointP set and p is the trivial surjectionP from { } P to B. The functional representation of M2(C), 1 f : M2(C) C∞(CP ) → t 2 1 is defined by fA([x]) xAx/¯ x for A M2(C) and [x] CP where -product on C (CP≡1) is definedk k by ∈ ∈ ∗ ∞ 1 l m l m + √ 1Xml l, m C∞(CP ) . ∗ ≡ · − ∈ Non-commutativity of this -product comes from the second term of the right hand side on the above∗ equation. We review the uniform Kählerbundle and the functional representation of non commutative C∗-algebras in section 2 intimately. Under the above consideration, we state the following theorem which is a version of the Serre- Swan theorem generalized to non commutative C∗-algebras under weaker condition.

Theorem 1.2 ((Main theorem) Serre-Swan theorem for non commutative C -algebras) Let X be a Hilbert C -module over a unital C -algebra , ∗ ∗ ∗ A ( , p, B) the uniform Kählerbundle of , u( ) the C -algebra of uni- P A K P ∗ form Kählerfunctions on and f : ∼= u( ) the Gel’fand representation of . P A K P A 2 (i) (Construction of vector bundle) There is a complex vector bundle X E on such that X has a flat connection D and a hermitian metric H. P E Furthermore there is a bundle map PX from a trivial bundle X × P on to X which image is dense in X at each fiber. P E E (ii) ( -action and hermitian metric) Let ΓX PX (Γconst(X )) ∗ ≡ ∗ × P ⊂ Γhol( X ) where Γconst(X ) is the set of all constant sections on E × P X and Γhol( X ) is the set of all holomorphic sections of X . Then ×P E E ΓX is a Hilbert u( )-module with right -action K P ∗

ΓX u( ) ΓX , × K P →

(s, l) s l s l + √ 1DX s ((s, l) ΓX u( )) 7→ ∗ ≡ · − l ∈ × K P

and a C∗-inner product

H ΓX ΓX :ΓX ΓX u( ) | × × → K P

where Xl is the holomorphic part of the complex Hamiltonian vector fields of l u( ) C ( ) with respect to the Kählerform of . ∈ K P ⊂ ∞ P P (iii) (Reconstruction) Under an identification f : ∼= u( ), ΓX is isomorphic to X as a Hilbert -module. A K P A Remark 1.1 In spite that the finitely generated projective property (=FP property) is ingredient of modules in the origianl Serre-Swan theorem (The- orem 1.1), reader may note that there is no condition about FP property of Hibert C∗-module in Theorem 1.2. We wish to explain for this natural question here. In the field of non commutative geometry [4, 5, 13], Serre-Swan theorem has been applied as a kind of guiding principle to make theory and give correspondence between modules of non commutative algebra and virtual vector bundles. In this context, the rigid mathematical statement is stretched as a guiding principle of “non commutative-commutative correspondence” along with quantum-classical correspondence in quantum mechanics. In analogy with the case of quantum mechanics, there are many ambiguities and unor- ganized interpretations about non commutative geometry and there is no ex- act theory which unifies them because the meaning of this correspondence is not clear and principle itself is not mathematics. Despite of these ambiguity, the non commutative geometry gives us strong interest and desire to study the virtual geometry associated with non commutative algebra. Therefore

3 we regard Serre-Swan theorem as a guiding principle of non commutative- commutative correspondence in the ﬁeld of non commutative geometry in this paper. We neglect the FP property here because we have no idea to treat the notion of FP property in our theory suitably. In this reason, we decide to use a word “Serre-Swan” in our paper in order to stress the sig- niﬁcance of the role of guiding principle in non commutative geometry. FP property may be a problem which may be studied in future. Inversely, we can regard the Hilbert C∗-structure is more essential than FP property according to our theorem. Here we would mention that there is a kind of FP property of Hilbert C∗-module by Kasparov stability theorem [10].

In subsection 3.2, we introduce X in Theorem 1.2. X is called the E E atomic bundle of a Hilbert C∗-module X which is a Hilbert bundle on the uniform Kählerbundle. We show its geometrical structure in subsection m 3 2m 3.4 (e.g. = M2(C) and X = M2(C) , then X ∼= S U(1) C which is an associatedA vector bundle of a Hopf bundleE (S3, µ,×CP 1,U(1))). In subsection 4.1, we give a flat connection D in Theorem 1.2. D is called the atomic connection of the atomic bundle. In subsection 4.2, by using a connection of the vector bundle, we define a -action of a function algebra on the vector space of holomorphic sections on∗ a vector bundle under more general situation than the case of Hilbert C∗-modules. In section 5, we give a proof of Theorem 1.2. Here we summarize correspondences between geometry and algebra.

Gel’fand representation Serre-Swan theorem

vector module space algebra bundle C(Ω) Γ(E) CG Ω point wise CG E Ω point wise product → action NCG B u( ) P → K P NCG X ΓX -product E → P ∗ -action ∗ where we call respectively, CG = commutative geometry as a geometry associated with commutative C∗-algebras, and NCG = non commutative geometry as a geometry associated with non commutative C∗-algebras by following A.Connes [4].

4 2 C∗-geometry

In a naive sense of correspondence between algebraic objects and geometric ones defined by algebraic equations, we propose a version of the algebraic geometry for non commutative C∗-algebras. Or, it may be considered as a generalized Gel’fand representation for non commutative C∗-algebraic objects ( = the automorphism group, modules, etc.) [3]. As a preparation, we review the basic for an assurance that such correspondence exits by the following [3]. What do we want to do ? It is often stated that usual geometry is commutative geometry in comparison to non commutative geometry ([4, 5, 13]). Commutative geometry means a topological, or geometrical space consisting points or its function algebra. On the other hand, non commutative geometry means the algebra which is non commutative and not function algebra in general, and it is considered as a virtual function space on a virtual topological, or virtual geometrical space. But there exist several kind of non-commutativity outside the region of genuine non commutative geometry, too. For example, the fundamental group of topological spaces, transformation group, the Lie algebra of vector fields of a differential manifold are generally non commutative. Furthermore, for a symplectic manifold M, the set C∞(M) of all smooth functions on M has the Poisson bracket determined by the symplectic form of M [6]. It is nothing but which is non commutative Lie bracket. In this sense, the function algebra is not always commutative. As a special case, for a Kählermanifold, the smooth function algebra has Kählerbracket from its Kählerform [11] which is the special case of symplectic form. For example, a complex projective space CP n is a Kählermanifold with a metric called Fubini-Study type. Cirelli, Maniàand Pizzocchero realize non-commutativity of C∗-algebra by Kählerbracket of projective space. We are trying to explain the non-commutativity appearing in non commutative geometry by that inherent in commutative geometry, such as the automorphism group, module, subalgebra, Atiyah-Singer index, K-group and so on in the theory of C∗-algebras.

2.1 Uniform K¨ahlerbundle We start from the geometric characterization of the set of all pure states and the spectrum of a C∗-algebra [3]. Assume now that E and M are topological spaces.

5 Definition 2.1 (E, µ, M) is called a uniform Kählerbundle if it satisfies the following conditions:

(i) µ is an open, continuous surjection between topological spaces E and M,

(ii) the topology of E is a uniform topology,

1 (iii) each fiber Em µ (m) is a Kählermanifold. ≡ − Remark 2.1 This definition of uniform Kählerbundle is weaker than the original one [3]. We do not use the condition of equivalence of Kählertopol- ogy and uniform topology in this paper.

The local triviality of uniform Kählerbundle is not assumed. M is always neither compact nor Hausdorff. We simply denote (E, µ, M) by E. For a uniform topology, see [2]. Any metric space is a uniform space. Examples and relations with C∗-algebra are given later. Roughly speaking, the fiber of uniform Kählerbundle is related to non-commutativity of C∗-algebra.

Deﬁnition 2.2 Two uniform K¨ahlerbundles (E, µ, M), (E0 , µ0 ,M 0 ) are isomorphic if there is a pair (β, φ) of homeomorphisms β : E E0 and → φ : M M 0 , such that µ0 β = φ µ → ◦ ◦ β 0 E ∼= E µ µ0 ↓ ↓ 0 M ∼= M φ

1 1 and any restriction β 1 : µ (m) (µ0 ) (φ(m)) is a holomorphic µ− (m) − − K¨ahlerisometry for any| m M. We call→(β, φ) a uniform K¨ahlerisomor- ∈ phism between (E, µ, M) and (E0 , µ0 M 0 ).

Example 2.1 (i) Any Kählermanifold N is a uniform Kählerbundle with a one-point set as the base space. In the same way, the direct n sum of Kählermanifolds Ni i=1 as a metric space is a uniform Kähler bundle with a n-point set{ as} the base space endowed with discrete topology.

6 (ii) Any compact Hausdorff space X is a uniform space. X is a uniform Kählerbundle with 0-dimensional fiber which itself as the base space [2]. We explain the nontrivial third example of uniform Kählerbundles as follows. Let be a unital C -algebra. Denote A ∗ : the set of all pure states on , and BP : the set of all equivalence classesA of irreducible representations of . A B is called the spectrum of . The weak∗ topology on is a uniform topology. By the GNS representationA of , there is a naturalP projection from onto B : A P p : B. P → If is commutative, then ∼= B ∼= “the set of all maximal ideals of ” as a compactA Hausdorff space.P We consider ( , p, B) as a map of topologicalA P spaces where is endowed with weak∗ topology and B is endowed with the Jacobson topologyP [12]. In Ref.[3], the following results are proved.

