C∗ -algebras of holonomy-diffeomorphisms & quantum gravity II

Journal of Geometry and Physics 99 (2016) 10–19

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C ∗ -algebras of holonomy-diffeomorphisms & quantum gravity II Johannes Aastrup ∗ , Jesper Møller Grimstrup Institut für Analysis, Leibniz Universität Hannover, Welfengarten 1, D-30167 Hannover, Germany

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Article history: Received 21 January 2015 Received in revised form 1 September 2015 Accepted 12 September 2015 Available online 28 September 2015 Keywords: Noncommutative geometry Quantum gravity Ashtekar variables

abstract We introduce the holonomy-diffeomorphism algebra, a C ∗ -algebra generated by flows of vector fields and the compactly supported smooth functions on a manifold. We show that the separable representations of the holonomy-diffeomorphism algebra are given by measurable connections, and that the unitary equivalence of the representations corresponds to measured gauge equivalence of the measurable connections. We compare the setup to Loop Quantum Gravity and show that the generalized connections found there are not contained in the spectrum of the holonomy-diffeomorphism algebra in dimensions higher than one. This is the second paper of two, where the prequel gives an exposition of a framework of quantum gravity based on the holonomy-diffeomorphism algebra. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Most fundamental theories of physics have connections among their basic variables, like the standard model of particle physics and the Ashtekar formulation of general relativity. It is therefore important, especially with respect to a quantization of these theories, to consider functions of connections, i.e. observables, within these theories. Probably the most well known example of such functions are the Wilson loops, i.e. traces of the holonomies of loops along closed paths; but also open paths have been considered, in particular when the observables have to act on fermions. One problem we have encountered in our attempt [1] to merge elements of canonical quantum gravity with noncommutative geometry is that variables like the Wilson loops, and related variables, tend to discretize the underlying spaces. Therefore in this paper we will commence the study of an algebra of ‘‘functions’’ of smeared objects in order to avoid this discretization. More concretely we will study a C ∗ -algebra generated by flows of vector fields on a manifold M and the smooth compactly supported functions on M. Flows of vector fields constitute a natural notion of families of paths, and when evaluated on a connection in the spin-bundle S, naturally gives an operator on the spinors, i.e. on L2 (M , S ), and not like a path just acting on one point in M. The holonomy-diffeomorphism algebra is defined as the C ∗ -algebra generated by the flows and the smooth function with norm given by the supremum over all the smooth connections. In this setup, smooth connections are viewed as representations of the holonomy-diffeomorphism algebra. It is the holonomy-diffeomorphism algebra which is our candidate for an algebra of observables. One test to see if a given algebra of observables is suitable is to look at the spectrum of the algebra, i.e. the space of irreducible representations modulo unitary equivalence. The main result in this paper is that all non-degenerate separable representations of the holonomy-diffeomorphism are given by so called measurable connections. These are objects, which

∗

Corresponding author. E-mail addresses: [email protected] (J. Aastrup), [email protected] (J.M. Grimstrup).

http://dx.doi.org/10.1016/j.geomphys.2015.09.007 0393-0440/© 2015 Elsevier B.V. All rights reserved.

