Groupoid Characterization of Locally Convex Partial ^*-Algebras
aa r X i v : . [ m a t h . OA ] J a n Groupoid Characterization of Locally ConvexPartial ∗ -Algebras N. O. Okeke Email: [email protected],
Physical and Mathematical Sciences, Dominican University, Ibadan
M. E. Egwe [email protected]
Department of Mathematics, University of Ibadan, Ibadan, Nigeria
Abstract
Given a locally convex space ( A , τ ) with a Hausdorff locally convex topol-ogy τ such that the following maps are continuous; u u ∗ for all u ∈ A , x x · y and x z · x for every left and right multipliers of A . In thispaper we re-characterized the locally convex partial ∗ -algebra ( A , Γ , · , ∗ , τ )arising from these continuous maps in terms of convolution algebra of a Liegroupoid Γ ⇒ A . This is advantageous because the pathologies of the under-lying spaces owing to their quantum mechanical nature are easily resolved ingroupoid terms. Introduction
The motivation for this work is basically to use the more conducive groupoid struc-ture to recharacterize the locally convex partial ∗ -algebra which was characterized Mathematics subject Classification (2010):
Key words and phrases:
Partial ∗ -algebras, Lie Groupoid, Local convexity, Groupoid equiva-lence, Unitary representation, ∗ -representation, Groupoid convolution algebra.
1y Ekhaguere in [7]. The re-characterization is basically through the relation Γ de-fined by the partial multiplication ( · ) on a locally convex linear space A , which isused to define a groupoid Γ ⇒ A , where the diagonal ∆ = { ( x, x ) ⊂ A × A} is thespace of objects isomorphic to A . The local convexity of the linear space A offersvarious advantages for the analysis of the resulting groupoid convolution algebra.These advantages, according to [27], include among other things, the following: • It is naturally compatible with the underlying topological framework; • It makes for automatic continuity of the smooth maps and differentials onthe framework which has important geometric application, especially in thedefinition of system of left Haar measures for the groupoid; • It aids the construction of a Lie group structure on the set of bisections whichis related and derivable from the canonical smooth structure on the manifoldof mappings.These accord with the fact that in analysis, unbounded operators frequently occurwhen symmetries are introduced using Lie groups. In this case the algebra arisingfrom a smooth net K , which is shown to be isomorphic to a Lie group (we sometimealso denoted the smooth algebra with K ), carries a global aspect of the partialsymmetry of the structure which complements with that of the relation Γ. Thus, theunbounded operators usually appear as open (unbounded) infinitesimal generatorswhich are mostly differential operators of infinite dimensional Lie pseudogroups. Wepresent the formulations of partial algebras in the sequel. Ekhaguere [7] shows the abundance of unbounded operators by re-characterizingthem in terms of linear subspaces open under a defined product operation. Theseopen and linear subspaces are the natural representations of unbounded operators.Thus, unbounded operators manifest as open linear subspaces under a product op-eration. Among such open linear subspaces mentioned that is of much interest for2his paper is the space of unbounded linear maps on locally convex topological vec-tor spaces, with composition of maps as product operation. Our intention is toreformulate this partial ∗ -algebra as a groupoid convolution algebra. Definition 1.1. [7] A partial algebra is a triplet ( A , Γ , · ), comprising a linear space A , a partial multiplication · on A , and a relation given byΓ = { ( x, y ) ∈ A × A : x · y ∈ A} such that ( x, v ) , ( x, z ) , ( y, z ) ∈ Γ implies ( x, αv + βz ) , ( αx + βy, z ) ∈ Γ and then( αx + βy ) · z = α ( x · z ) + β ( y · z ) and x · ( αv + βz ) = α ( x ◦ v ) + β ( x ◦ z ) for all α, β ∈ C . Proposition 1.2.