Theorem 2.1 (Reduced atomic realization) For any unital C∗-algebra , ( , p, B) is a uniform Kählerbundle. A P 1 For a fiber b p (b), let ( b, πb) be an irreducible representation P ≡ − H belonging to b B. ρ b corresponds [xρ] ( b) ( b 0 )/C ∈ ∈ P ∈ P H ≡ H \{ } × where ρ = ωx πb with ωx denoting a vector state ωx =< xρ ( )xρ >. ρ ◦ ρ ρ | · Then b has a Kählermanifold structure induced by this correspondence P from the projective Hilbert space ( b). We denote this correspondence by τ b: P H b b τ : b ( b); τ (ρ) [xρ]. (2.1) P → P H ≡ The Kählerdistance db of a fiber b is given by P db(ρ, ρ0 ) √2arcos < xρ x > ( ρ, ρ0 b) ≡ | | ρ0 | ∈ P which is the length of shortest geodesic between ρ and ρ0 in b. P Theorem 2.2 Let i be C -algebras with associated uniform Kählerbun- A ∗ dles ( i, pi,Bi), i = 1, 2. Then 1 and 2 are isomorphic if and only if P A A ∗ ( 1, p1,B1) and ( 2, p2,B2) are isomorphic as a uniform Kählerbundle. P P By this theorem ( , p, B) associated with is uniquely determined up to a uniform Kählerisomorphism.P From now,A we call ( , p, B) in Theorem 2.1 the uniform Kählerbundle associated with C -algebraP . ∗ A 7 2.2 A functional representation of non commutative C∗-algebras We reconstruct from the uniform Kählerbundle ( , p, B) associated with A1 P . Since b p (b) is a Kählermanifold for each b B, consider the A P ≡ − ⊂ P ∈ fiberwise smooth (= smooth in b for each b B) function. Let P ∈

C∞( ) : the set of all ﬁberwise smooth complex valued functions on . P P Deﬁne a product on C ( ) denoted by l m for l, m C ( ) by ∗ ∞ P ∗ ∈ ∞ P

l m l m + √ 1Xml (2.2) ∗ ≡ · − Define an involution on C ( ) by complex conjugate. Then (C ( ), ) ∗ ∞ P ∞ P ∗ becomes a ∗ algebra with unit which is not associative in general, where Xl is the holomorphic Hamiltonian vector field of l defined by an equation ¯ ωρ (Xl)ρ, Y¯ρ = ∂ρl( Y¯ρ )( Y¯ρ Tρ ) (2.3) ∈ P for each ρ , Kählerform ω on which is defined each fiber, ∂¯ is ∈ P P the anti-holomorphic differential operator on C∞( ), and Tρ is the anti- holomorphic tangent space of at ρ . By usingP Eq.2.3, Eq.2.2P can be written as follows: P ∈ P

l m = l m + √ 1 ω(X¯l,Xm). ∗ · − If , is the K¨ahlerbracket with respect to ω, then the following equality holds:{· ·} l m m l = √ 1 l, m ( l, m C∞( )). (2.4) ∗ − ∗ − { } ∈ P

Theorem 2.3 (Gel’fand representation of non commutative C∗-algebras) For a non commutative C -algebra , the Gel’fand representation ∗ A

fA(ρ) ρ(A), (A , ρ ) ≡ ∈ A ∈ P gives an injective ∗ homomorphism of unital ∗ algebras :

f : C ( ); A fA A → ∞ P 7→ where C∞( ) is endowed with -product deﬁned by (2.2). For a function l in the imageP f( ) of a map f, ∗ A 1 l sup ¯l l (ρ) 2 (2.5) k k ≡ ρ ∗ ∈P

8 defines a C∗-norm of ∗ algebra f( ). By this norm, an associative ∗ subalgebra (f( ), ) is a C -algebra whichA is isomorphic onto . A ∗ ∗ ∗ A Furthermore f( ) is equal to a subset u( )( C ( )) defined by A K P ⊂ ∞ P D2l = 0, D¯ 2l = 0, u( ) (l C∞( ): ) , K P ≡ ∈ P ¯l l, l ¯l, l are uniformly continuous on ∗ ∗ P (2.6) where D, D¯ are the holomorphic and anti-holomorphic part, respectively, of covariant derivative of Kählermetric defined on each fiber of . Hence, the P following equivalence of C∗-algebras holds:

= u( ). A ∼ K P

We call these objects C∗-geometry since any C∗-algebra can be recon- structed from the associated uniform K¨ahlerbundle [3] and, therefore, any C∗-algebra is determined by such a geometry.

Remark 2.2 In Theorem 2.3, the function fA of the image of a functional representation on A is not holomorphic. It is a complex quadratic form ∈ A on a Hilbert projective space like Example 1.1. Furthermore, fA satisfies not only continuous on across fiber but also more strong uniform continuity like the definition of u( ). K P Remark 2.3 (Non-commutativity and differential structure) It is not known yet whether the function space of uniform Kählerfunctions on a general uniform Kählerbundle satisfying conditions in Definition 2.1 becomes a C∗- algebra. In this sense, the class of uniform Kählerbundles may be larger than the class of C∗-algebras.

Remark 2.4 In [4], non commutative generalization of differential geometry is claimed by using the theory of C∗-algebras. A non commutative C∗-algebra is treated as a function algebra C∞(M) with respect to a virtual smooth manifold M. On the other hand, the smooth structure of the uniform Kählerbundle of a C∗-algebra appears only when is not commutative since statements above TheoremA 2.1. FurthermoreA it is known that a commutative C∗-algebra has no continuous derivation except 0. Un- der these considerations, our opinion is that the differential structure of a geometry makes a C∗-algebra non commutative. In commutative case, the differential structure do not arise from algebraic structure of a C∗-algebra.

9 Hence the geometry of non commutative C∗-algebras is necessarily diﬀer- ential geometry with the w∗ topology of the space of states, and that of commutative C∗-algebras is only (compact Hausdorﬀ)topology. By the above results, we obtain a fundamental correspondence between algebra and geometry as follows:

unital commutative C -algebra compact Hausdorﬀ space ∗ ⇔

TT unital generally non commutative uniform K¨ahlerbundle. ⇔ C∗-algebra associated with a C∗-algebra

The upper correspondence above is just the Gel’fand representation of unital commutative C∗-algebra. Since Remark 2.3, we must restrict the class of uniform K¨ahlerbundles to like the above.

Example 2.2 Assume that is a separable inﬁnite dimensional Hilbert space. H (i) When ( ) which is the algebra of all bounded linear operators A ≡ L H [0,1] on , the uniform K¨aherbundle of is ( ( ) , p, 2 b0 ) whereH ( ) is the projective HilbertA spaceP H of∪ P,− is the∪ union { } of projectiveP H Hilbert spaces of continuous dimensionalH P− Hilbert space indexed by 2[0,1], 2[0,1] is the power set of closed interval [0, 1] and b0 is the one-point set corresponding to the equivalence class of { } identity representation ( , id ( )) of ( ) on . Since the primitive H L H L H H [0,1] spectrum of ( ) is a two-point set, the topology of 2 b0 is L[0H,1] [0,1] ∪ { } equal to , 2 , b0 , 2 b0 [8]. In this way, the base space of the UKB{ ∅ is not always{ } a one-point∪ { }} set when an algebra is type I.

(ii) For a C∗-algebra generated by the Weyl form of 1-dimensional canonical commutationA relation

√ 1st U(s)V (t) = e − V (t)U(s)(s, t R), ∈ its uniform Kählerbundle is ( ( ), p, 1pt ). The spectrum is a one- point set 1pt since von NeumannP H uniqueness{ } theorem [1]. { } 10 (iii) CAR-algebra is a UHF algebra with the nest M2n (C) n N. The uniform KählerbundleA has the base space 2N and{ each fiber} ∈ on 2N is a separable infinite dimensional projective Hilbert space where 2N is the power set of the set N of all natural numbers with trivial topology, that is, the topology of 2N is just , 2N . In general, the Jacobson {∅ } topology of the spectrum of a simple C∗-algebra is trivial [8].

3 The atomic bundle of a Hilbert C∗-module

The aim of this section is the construction of a vector bundle by a given Hilbert C∗-module for a C∗-algebra.

3.1 Hopf bundles, Hilbert modules, deformation and canonical quantization

We start from a naive geometric example related to some Hilbert C∗-module in this section. In the text book of fiber bundles, a Hopf bundle is one of typical examples of U(1)-principal bundle where the total space is 3-sphere, the base space 1-dimensional complex projective space ( = 2-sphere) and the typical fiber 3 ∼ 1 U(1)(∼= circle). The Hopf bundle (S , µ, CP ) is defined as follows: Definition 3.1 (Hopf bundle) (S3, µ, CP 1) is called the Hopf bundle if

3 2 2 2 S (z1, z2) C : z1 + z2 = 1 , ≡ ∈ | | | | CP 1 (C2 0 )/C = S3/U(1), ≡ \{ } × µ : S3 CP 1, →

µ(z1, z2) [(z1, z2)] ≡ iθ iθ 3 = e (z1, z2): e U(1) ((z1, z2) S ). n ∈ o ∈ The base space CP 1 of a Hopf bundle is the total space of the uniform K¨ahler bundle of C∗-algebra M2(C) by Theorem 2.1. Some examples of vector 1 3 m bundles of CP can be obtained from associated vector bundles S U(1) C of the Hopf bundle deﬁned by the space of all U(1)-orbits in a direct× product space S3 Cm associated to a U(1)-action on Cm. Such vector bundle has ×

11 the structure group U(1). By Serre-Swan theorem for vector bundles of a 1 smooth compact manifold, a finitely generated projective C∞(CP )-module X corresponds to a vector bundle E of CP 1 unique up to isomorphisms [9, 1 13]. In this situation, C∞(CP )-action on Γ (E) is an action by pointwise product ∞ 1 Γ (E) C∞(CP ) (s, l) s l Γ (E), ∞ × 3 7−→ · ∈ ∞ (s l)(x) s(x)l(x)(x CP 1). · ≡ ∈ Let 1 1 1 , : C∞(CP ) C∞(CP ) C∞(CP ) {· ·} × → 1 1 be the Kählerbracket of CP . To define an action of a Lie algebra (C∞(CP ), , ), we define a -action by deforming the above action by a connection {·D of·} E (deformation∗ of action):

s l = s l s l + √ 1DX s (3.7) · ⇒ ∗ ≡ · − l 1 where Xl is deﬁned in (2.3). Rewrite a right action of C∞(CP )

Ml :Γ (E) Γ (E); sMl s l. ∞ → ∞ ≡ ∗ Then we obtain a deformed homomorphism of Lie algebras by curvature.