J. Aastrup, J.M. Grimstrup / Journal of Geometry and Physics 99 (2016) 10–19

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are similar to the generalized connections encountered in Loop Quantum Gravity (LQG), see [2], but which take the measure class of the Riemannian metrics into account instead of the measure class of the counting measure. The measurable connections of course contain the smooth connections. This means that the generalized connections, which dominate the spectrum found in LQG, are excluded from the spectrum of the holonomy-diffeomorphism algebra. We believe this is a very significant result since several open problems in LQG can be traced back to the singular nature of the generalized connections: for instance the non-separable kinematic Hilbert space, the lack of weakly continuous operators and the ensuing inability to construct an off-shell constraint algebra (see [3]). It is therefore highly desirable to find a natural algebra of observables, which avoids the generalized connections and has a spectrum, which is as close as possible to the classical configurations space. The algebra of holonomydiffeomorphisms meets this challenge and is furthermore, from a physical point of view, a natural choice of algebra since it simply encodes how ‘‘stuff’’ is moved around on a spatial manifold. This paper is the second of two papers. Where its prequel [4] is concerned with an exposition of a mathematical framework of quantum gravity based on the holonomy-diffeomorphism algebra this paper is solely concerned with the mathematical analysis of this algebra. The paper is organized as follows: In Section 2 we define the holonomy-diffeomorphism algebra. In Section 3 we define the flow algebra. This algebra is constructed as a quotient of the cross product of the group generated the flows of the vector fields and the compactly supported smooth functions on the manifold. The ideal in this cross product, which is divided out, is the relation of local reparametrization. In particular the representations defining the holonomy-diffeomorphism algebra also give representations of the flow algebra. We show that separable non degenerate representations of this flow algebra are given by so called measurable connections, and show the unitary equivalence between these measurable connection is given by measurable gauge equivalence. In Section 4 we compare our setup with the LQG setup. The generalized connections appearing in LQG also give rise to representations of the flow algebra, however non-separable representations. We show that the generalized connections can be obtained as the representations of a discretized version of the flow algebra. In Section 5 we study the properties of the representations of the holonomy-diffeomorphism algebra given by smooth connections. In particular we show that a connection is irreducible if and only if the corresponding representation of the holonomy-diffeomorphism algebra is irreducible, and give some structure of the separable part of the spectrum. In the second part of Section 5 we show that if the dimension of the manifold is bigger than 1 the representations defined in Section 4 coming from generalized connections are not contained in the spectrum of the holonomy-diffeomorphism algebra. We do, however, not know if there are other non-separable representations contained in the spectrum of the holonomydiffeomorphism algebra. 2. The holonomy-diffeomorphism algebra Let M be a connected manifold and S a vector bundle over M. We assume that S is equipped with a fiber wise metric. This 1

1

metric ensures that we have a Hilbert space L2 (M , Ω 2 ⊗ S ), where Ω 2 denotes the bundle of half densities on M. Given a 1 2

diffeomorphism φ : M → M, this acts unitarily on L2 (M , Ω ) via the pullback of forms. We denote the pullback of φ with φ ∗ . Let X be a vector field on M, which can be exponentiated, and let ∇ be a connection on S. Denote by t → etX the corresponding flow. Given m ∈ M let γ be the curve

γ (t ) = e(1−t )X (e1X (m)) running from e1X (m) to m. We define the operator 1

1

eX∇ : L2 (M , Ω 2 ⊗ S ) → L2 (M , Ω 2 ⊗ S ) in the following way: 1

Let ξ ∈ L2 (M , Ω 2 ⊗ S ) be locally over (e1X (m)) of the form S. The value of (eX∇ )(ξ ) in the point m is given as



i

1

ωi ⊗ si , where ωi ’s are elements in Ω 2 and si a section in

 ((e1X )∗ (ωi )(m)) ⊗ (Hol(γ , ∇)si (e1X (m))), i

where Hol(γ , ∇) denotes the holonomy of ∇ along γ . If the connection ∇ is unitary with respect to the metric on S, then eX∇ is a unitary operator. If we are given a system of unitary connections A we define an operator valued function over A via

A ∋ ∇ → eX∇ , 1

and denote this by eX . Denote by F (A, B (L2 (M , Ω 2 ⊗ S ))) the bounded operator valued functions over A. This forms a C ∗ -algebra with the norm

∥Ψ ∥ = sup {∥Ψ (∇)∥}, ∇∈A

1

Ψ ∈ F (A, B (L2 (M , Ω 2 ⊗ S ))).

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For a function f ∈ Cc∞ (M ) we get another operator valued function feX on A. Definition 2.0.1. Let C = span{feX |f ∈ Cc∞ (M ), X exponentiable vector field }. 1

The holonomy-diffeomorphism algebra HD(M , S , A) is defined to be the C ∗ -subalgebra of F (A, B (L2 (M , Ω 2 ⊗ S ))) generated by C . We will by H D (M , S , A) denote the ∗-algebra generated by C . It is this algebra that will be the object of study in this paper. We will in particular be interested in the spectrum and the representations of the holonomy-diffeomorphism algebra. 2.1. Formulation with a metric The construction above of the holonomy-diffeomorphism algebra shows that it is background independent. In some situations it is however convenient to have a formulation with a metric, and we will therefore in this subsection explain the construction given a fixed background metric g. Given an exponentiable vector field X and a unitary connection ∇ , the first attempt to define an operator associated to X would be to define

(eX∇ (ξ ))(m) = Hol(γ , ∇)ξ (e1X (m)),

ξ ∈ L2 (M , S , dg ).