The partial algebra ( A , Γ , · ) corresponds to the groupoid G =(Γ ⇒ A ) with arrows defined by the relation Γ = { ( x, y ) ∈ A × A : x · y ∈ A} .Proof. First, from the definition of the relation Γ on A , it follows that Γ is a sub-groupoid of groupoid of pairs G = A × A , and the maps are defined as follows. Thearrows are defined by the relation Γ = A (2) ⊂ A × A .(i) The source and target maps ( t, s ) : Γ → A are defined for any arrow ( x, y ) ∈ Γwith x · y ∈ A , as s ( x, y ) = y, t ( x, y ) = x ;(ii) The objection map o : A → ∆ ⊂ Γ is define as x ( x, x );(iii) The composition of arrows is partially defined, for ◦ : Γ × Γ → Γ , ( x, y ) ◦ ( u, z ) =( x, z ) whenever y = u ;(iv) The inverse map is defined as i : Γ → Γ of an arrow ( x, y ) is ( y, x ).Second, the groupoid satisfies the given linear conditions; for given ( x, v ) , ( x, z ) , ( y, z ) ∈ Γ, we have two linear subspaces defined by these arrows of the groupoid on A asfollowsΓ( x, · ) = { y ∈ A : ( x, y ) ∈ Γ } ⊂ A ; Γ( · , z ) = { x ∈ A : ( x, z ) ∈ Γ } ⊂ A . They are linear spaces, for given ( v, z ) ∈ Γ( x, · ) and ( x, y ) ∈ Γ( · , z ), it follows that( x, αv + βz ) ∈ Γ; which implies x · ( αv + βz ) = α ( x · v ) + β ( x · z ) ∈ A α ( x, v ) + β ( x, z ) ∈ Γ. This shows that Γ( x, · ) is a linear space.Also, given ( αx + βy, z ) ∈ Γ( · , z ), by linearity we have ( αx + βy ) · z = α ( x · z ) + β ( y · z ) ∈ A . Thus, we have α ( x, z ) + β ( y, z ) ∈ Γ( · , z ) for all α, β ∈ K ; showing alsothat Γ( · , z ) is a linear space. Remark 1.3.
The linear conditions imply that the subsets Γ( x, · ) , Γ( · , z ) of rightand left multipliers respectively, of any element of the linear space A are linearsubspaces of A determined by the relation Γ = { ( x, y ) : x · y ∈ A} . In groupoidterms, every arrow ( x, y ) ∈ Γ determines two linear subspaces Γ( x, · ) , Γ( · , y ) in A .These subspaces are related to the source and target fibres as follows s − ( y ) = Γ y = { ( u, y ) ∈ Γ : u ∈ Γ( · , y ) } ; t − ( x ) = Γ x = { ( x, v ) ∈ Γ : v ∈ Γ( x, · ) } . Thus, the right multipliers define the target fibre, while the left multipliers definethe source fibre. We extend this formulation to the definition of a partial ∗ -algebrain [7] as follows. Definition 1.4.
Given a partial algebra ( A , Γ , · ), a partial ∗ -algebra or an involutivepartial algebra is a quadruplet ( A , Γ , · , ∗ ) such that A is an involutive linear spacewith involution ∗ , with ( y ∗ , x ∗ ) ∈ Γ whenever ( x, y ) ∈ Γ and then ( x · y ) ∗ = y ∗ · x ∗ . Definition 1.5. A partial subalgebra (respectively partial ∗ -subalgebra ) is a subspace(respectively a ∗ -invariant subspace) B of A such that x · y ∈ B whenever x, y ∈ B and ( x, y ) ∈ Γ. Proposition 1.6.
Given the partial ∗ -algebra ( A , Γ , · , ∗ ) as defined above, there isa corresponding groupoid Γ ⇒ A defined by the equivalence relation Γ = { ( x, y ) ∈A × A : x · y ∈ A} on A , such that the compatibility of · and ∗ in Γ implies ( x · y ) ∗ = y ∗ · x ∗ .Proof. The proof follows immediately from that of the partial algebra, with theaddition of the involutive map ∗ : A → A compatible with the partial multiplicationand · : Γ → A defining the equivalence relation Γ.4 emark 1.7. We make the following remarks based on the formulation.(1) We can define the involutive map ∗ : A → A to be the inverse i : A → A . Then x ∗ = x − and ( x · y ) ∗ = ( x · y ) − = y − · x − = y ∗ · x ∗ . Thus, ( x, y ) ∗ = ( y, x ).(2) The groupoid of pairs G = A × A is also a partial algebra since it satisfies therelation. Thus, by the transitivity of the groupoid, it shows that every algebra is apartial ∗ -algebra, with inverse as involution.(3) Units in the groupoid are of the form ( x, x ) , ( y, y ) ∈ Γ, for they give identityarrows; and ( x, x ) ∗ = ( x, x ).We extend the definition of left and right multipliers of an element above to leftand right multipliers of the linear space A in terms of groupoid as follows. Definition 1.8.