1 Lemma 3.1 For s Γ (E) and l, m C∞(CP ), we have the following equation: ∈ ∞ ∈

(i) s[Ml,Mm] = √ 1sM l,m + RXl,Xms − { } where R is the curvature of D.

(ii) If M is associated with a ﬂat connection, then

[Ml,Mm] = √ 1M l,m (3.8) − { } on Γ (E). That is, M is a rescaled homomorphism between Lie alge- ∞ 1 bras C∞(CP ) and End(Γ (E)) with scaled factor √ 1. ∞ − Proof. See Lemma 4.2.

By a ﬂat connection D of E, we obtain a right action M of a Lie algebra 1 (C∞(CP ), , ) on Γ (E) rescaled by √ 1. Deﬁne a Lie algebra {· ·} ∞ − 1 1 h(C∞(CP )) Ml End(Γ (E)) : l C∞(CP ) . ≡ n ∈ ∞ ∈ o

12 1 Then h(C∞(CP )) is just the canonical quantization of a classical algebra 1 (C∞(CP ), , ) as operators on the space of sections by the quantum- classical correspondence{· ·} principle. 1 In this way, we obtain an operator Lie algebra h(C∞(CP )) on Γ (E). 1 ∞ By using Example 1.1 and Eq.2.4, M2(C) is embedded in C∞(CP ). Hence we obtain an injective homomorphism M f from a Lie algebra gl2(C) = 1 ◦ M2(C) to h(C∞(CP )) which is the following southeast arrow by composed other two arrows M and f:

f 1 gl2(C) C (CP ) → ∞

M f M ◦ & ↓ 1 h(C∞(CP )).

In fact, for A, B gl2(C), ∈

[(M f)(A), (M f)(B) ] = [Mf ,Mf ] ◦ ◦ A B = √ 1M fA,fB = M− { } √ 1 fA,fB − { } = MfA fB fB fA ∗ − ∗ = MfAB fBA − = MfAB BA = (M −f)([A, B]). ◦

We obtain an action of gl2(C) on Γ (E) by a flat connection D. We finish ∞ to explain a naive example. Note that there is no associativity of M2(C) in general for E in this example. In the general case, we construct a vector bundle E associated with a Hilbert C -module, and a isomorphism φ from a general C -algebra ∗ ∗ ∗ A with a uniform Kählerbundle ( , p, B) into an associative algebra E P ∗ B ≡ Alg < h(C∞( )) > and an isomorphism Ψ from Hilbert C∗-module X into Γ∗ (E) so as forP the following diagram to be commutative: ∞ X X × A →

Ψ φ Ψ × ↓ ↓

Γ (E) E Γ (E). ∞ × B → ∞

13 By this case, we can always construct E which satisﬁes the associativity of . On the other hand, we deﬁne the atomic bundle as a kind of vector A bundle from a Hilbert C∗-module and show the bundle structure by using a Hopf bundle.

3.2 The construction of the atomic bundle

Before starting to construct the atomic bundle of a Hilbert C∗-module, we state the deﬁnition of a Hilbert C∗-module.

Deﬁnition 3.2 ([7]) X is a Hilbert C∗-module over a C∗-algebra if X is a right -module and there is an valued-sesquilinear form A A A < >: X X ·|· × → A which satisﬁes the following conditions:

1/2 ξ X < ξ ξ > ( ξ X ). (3.9) k k ≡ k | k ∈

Let X be a Hilbert C∗-module over a unital C∗-algebra and ( , p, B) the uniform Kählerbundle associated with defined in TheoremA 2.1.P Defining A a closed subspace Nρ of X with ρ by ∈ P 2 Nρ ξ X : ρ( ξ ) = 0 (3.10) ≡ n ∈ k k o ( note ξ 2 =< ξ ξ > ), we consider the quotient vector space k k | ∈ A o X/Nρ EX,ρ ≡ o equipped with a sesquilinear form < >ρ on defined by ·|· EX,ρ o o < >ρ: C, ·|· EX,ρ × EX,ρ → o < [ξ]ρ [η]ρ >ρ ρ(< ξ η >)([ξ]ρ, [η]ρ ) | ≡ | ∈ EX,ρ

14 where o [ξ]ρ ξ + Nρ ( ξ X ). (3.11) ≡ ∈ EX,ρ ∈ o Then < >ρ becomes an inner product on X,ρ. Let X,ρ be the completion o ·|· 1E/2 E of by the norm ρ (< >ρ) . We obtain a Hilbert space EX,ρ k · k ≡ ·|· ( X,ρ, < >ρ ) from a Hilbert C -module X for each pure state ρ . E ·|· ∗ ∈ P Definition 3.3 (Atomic bundle) An atomic bundle X = ( X , ΠX , ) of a E E P Hilbert C -module X is defined as a fiber bundle X on : ∗ E P , X [ X,ρ E ≡ ρ E ∈P where the projection map ΠX : X is defined by ΠX (x) = ρ for x b E b→ Pb ∈ X,ρ. For b B, a B-fiber = ( , Π , b) of X is defined by E ∈ EX EX X P b X,ρ, EX ≡ [ E ρ b ∈P b b b ΠX : X b;ΠX ΠX b . E → P ≡ |EX The name of the atomic bundle comes from the (reduced)atomic representation of a C∗-algebra. The atomic bundle is a collection of its B-fibers: = b . X [ X E b B E ∈ In this way, X has a two-step fibration ( X , ΠX , ) and ( X , p ΠX ,B): E E P E ◦ ΠX X E → P

p ΠX p ◦ & ↓ B

1 b 1 where their fibers are X,ρ = ΠX− (ρ) and X = (p ΠX )− (b), respectively for ρ and b B. E E ◦ ∈ P ∈ 3.3 Unitary group action on the atomic bundle The aim of this subsection is to give the canonical definition of the typical fiber of the atomic bundle in order to state the structure theorem in subsection 3.4.

15 Let G be the group of all unitary elements in . Deﬁne an action χ of G on by A P χu(ρ) ρ Adu∗ (u G, ρ ). ≡ ◦ ∈ ∈ P We note that any pure state is moved to a pure state by any inner automorphism of . Hence χu maps b to b for each b B and u G. A P P ∈ ∈ Lemma 3.2 G acts on b by transitively. P Proof. Because the GNS representation of any element of b is irreducible, P any two unit vectors in b are transformed by πb(u) for some u G. Assume H ∈ that ρ1, ρ2 b. Then there are h, h0 b, h = 1 = h0 such that ρ1 =< ∈ P ∈ H k k k k h πb( )h > and ρ2 =< h0 πb( )h0 >, there is u G such that πb(u)h = h0 and| · | · ∈

χu(ρ1) = ρ1 Adu∗ =< h πb(u∗)πb( )πb(u)h >=< h0 πb( )h0 >= ρ2. ◦ | · | ·

Hence the action χ of G is transitive on b. P Next, define an action tb of G on o by EX b o t ([ξ]ρ) [ ξu∗ ] u G, [ξ]ρ . u ≡ χu(ρ) ∈ ∈ EX,ρ b t is well defined since a map ξ ξu maps Nρ to N . In fact, 7→ ∗ χu(ρ) 2 2 2 χu(ρ) ξu∗ = ρ (Adu∗) u ξ u∗ = ρ ξ . k k k k k k Then tb is a unitary map from o to o .Hence we can extend tb as a u EX,ρ EX,χu(ρ) u unitary map from X,ρ to . We note that E EX,χu(ρ) tb (x) =ct ¯ b (x)( u G, c U(1) ). (3.12) cu u ∈ ∈ b We define an action t of G on X by t b t , b B. Then T (t, χ) is an E |EX ≡ ∈ ≡ action of G on ( X , ΠX , ), that is, the following diagram is commutative for each u G: E P ∈ tu X = X E ∼ E

Π Π X ↓ ↓ X

χu = . P ∼ P

16 In fact,

(ΠX tu)([ξ]ρ) = ΠX ([ξu∗] ) = χu(ρ) = (χu ΠX )([ξ]ρ). ◦ χu(ρ) ◦ o Hence ΠX tu = χu ΠX holds on . By continuity, it holds on the whole ◦ ◦ EX,ρ X,ρ for each ρ . E ∈ P b b This action preserves B-fiber ( X , ΠX , b), b B, too. b b E P ∈ For two fibrations ( X , ΠX , b) and (S( b), µb, b), define a fiber prod- b,U(1) b E P H P uct S( b) of them by EX ⊂ EX × H b,U(1) b X X b S( b) E ≡ E ×P H b b = (x, h) S( b):Π (x) = µb(h) . n ∈ EX × H X o

For a Hopf bundle (S( b), µb, b), see a sentence after Theorem 2.1 and Appendix A. H P Deﬁne an action σb of G on b,U(1) by EX b b,U(1) σ (x, h) (tu(x), πb(u)h) (x, h) , u G . u ≡ ∈ EX ∈ In fact

b b Π (tu(x)) = χu(Π (x)) = χu(ρ) = ρ Adu∗ = µb(πuh) X X ◦ b,U(1) b b b,U(1) when (x, h) X and ΠX (x) = ρ = µb(h). Therefore σu(x, h) X . b ∈ E ∈ E Hence σu is well deﬁned. We note that a representation ( b, πb) of induces an action of G on H A S( b). H b,U(1) b Lemma 3.3 For (x, h) X and u G, if σu(x, h) = (y, h), then x = y. ∈ E ∈

o Proof. It is suﬃcient to show for the case x X,ρ. Assume x = [ξ]ρ. By assumption, ∈ E b (y, h) = σu(x, h) = (tu(x), πb(u)h)

= ( [ξu∗]χu(ρ), πb(u)h ).