(We have kept the notation from the previous section). Here dg denotes the Riemannian measure associated to g. The problem is, that since the flow of X might not preserve the measure dg, this is not a unitary operator, and even worse it might not define an operator at all, since ∥eX∇ (ξ )∥2 = ∞ can occur. To fix this let O ⊂ Rn be open and Φ1 : O → M a local coordinate system around m ∈ M. Furthermore let Φ2 : O → M be a different coordinate system around e1X (m) with Φ2 ◦ Φ1−1 = e1X where Φ2 ◦ Φ1−1 is defined. We define the operator as

 4 |g |(e1X (m)) Hol(γ , ∇)ξ (e1X (m)), ξ ∈ L2 (M , S , dg ), (e∇ (ξ ))(m) = √ 4 |g |(m) where |g |(m) denotes the determinant of the metric in the Φ1 coordinates and |g |(e1X (m)) the determinant of the metric in the Φ2 coordinates. This is unitary since   4 |g |(e1X (m)) Hol(γ , ∇)ξ (e1X (m)), √ 4 |g |(m) O   4 |g |(e1X (m)) 1X Hol (γ , ∇)ξ ( e ( m )) |g |(m)dx √ 4 |g |(m)   4 |g |(e1X (m)) 1X ξ (e (m)), = √ 4 |g |(m) O   4 |g |(e1X (m)) 1X ξ (e (m)) |g |(m)dx √ 4 |g |(m)    = ξ (e1X (m)), ξ (e1X (m)) |g |(e1X (m))dx, X

O

where m = Φ1 (x), and (·, ·) denote the fibrewise inner product. In this computation we have used that Hol(γ , ∇) is unitary. 3. An abstract algebra In order to study the spectrum and the representations of HD(M , S , A) we will introduce an algebra which is more abstract than HD(M , S , A) and which at the end carries the information of the representation theory of HD(M , S , A) plus some additional information. Let X be a vector field on M which can be exponentiated. We will denote the flow to time t as etX . By eX we denote the flow from 0 to 1, i.e. the map M × [0, 1] → M given by

(m, t ) → etX (m). Two such flows for two vector fields X1 , X2 can be composed via

t →

  (2tX )   e 1,   e((2t −1)X2 ) ,



t ∈ 0,

 t ∈

1 2

1



2

 ,1

This is of course usually not a flow.

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If two flows are the same modulo reparametrization we will identify the flows. Also we will identify eX e−X with the trivial flow I, and eX ◦ I, I ◦ eX with eX . We denote by F the group generated by these flows. The flow group F acts on M simply by considering for eX the diffeomorphism e1X . Note that this action is not faithful, since we can have a vector field X ̸= 0 with e1X = the identity diffeomorphism. For example M = S 1 = {eiθ |θ ∈ [0, 2π ]} and X = 2π ddθ . Here e1X is the identity, but etX θ = θ + 2π t. From the action on M we induce a left action on Cc∞ (M ) via eX (f )(m) = f (e−1X (m)). We form the cross product

F n Cc∞ (M ), where we consider F with the counting measure, see [5] page 171. This algebra consists of the linear span of formal products fF ,

f ∈ Cc∞ (M ), F ∈ F ,

with the multiplication relation f1 F1 f2 F2 = f1 F1 (f2 )F1 F2 and adjoint

(f1 F1 )∗ = F1∗ f 1 = F1−1 (f 1 )F1−1 . If we are given a vector bundle S with a metric over M and a unitary connection ∇ we get a ∗-representation ϕ∇ of 1

F n Cc∞ (M ) on L2 (M , Ω 2 ⊗ S ) via

ϕ∇ (feX ) = feX∇ . Therefore unitary connections give rise to representations of F n Cc∞ (M ). On the other hand a representation of F n Cc∞ (M ) will in general not have much to do with unitary connections; the reason being that if eX1 ̸= eX2 , but however coincide on X X supp f , then eX1 f ̸= eX2 f in F n Cc∞ (M ), but e∇1 f = e∇2 f for all connections. We thus need to add extra relations to the cross product. Let A be a subset of M. Let F1 , F2 ∈ F . We will consider the F1 and F2 restricted to A as maps F1 , F2 : A × [0, 1] → M . We will say that F1 is locally over A a reparametrization of F2 if for each a ∈ A exists a monotone growing piecewise smooth bijection