Hence h = πb(u)h. Equivalently we have

πb(u∗)h = h. (3.13)

17 By deﬁnition of ﬁber product,

χ (ρ) = Πb [ξu ] u X ∗ χu(ρ) = µb (πb(u)h) = µb(h) = ρ.

Therefore we have χu(ρ) = ρ and y = [ξu∗]ρ. By using the above results, we obtain the following equation:

2 2 x y ρ = ρ( ξ ξu∗ ) k − k k −2 k 2 = ρ( ξ ) + ρ( ξu∗ ) ρ(< ξ ξu∗ >) ρ(< ξu∗ ξ >) k k2 k k2 − | − | = ρ( ξ ) + ρ(u ξ u∗) ρ(< ξ ξ > u∗) ρ(u < ξ ξ >) k k2 k k 2 − | − | = ρ( ξ ) + χu(ρ)( ξ ) k k k k < h πb(< ξ ξ >)πb(u∗)h > < πb(u∗)h πb(< ξ ξ >)h > =− 2ρ( ξ |2) ρ(<| ξ ξ >) ρ(<− ξ ξ >) ( by| ( 3.13)| ) = 0. k k − | − |

We note that ρ =< h πb( )h > here. Hence we obtain x = y. | ·

Deﬁnition 3.4 F b is the set of all orbits of G in b,U(1). X EX Let (x, h) F b be an orbit containing (x, h) b,U(1). Since G acts on O ∈ X ∈ EX S( b) transitively, H (x, h) = σb (x, h): u G O { u ∈ } = (tu(x), πb(u)h): u G { ∈ } b,U(1) b for each (x, h) X . Hence FX is a family of spheres which is home- ∈ E b,U(1) omorphic to S( b) in . By Lemma 3.3, any element of (x, h) is H EX O written as (y , h0 ) where y is an element of b determined by h0 S( ) h0 h0 X b uniquely. E ∈ H

Lemma 3.4 For (y, h0 ) in (x, h), if y = x = 0, then h = h0 . O 6 0 b 0 Proof. By choice of (x, h ), there is u G such that σu(x, h) = (x, h ). b 0 ∈ tu(x) = x and πb(u)h = h . Since

0 b µb(h ) = ΠX (x) = µb(h),

18 there is c U(1) such that h0 = ch. Hence we can choose u = cI. If x = 0, then we have∈ 6 b b b x = tu(x) = tcI (x) =ct ¯ I (x) =cx ¯ by (3.12). Therefore c = 1 and we obtain h = h0 when x = 0. 6

Corollary 3.1 For c U(1), ∈ (x, ch) = (cx, h). O O Proof. If (y, h) (x, ch), then we can take u G as u =cI ¯ . Then ∈ O ∈

y = tu(x) = cx¯ = cx.

b Furthermore (0, h) = (0, h0 ): h0 S( b) because t is unitary for O { ∈ H } u each u G. Let (y, h0 ) (x, h) S( ) . Then there is u G X,µb(h) b ∈ ∈ O ∩ E × H 1 ∈ such that (y, h0 ) = σu(x, h). By choice of (y, h0 ), h0 µ− (µb(h)). Hence ∈ b there is c U(1) such that h0 = ch. ∈ We note that X,ρ and are equivalent as a Hilbert space when E EX,ρ0 ρ, ρ0 b. ∈ P b Proposition 3.1 F is a Hilbert space which is isomorphic to X,ρ for each X E ρ b. ∈ P Proof. Fix h0 S( ) such that µb(h0) = ρ. Deﬁne a map ∈ H b R : X,ρ F ; R(x) (x, h0). E → X ≡ O

Then R is surjective. If R(x) = R(y) for x, y X,ρ, then (x, h0) = ∈ E O (y, h0). By Lemma 3.4, x = y when x = 0. If x = 0, then y = 0. Hence R O 6 b is bijective. We deﬁne a structure of Hilbert space of FX from X,ρ by R. Then we have the statement of the proposition. E

b We denote calculations in FX .

0 (x, h) + (y, h ) = R(tuh (x)) + R(tu (y)) O O h0

19 0 where σuh (x, h) = (tuh (x), h0) and σu (y, h ) = (tu (y), h0). For k C, h0 h0 ∈

k (x, h) = kR(tu (x)) ·O h = R(k tu (x)) · h = R(tuh (kx)) = (kx, h). O Specially, c U(1), ∈ c (x, h) = (x, ch) ·O O by Corollary 3.1.

3.4 Structure of the atomic bundle We show that the atomic bundle has a Hilbert bundle structure in order to b define sections on it in the next section. Let ( S( b) U(1) F , π b , ( b)) X FX H b× P H be the associated bundle of (S( b), µb, ( b)) by FX where a Hilbert space b H P H FX is defined in Definition 3.4. By definition of the associated bundle(Appendix b A.3), an element of S( b) U(1) FX is written as the U(1)-orbit [(h, (x, k))] H × b O which contains (h, (x, k)) S( b) F . O ∈ H × X b Lemma 3.5 Any element of S( b) F can be written as [(h, (x, h))] H ×U(1) X O where (x, h) F b . O ∈ X b Proof. Take an element [(h, (y, k))] S( b) U(1) FX . By definition of (y, k) and the transitivity ofO the action∈ of HG on×S( ), there is u G such O b H ∈ that h = uk and (tu(y), h) (y, k). Denote x tu(y) Then (x, h) = (y, k). Hence ∈ O ≡ O O [(h, (y, k))] = [(h, (x, h))]. O O

For h S( ) and x b , we denote ∈ H ∈ EX b [h, x] [(h, (x, h))] S( b) F ≡ O ∈ H ×U(1) X from here. Recall for each b B, b is a K¨ahlermanifold which is isomorphic to a ∈ P b projective Hilbert space ( b) by a map τ in (2.1). P H

20 b b Theorem 3.1 For each b B, the B-fiber ( X , ΠX , b) at b is a local trivial ∈ E Pb Hilbert bundle which is isomorphic to ( S( b) U(1) FX , πF b , ( b)): H × X P H b b b ( X , ΠX , b) = ( S( b) U(1) FX , πF b , ( b)) E P ∼ H × X P H b where FX is a complex Hilbert space defined in Definition 3.4 which is isomorphic to X,ρ for each ρ b. E ∈ P b b b Proof. Define a map Ψ : S( b) F by EX → H ×U(1) X b b Ψ (x) [hx, x](x ) ≡ ∈ EX 1 b where hx µb− (ΠX (x)). We show this definition is independent in the ∈ 1 b choice of hx. If h0 µ− (Π (x)), then there is c U(1) such that h0 = chx. ∈ b X ∈ We denote h = hx for simplicity. Then we have

(h0 , (x, h0 )) = (ch, (x, ch)) O O = (hγc¯, c (x, h)) = (h, (x,O h)) c¯ O · where γ is an action of U(1) deﬁned in Appendix A.3. Hence

[h, x] = [(h, (x, h)] = [(h0 , (x, h0 )] = [h0 x]. O O In this way, Ψb is well deﬁned. If Ψb(x) = Ψb(y) for x, y b , then [h, x] = ∈ EX [h0 , y]. Therefore there is c U(1) such that (h, (x, h))c = (h0 , (y, h0 )). ∈ O O By h0 =ch ¯ and Corollary 3.1,

(y, ch¯ ) = (y, h0 ) O =Oc ¯ (x, h) = O(x, ch¯ ). O b b b By Lemma 3.3, x = y. Hence Ψ is injective. By deﬁnition of FX ,Ψ is surjective. Ψb is a bijection. Furthermore the following diagram is commutative: Ψb b b = S( b) F EX ∼ H ×U(1) X

b π Π F b X ↓ ↓ X

τ b b = ( b) . P ∼ P H 21 b b b We obtain a set-theoretical isomorphism (Ψ , τ ) of fibrations between ( X , b b b E ΠX , b) and ( S( b) U(1) FX , πF b , b ) such that any restriction Ψ X,ρ P H × X P |E of Ψb at a fiber is unitary between and π 1 (ρ) for ρ . X,ρ X,ρ F−b b E E X ∈ P b b We define a Hilbert bundle structure of ( X , ΠX , b) from ( S( b) U(1) b b b E P H × FX , πF b , b ) by (Ψ , τ ). We have the statement. X P

We have constructed a Hilbert bundle from a Hilbert C∗-module by Definition 3.3 and Theorem 3.1. Let (X , t, ) be a rivial complex vector bundle on . Then there is a map × P P P PX : X X , (3.14) × P → E defined by PX (ξ, ρ) [ξ]ρ ((ξ, ρ) X ). ≡ ∈ × P The map PX has a dense image in X at each B-fiber. For ρ and E ∈ P x PX (X ) X,ρ, ∈ × P ∩ E 1 (PX )− (x) = Nρ where Nρ is a vector space defined in (3.10). The following diagram is commutative:

PX X X × P → E

t Π ↓ ↓ X id . P → P Hence (PX , id) is a bundle map from (X , t, ) to ( X , ΠX , ). × P P E P By Theorem 3.1 and Definition 3.3, the atomic bundle of a Hilbert C∗- module is a family of associated bundles of Hopf bundles indexed by spectrum B: = S( ) F b . X ∼ [ b U(1) X E b B H × ∈ Example 3.1 If is commutative, then Ω = B and each B-fiber is A ≡ P ∼ b b = b . EX → P ∼ { } Hence the smooth structure of B-fiber collapses at each b B. Furthermore, n n n ∈ n if X = = C(Ω) , then Nω = fi i=1 C(Ω) : fi(ω) = 0 A ≡ A⊕· · ·⊕A ∼ ∼n {{ } ∈ n } for each ω Ω. Since /Nω = C, X,ω = C . Hence X = Ω C . ∈ A ∼ E ∼ E ∼ × 22 n 1 Example 3.2 If Mn(C), then ∼= CP − ( = n 1 dimensional complex projectiveA space), ≡ B is a one-pointP set. Hence the atomic− bundle of a Hilbert C∗-module over Mn(C) is a locally trivial smooth Hilbert bundle n 1 on CP − . Furthermore the atomic bundle of a Hilbert C∗-module over a finite dimensional C∗-algebra is a family of locally trivial smooth Hilbert bundles on complex projective spaces.

4 Connection and -action ∗ In this section, we define a flat connection D on the atomic bundle and show a relation between the associativity of -action defined by D and the flatness of D. ∗

4.1 The atomic connection of the atomic bundle To define the -action of (C ( ), ) on the smooth sections of the atomic ∗ ∞ P ∗ bundle of a Hilbert C∗-module X, we define some connection D of X which is called the atomic connection. E Let X = ( X , ΠX , ) be the atomic bundle of a Hilbert C -module X E E P ∗ over a C -algebra defined in section 3. Let Γ( X ) be the set of all bounded ∗ A E sections of X , that is, Γ( X ) s : X is a right inverse of ΠX and satisfies E E 3 P → E s sup s(ρ) ρ < . (4.15) k k ≡ ρ k k ∞ ∈P By the following operations, Γ( X ) is a complex linear space: E

(s + s0 )(ρ) s(ρ) + s0 (ρ)( ρ , s, s0 Γ( X )), ≡ ∈ P ∈ E

(ks)(ρ) ks(ρ)( ρ , s Γ( X ), k C ). ≡ ∈ P ∈ E ∈

Furthermore Γ( X ) is isometric onto a Banach space ρ X,ρ of the direct E L ∈P E sum of X,ρ ρ . By Theorem 3.1, we can consider the differentiability of {E } ∈P s Γ( X ) at each B-fiber ∈ E b s b : b X |P P → E for each b B in the sense of Fréchet differential of Hilbert manifolds. ∈ Define Γ ( X ) the set of all B-fiberwise smooth sections in Γ( X ). ∞ E E

23 Let X( ) be the set of all B-fiberwise smooth vector fields of . H is a P P hermitian metric of X defined by E

0 0 Hρ(s, s ) Ds(ρ) s (ρ)E (4.16) ≡ ρ for ρ , s, s0 Γ ( X ) [11]. ∈ P ∈ ∞ E Deﬁnition 4.1 D is a connection of X if D is a bilinear map of complex vector spaces E D : X( ) Γ ( X ) Γ ( X ) P × ∞ E → ∞ E which is C ( )-linear with respect to X( ) and satisﬁes the Leibniz law ∞ P P with respect to Γ ( X ): ∞ E

DY (s l) = ∂Y l s + l DY s · · · for s Γ ( X ), l C∞( ) and Y X( ). ∈ ∞ E ∈ P ∈ P h For Y X( b), h S( b) and ρ h, we denote Y a tangent vector ∈ P ∈ H ∈ V ρ at ρ in a local coordinate h. We deﬁne a linear map H Ah : F b F b Y,ρ X → X by multiplying a number

h 1 < βh(ρ) Yρ > | 2 . −2 1 + βh(ρ) k k That is, for z βh(ρ) h, ≡ ∈ H 1 < z Y h > Ah e = | ρ e ( e F b ). (4.17) Y,ρ −2 1 + z 2 ∈ X k k For the space of sections of b , we show that EX Proposition 4.1

h h D ∂ h + A Y,ρ ≡ Yρ Y,ρ deﬁnes a ﬂat connection D of b . EX Proof. Appendix B.

24 Deﬁnition 4.2 We call the connection in Proposition 4.1 the atomic connection of the atomic bundle.

We note that the atomic connection of the atomic bundle X is independent E in a Hilbert C∗-module X. We prepare some equations for the main theorem. For ρ h, deﬁne a h ∈ V vector in Ωρ in b by H β (ρ) + h Ωh h . ρ 2 ≡ 1 + βh(ρ) p k k Assume that ρ = ωx πb for a x b. Then ◦ ∈ H 1 x x − < h x > Ωh = = | | |x. ρ < h x > < h x > < h x >

| | | h Hence [x] = [Ωρ ] and < h Ωh > > 0. | ρ Let s be a section in Γ( X ) such that for each ρ b , there is ξρ X E ∈ P ⊂ P ∈ s(ρ) = [ξρ]ρ X,ρ. Let z = βh(ρ) for h S( b) such that ρ h. ∈ E ∈ H ∈ V Lemma 4.1 The following equations hold:

h Ω πb(< ξ0 ξρ >)(z + h) D ρ0 E < e ψα,h(s(ρ)) >= | (4.18) | 1 + z 2 p k k for e = ([ξ0 ] , h) F b , O ρ0 ∈ X

ˆ h < z Y > ∂Y φh(ρ)(s(ρ)) = ∂Y ξρ + ξρ K | , h ! (4.19) O Y,ρ − 2(1 + z 2) k k ρ where an element Kh is deﬁned by Y,ρ ∈ A h πb(KY,ρ)(h + z) = Y (4.20) ˆ and [∂Y ξρ]ρ X,ρ is deﬁned by ∈ E ˆ D [η]ρ [∂Y ξρ]ρ E ρ(∂Y < η ξρ >) ρ ≡ | for [η]ρ X,ρ. ∈ E Proof. Appendix C.

25 4.2 The -action of a function algebra on sections of the atomic∗ bundle

By (2.2), the function space C∞( ) is a ∗ algebra with -product which is not generally associative. We defineP the -action of (C∗ ( ), ) on the ∗ ∞ P ∗ smooth sections of the atomic bundle of a Hilbert C∗-module by using the atomic connection D of X . And we characterize algebraic properties, com- E mutativity, associativity, of -action by D and the curvature of X with respect to D. ∗ E Let D be any connection of X . E Definition 4.3 We define the (right) -action of C∞( ) on Γ ( X ) by ∗ P ∞ E

s l s l + √ 1DX s ∗ ≡ · − l for l C∞( ) and s Γ ( X ) where Xl is the holomorphic Hamiltonian vector∈ field ofP l with respect∈ ∞ toE the Kählerform of . P Remark 4.1 (i) By this lemma, the non-commutativity of -action of ∗ C∞( ) on Γ ( X ) depends on the connection D of X and the Kähler formP of . ∞ E E P (ii) We use three different notions, -action, -product and involution by the same symbol “ ” when they∗ do not make∗ confusions. ∗ We show the geometric characterization of -action. ∗ Lemma 4.2 For each s Γ ( X ) and l, m C∞( ), the following equation holds: ∈ ∞ E ∈ P

(s l) m (s m) l = √ 1 l, m + [DX ,DX ] s, ∗ ∗ − ∗ ∗ − { } l m

s (l m) s (m l) = √ 1 l, m + D s. ∗ ∗ − ∗ ∗ − { } [Xl,Xm] Proof. According to Deﬁnition 4.3, we have

(s l) m (s m) l ∗ ∗ − ∗ ∗

26 = s l + √ 1DX s m s m + √ 1DX s l · − l ∗ − · − m ∗

= √ 1DX (s l) + √ 1 (DX s) m + √ 1DX DX s − m · − l · − m l √ 1DX (s m) √ 1 (DX s) l + √ 1DX DX s − − l · − − m · − l m

= √ 1s (Xml Xlm) DX DX s + DX DX s − · − − m l l m

= √ 1 l, m s + [DX ,DX ]s − { } l m since ∂l(Xm) ∂m(Xl) = ω(Xl, Xm) ω(Xm, Xl) = l, m . − − { } Similarly we see s (l m) s (m l) = s (l m m l) ∗ ∗ − ∗ ∗ = √∗ 1s∗ l,− m ∗ − ∗ { } = √ 1 l, m s + √ 1D s X l,m − { } − { } = √ 1 l, m s D [Xl,Xm]s − { } − − = √ 1 l, m + D s − { } [Xl,Xm] since X l,m = [Xl,Xm] { } − for l, m C ( ). ∈ ∞ P Let the associator a(l, m) of l, m C ( ) be an operator ∈ ∞ P a(l, m):Γ ( X ) Γ ( X ) ∞ E → ∞ E deﬁned by

a(l, m)s (s l) m s (l m)(s Γ ( X )). ≡ ∗ ∗ − ∗ ∗ ∈ ∞ E Then we have a relation between associativity and curvature.

Proposition 4.2 On Γ ( X ) and for l, m C∞( ), the following equation holds: ∞ E ∈ P a(l, m) a(m, l) = RX ,X , − l m where R is the curvature of X with respect to D deﬁned by E RY,Z [DY ,DZ ] D (Y,Z X( )). ≡ − [Y,Z] ∈ P

27 5 A sectional representation of Hilbert C∗-modules

Before starting the main part of Theorem 1.2, we summarize notations in this article. Let X be a Hilbert C -module over a unital C -algebra , ∗ ∗ A u( ) the image of the Gel’fand representation of and X = ( X , ΠX , ) K P A E E P the atomic bundle of X. For the map PX deﬁned in (3.14), deﬁne a linear map PX : Γ(X ) Γ( X ), ∗ × P → E

(PX (s)) (ρ) PX (s(ρ)) ( s Γ(X ), ρ ). ∗ ≡ ∈ × P ∈ P We deﬁne a subspace ΓX of Γ( X ) as follows: E Deﬁnition 5.1

ΓX PX (Γconst(X )) ≡ ∗ × P where Γconst(X ) is the subspace of Γ(X ) consisting of all constant sections. × P × P

Remark 5.1 ΓX is quite smaller that the set of all holomorphic sections of X . In fact, by Theorem 5.1, the hermitian form restricted on ΓX is in E u( ). Hence ΓX is small as same as u( ). K P K P We state a reconstruction theorem of a Hilbert C∗-module by the atomic bundle.

Theorem 5.1 (i) Any element in ΓX is holomorphic.

(ii) ΓX is a Hilbert C -module over u( ). ∗ K P (iii) There is an isomorphism between two Banach spaces

Ψ: X ∼= ΓX such that the following diagram is commutative: X X × A →

Ψ f Ψ × ↓ ↓

ΓX u( ) ΓX , × K P → where two horizontal arrows mean module actions. Hence, under the identiﬁcation f : ∼= u( ), ΓX is isomorphic onto X as a Hilbert -module. A K P A 28 We prepare some lemmata for the proof of Theorem 5.1 and explain how the structure of Hilbert C∗-module is interpreted as the geometrical structure of the atomic bundle. For ξ X, we deﬁne a section sξ Γ( X ) of X by ∈ ∈ E E

sξ(ρ) [ξ]ρ ( ρ ). ≡ ∈ P Consider -action defined in Definition 4.3 by the atomic connection D in ∗ Proposition 4.1 of X . Recall that X and Γ( X ) have norms defined by (3.9) and (4.15), respectively.E Then, the followingE lemma holds.

Lemma 5.1 For each ξ X, ∈ (i) ξ sξ is linear and isometric, 7→ (ii) sξ Γ ( X ) and it is holomorphic, ∈ ∞ E (iii) sξ fA = sξ A for A . ∗ · ∈ A Proof. (i) ξ sξ is linear by definition of linear structure of each fiber of 7→ X , and we have E sξ = sup sξ(ρ) ρ k k ρ k k = sup∈P ρ(< ξ ξ >) 1/2 ρ | | | ∈P = < ξ ξ > 1/2 k | k = ξ X . k k Hence sξ is bounded on and a map s is an isometry. P (ii) Let ρ . Assume ρ b for some b B. Let ( , π) be a representative ∈ P ∈ P ∈ H irreducible representation of b. Take local trivialization ψα,h at ( h, βh, h) b V H and at ρ with a typical fiber FX defined in Definition 3.4. By (4.19), we obtain

h < z Y > ∂Y φh(ρ)(sξ(ρ)) = ξ K | , h ! . (5.21) O Y,ρ − 2(1 + z 2) k k ρ Owing to (4.20), the right-hand side of (5.21) is smooth with respect to z βh(ρ) h, and hence, sξ is smooth at b for each b B. For ρ0 b, ≡ ∈ H P ∈ ∈ P we can choose h0 S( b) such that ∈ H

ρ0 =< h0 πb( )h0 > . | ·

29 Then βh0 (ρ0) = 0. By (4.18), we have

h0 Ω πb(< ξ0 ξ >)(z + h0) D ρ0 E < e φh (ρ)(sξ(ρ)) >= | | 0 1 + z 2 p k k ¯ for z = βh0 (ρ), ρ h0 . For an anti-holomorphic tangent vector Y of b, we have ∈ V P ¯ < Y z > ∂ ¯ φh(ρ)(sξ(ρ)) = ξ | , h ! Y O − 2(1 + z 2) k k ρ from which follows ∂¯¯ φ (ρ)(s (ρ)) = 0. Y h ξ z=0

We see that the anti-holomorphic derivative of sξ vanishes at each point in b. Hence sξ is holomorphic. P (iii) Let A . For b B and ρ0 b, take local coordinate ( h, βh, h) ∈ A ∈ ∈ P V H at ρ0 where h is a unit vector in and ( , π) is a representative irreducible H H representation of b. Then for z h, we have ∈ H 1 < (z + h) π(A)(z + h) > fA β− (z) = | . ◦ h 1 + z 2 k k Then the representation Xh of the Hamiltonian vector ﬁled X at ( , β , ) fA fA h h h is V H h Xf = √ 1 π(A)(z + h) < h π(A)(z + h) > (z + h) A z − − − | for z h. If we take h such that βh(ρ0) = 0, then it holds that ∈ H h Xf = √ 1 π(A)h < h π(A)h > h . A 0 − − − | D satisﬁes

< v (DX s)(ρ0) >ρ = ∂ρ < v s( ) >ρ (Xf ) | fA 0 0 | · 0 A for v Eh, s Γ ( X ). Hence we have ∈ ∈ ∞ E

(DXf sξ)(ρ0) = ξaXf ,0 A h A iρ0 where aX ,0 satisﬁes fA ∈ A

π aX ,0 h = X fA fA = √ 1 π(A) < h π(A)h > h. − − − |

30 Therefore we have

√ 1(DX sξ)(ρ0) = √ 1[ ξaX ,0 ]ρ − fA − fA 0 = √ 1 h ξ √ 1A < h π(A)h >i − · − − − | ρ0 = [ ξA]ρ [ξ]ρ < h π(A)h > 0 − 0 | = sξA(ρ0) sξ(ρ0)fA(ρ0) − from which follows

(sξ fA)(ρ0) = sξ(ρ0)fA(ρ0) + √ 1(DX sξ)(ρ0) ∗ − fA = sξA(ρ0). Therefore we arrive at

(sξ fA)(ρ) = sξA(ρ)( ρ ) ∗ ∈ P which proves lemma (iii).

Proof of Theorem 5.1 (i). Deﬁne a map

Ψ: X Γ ( X ); Ψ(ξ) sξ ( ξ X ). → ∞ E ≡ ∈ Then Ψ is a linear isometry by Lemma 5.1. For each τ Γconst(X ), ∈ × P there is ξ X such that τ(ρ) = (ξ, ρ) for ρ . We denote such τ bys ˆξ. Then ∈ ∈ P s ΓX there is ξ X such that s = PX sˆξ ∈ ⇔ ∈ ∗ s(ρ) = [ξ]ρ ( ρ ) ⇔ ∈ P s = sξ ⇔ s = Ψ(ξ) ⇔ s Ψ(X). ⇔ ∈ Hence ΓX = Ψ(X). Therefore Theorem 5.1 (i) follows from Lemma 5.1 (ii).

Lemma 5.2 (i) ΓX is a right u( )-module by -action deﬁned in Def- inition 4.3. K P ∗

(ii) For a hermitian metric H of X which is deﬁned by Equation (4.16), E let h be the restriction H ΓX of H of X on ΓX . Then a function- valued sesquilinear form | E

h :ΓX ΓX C∞( ) × → P 31 satisﬁes

h(s, s0 ) u( )( s, s0 ΓX ), ∈ K P ∈ h(s, s0 ) = h(s0 , s)( s, s0 ΓX ), ∈ h(s, s) 0 ( s ΓX ), (5.22) ≥ ∈ h( s, s0 f ) = h(s, s0 ) f ( s, s0 ΓX , f u( )), ∗ 1/2 ∗ ∈ ∈ K P h(s, s) = s ( s ΓX ) k k k k ∈ where the positivity in (5.22) means h(s, s) being a positive-valued function on and the norm of h(s, s) is the one deﬁned in (4.15). P (iii) The following equation holds:

h (Ψ(ξ), Ψ(η)) = ρ(< ξ η >)( ξ, η X, ρ ). ρ | ∈ ∈ P Proof. (i) From the proof of Theorem 5.1 (i), ΓX = Ψ(X). Since Lemma 5.1 (iii) and u( ) = f( ), the following map K P A ΓX u( ) = Ψ(X) f( ) (s, l) s l Ψ(X) = ΓX × K P × A 3 7−→ ∗ ∈ is bilinear. Hence, ΓX is a right u( )-module. Thus (i) is veriﬁed. (ii) and (iii): Next, we have the followingK P equations

(Ψ(ξ), Ψ(ξ0 )) = (s , s ) hρ hρ ξ ξ0 = H (s , s ) ρ ξ ξ0 = < sξ(ρ) s (ρ) >ρ | ξ0 = ρ(< ξ ξ0 >), | which proves (iii). Furthermore,

32 Hence it is shown that

h(s, s0 l) = h(s, s0 ) l (s, s0 ΓX , l u( )). ∗ ∗ ∈ ∈ K P Remain equations appearing in the statement (ii) follow from the property of C∗-valued inner product of X and the proof of Lemma 5.1 (i). This com- pletes the proof of (ii).