ϕa : [0, 1] → [0, 1] such that F1 (a, t ) = F2 (a, ϕa (t )). Definition 3.0.1. Let I be the subset of F n Cc∞ (M ) given by

{F1 f − F2 f |F1 is a local reparametrization of F2 over supp f } Lemma 3.0.2. I is a ∗-ideal in F n Cc∞ (M ). Proof. Multiplying I from the right with a function g ∈ CC∞ (M ) preserves I. Multiplying from the left we get g (F1 f − F2 f ) = F1 F1−1 (g )f − F2 F2−1 (g )f . Since F1 is a local reparametrization of F2 we have F2−1 (g )f = F1−1 (g )f , and thus g (F1 f − F2 f ) ∈ I. When F1 is a local reparametrization of F2 , then so is FF1 of FF2 . Hence multiplication from the left with F preserves I. Right multiplication yields

(F1 f − F2 f )F = (F1 F − F2 F )F (f ), and since F1 F is a local reparametrization of F2 F over supp F (f ) we have IF ⊂ I.



Definition 3.0.3. The flow algebra F M is defined as

F n Cc∞ (M )/I . Note that for a unitary connection ∇ we have ϕ∇ (I ) = 0, and hence it descends to a ∗-representation, also denoted ϕ∇ , 1

of F M on L2 (M , Ω 2 ⊗ S ). It follows that Cc∞ (M ) embeds into F M as f → f · e0 .

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3.1. Separable representations We will now study the separable ∗-representations of F M, i.e. ∗-homomorphisms from F M to B (H ), H separable. Therefore let ϕ : F M → B (H ) be such a representation, and we will also assume that ϕ is non-degenerate. In particular we get a ∗-representation, also denoted ϕ , of Cc∞ (M ) on H . Lemma 3.1.1. Let ϕ : Cc∞ (M ) → B (H ) be a ∗-representation. Then ϕ has a unique extension to a ∗-representation

ϕ˜ : C0 (M ) → B (H ). Proof. We only need to show that ϕ is continuous. We first extend ϕ to a unital ∗-homomorphism

ϕ ∼ : Cc∞ (M )∼ → B (H ), where Cc∞ (M )∼ denotes Cc∞ (M ) with a unit added. The norm of ϕ ∼ (f ) is given via the spectral radius. If λ ̸∈ f (M ) then λ − f has an inverse in Cc∞ (M )∼ . This implies that λ ̸∈ spec(ϕ ∼ (f )), and ϕ is therefore continuous.  We will also denote the extension by ϕ . The commutant ϕ(C0 (M ))′ of ϕ(C0 (M )) in B (H ) is a type I von Neumann algebra. This follows since ϕ(C0 (M ))′ = (ϕ(C0 (M ))′′ )′ , ϕ(C0 (M ))′′ is a commutative von Neumann algebra (and hence type I) and the commutant of a type I von Neumann algebra is again a type I von Neumann algebra (see Theorem 9.1.3.  in [6]). This means that there exists an orthogonal family of projections {Pn }n∈{1,...,∞} in the center of ϕ(C0 (M ))′ with Pn = 1H such that Pn ϕ(C0 (M ))′ is a type In . Let F ∈ F and A ∈ ϕ(C0 (M ))′ . Then ϕ(F )Aϕ(F −1 ) ∈ ϕ(C0 (M ))′ since

ϕ(f )ϕ(F )Aϕ(F −1 ) = ϕ(F )ϕ(F (f ))Aϕ(F −1 ) = ϕ(F )Aϕ(F (f ))ϕ(F −1 ) = ϕ(F )Aϕ(F −1 )ϕ(f ). Conjugating with ϕ(F ) is therefore an automorphism of ϕ(C0 (M ))′ , and it follows that conjugating with ϕ(F ) preserves the type structure, and thus ϕ(F ) commutes with each Pn . This means that studying the representations of F M we can restrict to the case of ϕ(C0 (M ))′ being of type In for a fixed n ∈ {1, . . . , ∞}. In fact it follows from what will come that all representations are of type In for a fixed n. Since ϕ(C0 (M ))′ is of type In , ϕ(C0 (M ))′ is isomorphic to L∞ (X ) ⊗ B (H ), with dimH = n, and X being some σ -finite measure space with measure µ. This means that the representation ϕ is given via a representation

ψ : Cc∞ (M ) → B (L2 (X , µ)) with ψ(Cc∞ (M )) ⊂ L∞ (X ) and ϕ = ψ ⊗ 1. Since ϕ is non-degenerate, we can assume that X = M, (ψ(f )ξ )(m) = f (m)ξ (m), and that µ is a regular Borel measure on M. Furthermore we can also assume that (M , µ) is finite. All together we have Proposition 3.1.2. Let ϕ : F M → B (H ) be a separable non-degenerate representation. There exists a finite Borel measure µ on M such that ϕ is unitarily equivalent to a representation ϕ1 on L2 (M , µ) ⊗ Hn , dimH = n, n = 1, . . . , ∞, with

(ϕ1 (f )ξ ⊗ η)(m) = f (m)ξ (m)η,

f ∈ Cc∞ (M ).

Let ϕ be a representation of the same form as ϕ1 in the above proposition. We will identify L2 (M , µ) ⊗ Hn with L (M , µ, Hn ). We assume, that µ(M ) = 1. We define the measure µF as 2

µF (A) = µ(F (A)), for all measurable subsets A of M. Let F be a flow and let f ∈ C0 (M ). We have

(ϕ(F )ϕ(f )ξ )(m) = (F (f ))(m)(ϕ(F )ξ )(m).

(1)

Let ξ be a vector with ∥ξ (m)∥2,n = 1, where ∥ · ∥2,n denotes the norm on Hn . We define kF (m) = ∥ϕ(F )(ξ )(m)∥22,n . We have

∥f ∥22 = ⟨ϕ(f )ξ , ϕ(f )ξ ⟩ = ⟨ϕ(F )ϕ(f )ξ , ϕ(F )ϕ(f )ξ ⟩ = ⟨ϕ(F (f ))ϕ(F )ξ , ϕ(F (f ))ϕ(F )ξ ⟩  = |f (F −1 (m))|2 ∥ϕ(F )ξ (m)∥22,n dµ(m)

(2)

M

for all f ∈ C0 (M ). In particular the function λ : C0+ (M ) → R given by λ(f ) = of the choice ξ , and therefore kF is independent of ξ µ-almost everywhere.

 M

|f (m)|∥ϕ(F )ξ (m)∥22,n dµ(m) is independent

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Since formula (2) holds for all bounded measurable functions we get

µF (F −1 (A)) = µ(A) =



12A dµ = M



|F (1A )|2 kF dµ =

 M

M

1F −1 (A) kF dµ,

thus µF = kF µ, and therefore µF ≪ µ. The same holds for F −1 and we get Lemma 3.1.3. µ is equivalent to µF . With this lemma we can now prove Theorem 3.1.4. µ is equivalent to a measure induced by a Riemannian metric on M. Proof. This follows from Lemma 3.1.3 and the result that a quasi-invariant Borel measure on a locally compact group is equivalent to the Haar measure.  Because of (1) we can consider ϕ(F ) as a measurable family m → ϕ(F )(m) in U (n), the group of unitary operators on

Hn , i.e.

(ϕ(F )ξ )(m) =



kF (m)ϕ(F )(m)(ξ (F −1 (m))).

This gives rise to the following Definition 3.1.5. A measurable U (n)-connection, n = 1, . . . , ∞, is a map ∇ from F to the group of measurable maps from M to U (n) satisfying 1. ∇(1) = 1. 2. ∇(F1 ◦ F2 )(m) = ∇(F1 )(m) ◦ ∇(F2 )(F1−1 (m)). 3. If F1 and F2 are the same up to local reparametrization over some set U ⊂ M, then

∇(F1 )U = ∇(F2 )U . 1

A measurable U (n)-connection ∇ gives rise to a representation of F M on L2 (M , Ω 2 ⊗ Hn ) via

(ϕ∇ (f )(ξ ))(m) = f (m)ξ (m)   ϕ∇ (F )(ξ )(m) = (F −1 )∗ (F −1 (m)) ⊗ ∇(F )(m) ξ (F −1 (m)). With the work done so far in this section we have Theorem 3.1.6. Any non-degenerate separable representation of F M is unitarily equivalent to a representation of the form ϕ∇ , where ∇ is a measurable U (n)-connection. 3.2. Equivalence of representations Definition 3.2.1. Two representations ϕ1 , ϕ2 of F M on H1 , H2 are called unitary equivalent if there exist a unitary U : H1 → H2 with