Proof of Theorem 5.1 (ii), (iii): (ii) By Lemma 5.2 (i), (ii) and Deﬁnition 3.2, h :ΓX ΓX u( ) (5.23) × → K P is a positive deﬁnite C∗-inner product of ΓX . Hence ΓX is a Hilbert C∗- module over a C∗-algebra u( ). (iii) By the proof of LemmaK 5.1P (i) and Lemma 5.2 (i), Ψ is an isomorphism between X and ΓX . If we rewrite module actions φ and ψ of X and ΓX , respectively, by φ(ξ, A) = ξA, ψ(s, l) = s l ∗ for ξ X, A , s ΓX and l u( ), then we have ∈ ∈ A ∈ ∈ K P

(ψ (Ψ f)) (ξ, A) = Ψ(ξ) fA ◦ × ∗ = sξ fA ∗ = sξA = Ψ(ξA) = (Ψ φ)(ξ, A) ◦ by Lemma 5.1 (iii). Hence we obtain the following equation:

ψ (Ψ f) = Ψ φ. ◦ × ◦ Therefore the diagram in the statement (iii) is commutative.

We summarize our results. The functional representation f of a non commutative unital C∗-algebra with the set of all pure states on , and the sectional representationAs of a Hilbert P -module X are givenA as A

33 follows: f = u( ) C ( ), A ∼ K P ⊂ ∞ P

A fA; fA(ρ) ρ(A), 7→ (A ≡ , ρ ), ∈ A ∈ P s X = ΓX Γholo( X ), ∼ ⊂ E

ξ sξ; sξ(ρ) [ξ]ρ, 7→ (ξ ≡X, ρ ) ∈ ∈ P where X is the atomic bundle of X and Γholo( X ) is the set of all holomor- E E phic sections on X . The correspondence of module structures of them is the following: E

(ξ, A) X = ΓX u( ) (sξ, fA) ∈ × A ∼ × K P 3

↓ ↓

ξA X = ΓX sξ fA. ∈ ∼ 3 ∗ Remark 5.2 Compairing the characterization of non commutative Gel’fand representation of C∗-algebra by [3], it seems that our characterization is not sufficient. Reader may request that another characterization which is defined by uniformity and etc, is necessary. Of course, we have tried to find more suitable characterization of section. Despite of our effort, we have not suceeded yet. The difficulty of characterization is similar to Remark 2.3. This is a problem in our future study.

Acknowledgement I would like to thank Prof. I.Ojima for a critical read- ing of this paper.

Appendix

A Hopf bundle

We summarize about Hopf bundle and its associated bundle.

34 A.1 Deﬁnition

We denote a Hilbert space over C such that dim 1. Denote C× C 0 . DefineH H ≥ ≡ \{ } S( ) z : z = 1 , H ≡ { ∈ H k k } ( ) ( 0 )/C . P H ≡ H\{ } × We call S( ) and ( ) a Hilbert sphere and a projective Hilbert space over , respectively.H WeP denoteH an element of ( ) by [z] for z 0 . We defineH a topology of S( ) the relative topologyP H of , and that∈ H of \ {( }) the quotient topology fromH 0 by the naturalH projection.P DefineH a projection µ from S( )H\{ to ( }) ⊂ by H H P H µ : S( ) ( ), H → P H µ(z) [z](z S( )). ≡ ∈ H Definition A.1 We call (S( ), µ, ( )) a Hopf (fiber)bundle over . H P H H Clearly, µ 1([z]) = S1 for each [z] ( ). − ∼ ∈ P H Example A.1 When = C, then H S( ) = S1, ( ) = 1pt . H P H { } When = C2, H S( ) = S3, ( ) = CP 1. H P H We define local trivial neighborhoods of a Hopf bundle ([3]). Fix h S( ) and define ∈ H

h z S( ):< h z > > 0 , W ≡ { ∈ H | }

h [z] ( ):< h z >= 0 , V ≡ { ∈ P H | 6 }

h z :< h z >= 0 , H ≡ { ∈ H | } z βh : h h; βh([z]) h ([z] h). V → H ≡ < h z > − ∈ V | Then ( h, βh, h) h S( ) is a system of local coordinates of ( ). ( ) is a K¨ahlermanifold{ V H } ∈ byH this local coordinate system [3]. P H P H

35 Let ψh be the local trivialization of S( ) at h deﬁned by H V 1 ψh : µ− ( h) = h U(1) V ∼ V × ψh(z) ([z], φh(z)), ≡

< z h > 1 φh(z) | ( z µ ( h)), ≡ < h z > ∈ − V | | |

1 < h z > ψ− ([z], g) z | g ([z] h, g U(1) ). h ≡ < h z > ∈ V ∈ | | | Hence h h S( ) is a system of local trivial neighborhoods of ( ) for (S( ),{V µ, }( ∈)).H Let R be a right action of U(1) on S( ) deﬁnedP byH H P H H S( ) U(1) S( ); (z, c) z c = Rcz cz.¯ H × → H 7→ · ≡ Then the following conditions are satisﬁed:

(i) µ(Rcz) = µ(z),

(ii) R is free, that is, if Rcz = z, then c = 1, (iii) for each h S( ), ∈ H < z h > φh(Rcz) = | c (z S( ), c U(1)). < h z > ∈ H ∈ | | | Hence (S( ), µ, ( )) is a principal U(1)-bundle. H P H Lemma A.1 For h, h0 S( ), assume = . For z, X , we h0 h h have ∈ H V ∩ V 6 ∅ ∈ H h + z 1 0 (βh βh− )(z) = h , 0 ◦ < h0 h + z > − |

0 1 1 < h X > ∂z(βh βh− )(X) = X | (h + z). 0 ◦ < h0 h + z > − < h0 h + z >2 | | Definition A.2 For a local trivial neighborhood h, V Ωh : h S( ) V → H is a local section defined by < z h > Ωh([z]) | z ([z] ( )) ≡ < z h > ∈ P H | | | where z S( ). ∈ H 36 By definition, < h Ωh(ρ) > > 0 for ρ h. | ∈ V A.2 Transition function

If h, h0 S( ) such that h0 h, then the transition function ∈ H ∈ V Q : h U(1) h0 h V ∩ Vh0 → is deﬁned by

Qhh([z]) = 1 ([z] h). ∈ V (ii) If h, h0 S( ) satisfy < h0 h >= 0, then ∈ H | 6 1 Q = Q− . h0 h hh0

(iii) If h, h0 , h00 S( ) are mutually non orthogonal, then ∈ H Q ([z]) Q ([z]) = Q ([z]) ([z] h ). h00 h0 · h0 h h00 h ∈ V ∩ Vh0 ∩ Vh00 Lemma A.2 Let X be a tangent vector of ( ) at ρ which is h h0 realized in and β (ρ) = z. Then we haveP H ∈ V ∩ V Hh0 h0 0 2 1 1 1 < z + h h > < h X > ∂z Q− β− (X) = | | . h0 h ◦ h0 −2 < h z + h0 > 3 | | | Proof. For w , we have ∈ Hh0 1 1 1 Q− β− (w) = Q β− (w) h0 h ◦ h0 hh0 ◦ h0

< w + h0 h > < h0 w + h > = | | < h w + h0 > < h0 w + h > | | | | | | < w + h0 h > = | < h w + h0 > | | | 37 because < h0 w >= 0 by deﬁnition of . | Hh0

Lemma A.3

1 1 1 1 < h X > Qh h β− (w) ∂w Q− β− (X) = | . 0 ◦ h0 · h0 h ◦ h0 −2 < h w + h0 > | Proof. By the previous lemma,

1 1 1 Q β− (w) ∂w Q− β− (X) h0 h ◦ h0 · h0 h ◦ h0

1 < h w + h0 >< h X >< w + h0 h >2 = | | | −2 < w + h0 h >2 < h w + h0 >2 | | 1 < h X > = | . −2 < h w + h0 > |

A.3 Associated bundles of Hopf bundles

Let F be a C∞-manifold with left U(1)-action α and S( ) F the direct product space of S( ) and F . Deﬁne a right U(1)-actionHγ on× S( ) by H H

zγc cz¯ ( c U(1), z S( )). ≡ ∈ ∈ H

We deﬁne S( ) U(1) F by the set of all orbits of U(1) in S( ) F where a U(1)-actionH × is deﬁned by H ×

(z, f)c (zγc, α(¯c)f)( c U(1), (z, f) S( ) F ). ≡ ∈ ∈ H ×

38 The topology of S( ) F is induced from S( ) F by the natural pro- H ×U(1) H × jection π : S( ) F S( ) U(1)F . We denote the element of S( ) U(1)F containing (x,H f)× by→ [(x, fH)].× Deﬁne a projection H ×

πF : S( ) F ( ) H ×U(1) → P H by πF ([(x, f)]) µ(x) [(x, f)] S( ) F . ≡ ∈ H ×U(1)

Definition A.3 A fibration F S( ) F, πF , ( ) is called the ≡ H ×U(1) P H associated bundle of ( S( ), µ, ( )) by F . H P H For h S( ), define a map ∈ H 1 ψα,h : π− ( h) h F F V → V × by ψα,h ([(z, f)]) ( µ(z), φα,h([(z, f)]) ) , ≡ 1 φα,h([(z, f)]) α(φh(z))f ( [(z, f)] π− ( h)). ≡ ∈ F V Hence we have

< z h > 1 ψα,h([(z, f)]) = [z], α | f ( [(z, f)] π− ( h)), < h z > ∈ F V | | |

1 < h z > ψ− ([z], f) = z, α | f ( ([z], f) h F ). α,h < h z > ∈ V × | | |

The deﬁnition of ψα,h is independent choice of (z, f). In fact, if (z0 , f 0 ) = (z, f)c for c U(1), then (z0 , f 0 ) = (¯cz, α(¯c)f) and ∈ < z0 h > ψα,h([(z0 , f 0 )] = [z0 ], α | ! f 0 ! < h z0 > | | | < z h > = [z], α c | α(¯c)f < h z > | | | < z h > = [z], α | f < h z > | | |

= ψα,h([(z, f)]).