ϕ1 (a) = U ∗ ϕ2 (a)U ,

for all a ∈ F M . 1

According to 3.1.6 any two non-degenerate separable representations are of the form ϕ∇1 , ϕ∇2 and H1 = L2 (M , Ω 2 ⊗ 1 2

Hn1 ), H2 = L2 (M , Ω ⊗ Hn2 ). If they are unitarily equivalent we have

(ϕ∇1 (f )ξ )(m) = f (m)ξ (m) = (ϕ∇2 (f )ξ )(m), and therefore the unitary U is given as a measurable map m → u(m), with u(m) : Hn1 → Hn2 unitary. Consequently we have n1 = n2 . This leads to the following Definition 3.2.2. A measurable U (n)-gauge transform is a measurable map M ∋ m → U(Hn ). Two measurable U (n)-connections ∇1 , ∇2 are called gauge equivalent if there exists a measurable U (n)-gauge transform m → u(m) with

∇1 (F )(m) = u(m)∇2 (F )(u(F −1 (m)))∗

for all F ∈ F .

Proposition 3.2.3. Two representation of the form ϕ∇1 , ϕ∇2 , ∇1 , ∇2 measurable U (n) connections, are unitarily equivalent if and only if ∇1 , ∇2 are measurable gauge equivalent.

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4. Comparison to the LQG spectrum In this section we will compare the setup we have so far to that of LQG, and also give some non-separable representation of F M. For simplicity we will only work with piecewise analytic flows and paths. When we talk about paths we will identify l−1 ◦ l with the trivial path starting and ending in the start point of l. We will also identify two path which are the same up to reparametrization. Definition 4.0.4. Let G be a connected Lie-group. A generalized connection is an assignment ∇(l) ∈ G to each piecewise analytic path l, such that

∇(l1 ◦ l2 ) = ∇(l1 )∇(l2 ). For details on generalized connections see [7,8], see also [9]. Let us now further assume that we have a representation of G as a subgroup of U (n). Note that we can in general not 1

use a generalized connection to define a representation of F M on L2 (M , Ω 2 ⊗ Hn ) like we did for a smooth or measurable connection. The problem is, that eX∇ need not be measurable. On the other hand, if we equip M with the counting measure, we can use a generalized connection ∇ to define a representation of F M on L2 (M , Hn ). Here we see, however, that a measurable connection does not define a representation on L2 (M , Hn ), since a measurable connection is only defined up to measure zero sets, i.e. not in single points. Definition 4.0.5. A generalized unitary gauge transformation is a map U : M → U (n). Two generalized connections ∇1 and ∇2 are said to be unitarily gauge equivalent if for all paths l we have U (e(l))∇1 (l)U ∗ (s(l)) = ∇2 (l), where e(l) denotes the end point of l and s(l) the start point. In order to see the generalized connections as related to the spectrum of an algebra similar to the flow algebra, we will define a discrete version of it. Let Cd (M ) be the algebra of functions on M with finite support. We define Fd M like F M but with Cc∞ (M ) replaced by Cd (M ). For a given point m ∈ M we denote by 1m the function with value 1 in m and zero elsewhere. This is a projection, and due to the relations defining Fd M we have F 1m F −1 = 1F (m) . Given a non-degenerate representation ϕ : Fd M → B (H ) we define Hm = ϕ(1m )H . Since ϕ(F ) is a unitary operator, then, due to the conjugation relation with 1m above, ϕ(F ) : Hm → HF (m) is a unitary operator. In particular we have

H=



Hm ,

m∈M

and all the Hm ’s have the same dimension, and we can therefore write the Hilbert space as

H=



Hn = L2 (M , Hn ),

m∈M

where M is equipped with the counting measure and Hn is a Hilbert space of dimension n, n being a cardinal number. We thus have Proposition 4.0.6. To every non-degenerate representation ϕ of Fd M, there exists a generalized connection ∇ , such that ϕ is of the form

ϕ : Fd M → B (L2 (M , Hn )), with

(ϕ(f )ξ )(m) = f (m)ξ (m), and

(ϕ(F )ξ )(F (m)) = ∇(Fm )ξ (m), where Fm is the edge F defines between m and F (m). Two non-degenerate representations ϕ1 , ϕ2 , associated to two generalized connections ∇1 , ∇2 , are equivalent if and only if they are generalized gauge equivalent.

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5. The spectrum of the holonomy-diffeomorphism algebra We will in this section restrict attention to HD(M , S , A), where S is the trivial two-dimensional bundle, and A is the set of SU (2)-connections. 5.1. Properties of the representations We remind the reader of the following two Definition 5.1.1. A connection is called irreducible if at a given point m the holonomy group in this point acts irreducible on the fiber at m. Definition 5.1.2. A ∗-representation ϕ : A → B of a C ∗ -algebra A on a Hilbert space H is called irreducible if ϕ(A) acts irreducibly on H . There is the following well known characterization of irreducible representations, see [10]. Theorem 5.1.3. A representation ϕ : A → B (H ) of a C ∗ -algebra on a Hilbert space is called irreducible if one of the following equivalent conditions are fulfilled: 1. ϕ is irreducible. 2. The commutant ϕ(A)′ = {b ∈ B (H )|bϕ(A) = ϕ(A)b for all a ∈ A} is equal to C1H . 3. Every nonzero vector ξ ∈ H is cyclic for ϕ(A), or ϕ(A) = 0 and H = C. We can now connect the two notions of irreducibility Proposition 5.1.4. A representation ϕ∇ is irreducible if and only if ∇ is irreducible. When ∇ is reducible, the representation ϕ∇ splits into two irreducible representations, corresponding to U (1) connections. Proof. Clearly the representations ϕ∇ are not zero. Let us assume that ∇ is irreducible. Since the holonomy group acts irreducibly in one point it acts irreducibly in all points, and for every two points in M the holonomy paths between these two points acts irreducibly between the fibers in the points. Since we have the flows as operators, and we can multiply these with compactly supported smooth functions, it follows that every nonzero vector is cyclic for ϕ∇ (HD(M , S , A)), i.e. ϕ∇ acts irreducibly. On the other hand, if ∇ is not irreducible, we can split the bundle S into two line bundle, each being invariant under the action of the holonomy groupoid. Consequently ϕ∇ splits into two irreducible representations, each corresponding to a U (1) connection. This motivates the following Definition 5.1.5. A measurable U (n)-connection ∇ is called irreducible if the corresponding representation ϕ∇ is irreducible. We remind the reader of the following Definition 5.1.6. Let A be a C ∗ -algebra. The spectrum of A, spec(A), is defined as spec(A) = {Irreducible representations}/Unitary equivalence . We have Proposition 5.1.7. Put

U1 = {Measurable U (1)-connections} and

U2 = {Irreducible measurable U (2)-connections}. The separable part of the spec (HD(M , S , A)) is contained in

(U1 ∪ U2 )/{Measurable gauge equivalence}. Proof. The only statement that remains to be proved is that irreducible measurable U (n)-connections, n ≥ 3, do not appear in the spectrum. In order to prove this we recall some facts on the primitive spectrum. Let A be a C ∗ -algebra. An Ideal I ⊂ A

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is called primitive if I is the kernel of some irreducible representation. Let Aˇ be the set of primitive ideals in A. There is a natural surjective map Ξ from spec(A) to Aˇ by mapping an irreducible representation (π , H ) to Ker π . A subset F ⊂ Aˇ is called closed, if F = hull(J ) := {I ∈ Aˇ | J ⊂ I } for some subset J of A. The system of closed subsets of Aˇ forms the closed ˇ called the Jacobson topology, see Theorem 4.1.3 in [11]. Furthermore the closure of a set F ⊂ Aˇ subsets of a topology on A, − is given by F = hull(∩I ∈F I ). Let Aˇ n be the subset of Aˇ of representations of dimension ≤ n. According to proposition 4.4.10 ˇ Also if I ⊂ A is a closed ideal in A we have hull(I ) = spec(A/I ) (Theorem 4.1.11. in [11]). This shows in [11], Aˇ n is closed in A. that a C ∗ -algebra generated by representations of dimension ≤ n only has irreducible representations of dimension ≤ n. In F M we consider the ∗-subalgebra D generated by Cc∞ (M ) and products of flows, whose action is the identity on M. For each irreducible measurable U (n) connection ∇ the weak closure of ϕ∇ (D) is L∞ (M ) ⊗ Mn . This follows, since D is the ∗-subalgebra of F M, which fixes the points in M, and since ∇ is irreducible. We will also denote the C ∗ -subalgebra of HD(M , S , A) generated by D by D. For each of the representations generating D the weak closure of ϕ∇ (D) is L∞ (M ) ⊗ M2 or L∞ (M ) ⊗ C2 , since D is generated by SU (2)-connections. If we therefore had a representation ϕ∇ : HD(M , S , A) → L2 (M , Hn ) given by a irreducible measurable U (n) connection, n ≥ 3, this would result in a morphism ϕ∇ : D → L∞ (M ) ⊗ Mn commuting with the action of C0 (M ), i.e. ϕ∇ would be a fibered map over M mapping fibers generated by one and two dimensional representations onto a factor of dimension ≥ 3. This cannot be the case due to the above mentioned theorems on the spectrum.  5.2. The non-separable part of the spectrum of the holonomy-diffeomorphism algebra In general we do not know what the non-separable part of the spectrum looks like. We have, however, the following Proposition 5.2.1. Let dim(M ) > 1. The representations ψ∇ , where ∇ is a generalized connection given by representing the flow-algebra on L2 (M , Hn ) with the counting measure, are not contained in the spectrum of HD(M , S , A). 1