39 ψα,h is a local trivialization of F at h. The transition function is given by V 1 Qˆ ψ ψ− :( h) F ( h) F, α,h0 ,h ≡ α,h0 ◦ α,h Vh0 ∩ V × → Vh0 ∩ V × ([z], f) [z], α(Q ([z]))f . 7→ h0 h Each vector space V over C has the scalar multiplication as U(1)-action α. If F = V , then we have

([z], f) ( h) F . ∈ Vh0 ∩ V × A.4 Recovery of the typical ﬁber Let (S( ), µ, ( )) be a Hopf bundle and F a complex Hilbert space. We considerHS( )P H F by the scalar multiple of U(1). H ×U(1) Proposition A.1 There is the following equivalence relation of ﬁber bundles on ( ): P H

S( ) U(1) F ( ) S( ) = S( ) F H × ×P H H ∼ H × where the left hand side is the ﬁber product of (S( ) F, πF , ( )) and H ×U(1) P H (S( ), µ, ( )), and (S( ) F, µF , ( )) is the trivial bundle. H P H H × P H Proof. Let

X1 S( ) U(1) F ( ) S( ). ≡ H × ×P H H We note that any element of X1 is written as ([(h, v)], h) where [(h, v)] S( ) F because ∈ H ×U(1) πF ([(h, v)]) = µ(h) and we can choose phase factor of (h, v) according to h. Let

πˆF : X1 ( );π ˆF ([(h, v)], h) h. → P H ≡

40 Deﬁne a map

Φ: X1 S( ) F ; Φ ([(h, v)], h) (h, v). → H × ≡ If (h0 , v0 ) [(h, v)], then there is c U(1) such that ∈ ∈

(h0 , v0 ) = (h, v)c = (hγc, cv¯ ) = (¯ch, cv¯ ).

But our notation restricts c = 1 since h0 = h. Hence Φ is well deﬁned. Φ is bijective. Furthermore

(µF Φ)([(h, v)], h) = µF (h, v) ◦ = h =π ˆF ([(h, v)], h).

Therefore µF Φ =π ˆF and (Φ, id) is a bundle map between X1 and ◦ (S( ) F, µF , ( )). Hence we have the statement. H × P H Let G be a group such that G acts on S( ) transitively and acts on F trivially. Then we have the following proposition.H

Proposition A.2 There is the following equivalence of linear spaces:

S( ) U(1) F ( ) S( ) G = F. H × ×P H H . ∼ Proof. We use the symbol in the previous proposition. Let α and β be actions of G on ( ) and F , respectively. Denote the action S H

αˆ (α U(1) 1) ( ) α ≡ × ×P H of G on X1 and

Y1 S( ) U(1) F ( ) S( ) G. ≡ H × ×P H H .

For [x] = [([(h, v)], h)] Y1, ∈

[x] = ([(αgh, v)], αgh): g G . { ∈ } Hence we can extend Φ as

Φ:ˆ Y1 S( ) F ; Φ([ˆ x]) [Φ(x)]. → H × ≡

41 Then Φ([([(ˆ h, v)], h)]) = [Φ([([(h, v)], h))] = Φ ([(αgh, v)], αgh): g G { ∈ } = (αgh, v): g G = {S( ) v . ∈ } H × { } Note Φˆ is bijection, too. Hence we denotev ˆ Φ([([(ˆ h, v)], h)]).] We deﬁne ≡ a linear structure for Y1 by avˆ + bwˆ (av + bw)(v, w F, a, b C). ≡ d ∈ ∈ Then ˆ: Y1 F → is a linear isomorphism.

A.5 Connections of an associated bundle of a Hopf bundle

Let F (S( ) U(1) F, πF , ( )) be an associated vector bundle of a Hopf bundle≡ (S( H), µ,× ( )) by aP complexH Hilbert space F . Let Γ(F) be the set of all smoothH sectionsP H of F, that is the set of right inverses of a projection πF . By the following operations, Γ(F) is a complex linear space:

(s + s0 )(ρ) s(ρ) + s0 (ρ)(ρ ( ), s, s0 Γ(F)), ≡ ∈ P H ∈ (ks)(ρ) ks(ρ)(ρ ( ), s Γ(F), k C). ≡ ∈ P H ∈ ∈ Deﬁnition A.4 D is a connection of F if D is a bilinear map of complex vector spaces D : X( ( )) Γ(F) Γ(F) P H × → which is C∞( ( ))-linear with respect to X( ( )) and satisﬁes the Leibniz law with respectP H to Γ(F): P H

DY (s l) = ∂Y l s + l DY s · · · for s Γ(F), l C ( ( ) and Y X( ( )). ∈ ∈ ∞ P H ∈ P H h For Y X( ( )), h S( ) and ρ h, we denote Y a tangent vector at ∈ P H ∈ H ∈ V ρ ρ in a local coordinate h. We consider a linear map H Ah : F F Y,ρ → h h such that ∂Y + A is a connection of F. |ρ Y,ρ

42 Fact A.2

D ∂ + A ≡ h is a connection of F if and only if a family A h S( ) satisﬁes the following equality: { } ∈ H h0 1 < h Y > h AY,ρ = | + AY,ρ ( ρ h h ) (A.24) −2 < h z + h0 > ∈ V 0 ∩ V | where Y is a holomorphic tangent vector of ( ) at ρ which is realized on and z = β (ρ). P H Hh0 h0 Proof. By using Leibniz rule and Lemma A.3,

h0 1 1 < h Y > h 1 AY β− (z) = | + AY β− (z)( z βh ( h h)). ◦ h0 −2 < h z + h0 > ◦ h0 ∈ 0 V 0 ∩ V | We have the statement.

B The atomic connection

Proof of Proposition 4.1. h At the beginning, we show the cocycle condition for A A h S( b) ≡ { } ∈ H deﬁned by (4.17). For ρ , choose h, h0 S( ) such that ρ . b b h h0 The cocycle condition for∈A Pis given by (A.24)∈ inH Appendix A.2.∈ V ∩ V

43 Let z0 β (ρ), z βh(ρ). Then we have ≡ h0 ≡ < z Xh > 2 Ah = | ρ − · X,ρ 1 + z 2 k k

1 0 1 h0 < (βh β− )(z ) ∂ (βh β− )(Xρ ) > = h0 z0 h0 ◦ | 1 ◦ 2 1 + (βh β− )(z0 ) k ◦ h0 k

z0 + h0 Xh0 < h Xh0 > (z0 + h0 ) * ρ | ρ + < h z0 + h0 > < h z0 + h0 > − < h z0 + h0 >2 = | | 2 | 0 0 z + h

< h z0 + h0 > |

0 0 2 h0 0 0 h0 z + h < h Xρ > < z + h Xρ > k k | = | − < h z0 + h0 > | z0 + h0 2 k k

0 h0 h0 < z Xρ > < h Xρ > 0 h0 = | | ( since < h Xρ >= 0) 1 + z0 2 − < h z0 + h0 > | k k |

h0 h0 < h Xρ > = 2 AX,ρ | − · − < h z0 + h0 > . | We obtain (A.24). Therefore D which is deﬁned in Proposition 4.1 is a connection. The curvature R of D is written by using A deﬁned in (4.17) as follows:

RX,Y = (dA)(X,Y ) + (A A)(X,Y )(X,Y X( b)). ∧ ∈ P Since A is scalar, A A = 0. ∧

44 In a local coordinate ( h, βh, h) of ρ b and z = βh(ρ) h, we have V H ∈ P ∈ H h h h (dzA)(X,Y ) = XA YA A Y,z − X,z − [X,Y ],z X < z Y > Y < z X > = | − | 2(1 + z 2) k k < z X >< z Y > < z X >< z Y > + | | − | | 2(1 + z 2)2 k k < z [X,Y ] > | − 2(1 + z 2) k k

Thus D is ﬂat.

C Proof of Lemma 4.1

b 1 b Proof. Let φα,h : (Π ) ( h) F be a map deﬁned by X − V → X

ψα,h(x) = (µb(h), φα,h(x)) .

45 b 1 For e = ([ξ0 ] , h) F such that h µ− (ρ), we have O ρ0 ∈ X ∈ b < e φα,h(s(ρ)) >= < ([ξ0 ] , h0 ) ([ξ]ρ, h) > | O ρ0 |O

h = [ξ0 ]ρ ξu h ρ,ρ0 iρ 0 ρ0

h = ρ < ξ0 ξρu > | ρ0 ,ρ0

h h h = Ω πb(< ξ0 ξρu >)Ω D ρ0 | | ρ0 ,ρ ρ0 E

h h h = Ω πb < ξ0 ξρ > πb u Ω D ρ0 | | ρ0 ,ρ ρ0 E

h 0 h = Ω πb < ξ ξρ > Ωρ D ρ0 | | E

h Ω πb(< ξ0 ξρ >)(z + h) = D ρ0 | | E . 1 + z 2 p k k From this, the following equality holds:

< e ∂Y φh(ρ)(s(ρ)) >= ∂Y < e φh(ρ)(s(ρ)) > | |

h 0 DΩ πb(< ξ ξρ >)(z + h)E = ∂Y ρ0 | | ! 1 + z 2 p k k

h 0 DΩ πb(∂X < ξ ξρ >)(z + h)E = ρ0 | | 1 + z 2 p k k

ˆ b < z Y > ∂Y φh(ρ)(s(ρ)) = ∂Y ξρ + ξρ K | , h ! . (C.25) O Y − 2(1 + z 2) k k ρ

46 We ﬁnish to prove Lemma.

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