Proof. We will show that ψ∇ cannot be bounded by the representations of the form ϕ∇1 : F M → B (L2 (M , Ω 2 ⊗ S )), where ∇1 is a smooth connection. We choose an open subset U of M diffeomorphic to Rn , n ≥ 2. We consider a subgroup of the flow group, which acts on n U like Gl+ n (R) on R . 2 n The Gl+ ( R ) -part of the representation ψ∇ is given by representing Gl+ n n (R) on L (R , c ), c being the counting measure. 2 n The subspace C10 of L (R , c ) is invariant under this representation, and therefore the trivial representation of Gl+ n (R) is contained in this representation. The representation ϕ∇ is equivalent to two copies of a representation π on L2 (Rn ), where the Gl+ n (R)-part of the representation is given by 1

π (g )(ξ )(x) = |detg |− 2 ξ (xg −1 ),

g ∈ Gl+ n (R).

The n-fold tensor product of π is given by n

(π ⊗n (ξ ))(x) = |detg |− 2 ξ (xg −1 ),

x ∈ Mn (R)

on L (Mn (R), λ). Since the Haar measure on Gln (R) is given by 2

µ(S ) =



|detx|−n dx, S

where S is a subset of Mn (R), the n-fold tensor product of π is equivalent to the left regular representation of Gl+ n (R) on L2 (Gln (R), µ). In particular π is bounded by the left regular representation. We will now show that the trivial representation of Gl+ n (R) is not weakly contained in the left regular representation, thereby concluding that ψ∇ cannot be bounded by the representations of the form ϕ∇1 , where ∇1 is the trivial connection on Rn . 2 Denote by λ the left regular representation of Gl+ n (R) on L (Gln (R)). We want to show that the trivial representation is not weakly contained in this representation when we consider Gl+ n (R) as discrete group. Since λ ⊗ λ is equivalent to a multiple of λ it follows that the trivial representation is weakly contained in λ if and only if the trivial representation is weakly contained in λ ⊗ λ. According to Theorem 5.1 in [12] this is equivalent to the trivial representation being weakly contained in λ ⊗ λ when Gl+ n (R) is considered as a continuous group with the usual topology. This is the again equivalent to the trivial representation being weakly contained in λ when Gl+ n (R) is considered as a continuous group with the usual topology. Restricting to λ to Gl+ n (R) we get that the trivial representation is contained in the left regular representation of + Gl+ n (R) as continuous group. This is however in contradiction to the Gln (R) being non-amenable for n ≥ 2, see [13]. If ∇2 is an arbitrary smooth SU (2)-connection we can proceed as follows: Since the trivial representation of Gl+ n (R) is not weakly contained in the left regular representation, there exists, according to Proposition G.4.2 in [14], positive numbers a1 , . . . , ak and elements g1 , . . . , gk in the flow group with

∥ϕ∇1 (a1 g1 + · · · + ak gk )∥ < ∥ψ∇ (a1 g1 + · · · + ak gk )∥.

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However since a1 , . . . , ak are positive, we have

∥ϕ∇2 (a1 g1 + · · · + ak gk )∥ ≤ ∥ϕ∇1 (a1 g1 + · · · + ak gk )∥. Hence ψ∇ cannot be bounded by ϕ∇2 .



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