We define the left and right multipliers of A respectively as:Γ( · , A ) = { x ∈ A : ( x, u ) ∈ Γ , ∀ u ∈ A} Γ( A , · ) = { y ∈ A : ( v, y ) ∈ Γ , ∀ v ∈ A} . Since the source fibre s − ( y ) = Γ y is defined by the left multipliers of y , namely,Γ( · , y ); we let s − ( y ) = Γ( · , y ). Similarly, we let the target fibre t − ( x ) = Γ( x, · ).Then for all x ∈ Γ( · , A ) , y ∈ Γ( A , · ) respectively Γ( x, · ) = Γ( · , y ) = A .From the formulation, it follows that for any other element u Γ( · , A ) or v Γ( A , · ), we have proper subsets Γ( u, · ) ⊂ A and Γ( · , v ) ⊂ A . It is also true that A is invariant under the iteration of a left multiplier x ∈ Γ( · , A ) (respectively a rightmultiplier y ∈ ( A , · )). This gives rise to the following result on the ideal structureof the left and right multipliers. Proposition 1.9.
Given the left Γ( · , A ) and right Γ( A , · ) multipliers of the lin-ear space, their intersection Γ( · , A ) ∩ Γ( A , · ) form an ideal of the partial ∗ -algebra ( A , Γ , · , ∗ ) .Proof. By definition Γ( · , A ) (respectively Γ( A , · )) is closed (or invariant) under right(respectively left) multiplication by A ; that is, Γ( · , A ) × A → Γ( · , A ) and A × Γ( A , · ) → Γ( A , · ). Thus, the restriction of the partial multiplication A × A → A tothese subsets Γ( · , A ) × A → A and A × Γ( A , · ) → A makes it a full multiplicationor product. 5 orollary 1.10. The left multipliers Γ( · , A ) (respectively right Γ( A , · ) ) is a left(respectively right) module of the partial ∗ -algebra ( A , Γ , · , ∗ ) or left (right) Γ -module. When the linear space A is locally convex with a Hausdorff locally convex topol-ogy τ , a locally convex partial ∗ -algebra is defined by [7] on A as follows. Definition 1.11. [7] A locally convex partial algebra (respectively a locally convexpartial ∗ -algebra ) is a quadruplet ( A , Γ , · , τ ) (respectively a quintuplet ( A , Γ , · , ∗ , τ ))comprising a partial algebra ( A , Γ , · ) (respectively a partial ∗ -algebra ( A , Γ , · , ∗ ))and a Hausdorff locally convex topology τ such that ( A , τ ) is a locally convex spaceand the maps x x · y and x z · x are continuous for every y ∈ Γ( A , · ) and z ∈ Γ( · , A ) (respectively the maps u u ∗ , x x · y and x z · x are continuousfor every u ∈ A , y ∈ Γ( A , · ) and z ∈ Γ( · , A )).To realize the locally convex partial ∗ -algebra in groupoid terms, we need the fol-lowing definitions of the topological groupoid and locally convex topological groupoid.The latter is derived from a modification of the definition of locally convex Liegroupoid given in ([27], 1.1). Definition 1.12.
A topological groupoid is a groupoid Γ ⇒ A such that its setof morphisms Γ and set of objects A are topological spaces, and its composition m : Γ × Γ → Γ, source and target t, s : Γ → A , objection o : A →
Γ, and inversion i : Γ → Γ maps are continuous, with the induced topology on the set of composablearrows Γ (2) from Γ × Γ. Definition 1.13. [27] Let G = (Γ ⇒ A ) be a groupoid over A with the source andtarget t, s : Γ → A projections. Then G is a locally convex (and locally metriz-able) topological groupoid over A if (i) A and Γ are locally convex spaces; (ii) thetopological structure of G makes s and t continuous; i.e. local projections; (iii) thepartial composition m : Γ × s,t Γ → Γ, objection o : A →
Γ, and inversion i : Γ → Γare continuous maps.
Proposition 1.14.
The locally convex partial ∗ -algebra ( A , Γ , · , ∗ , τ ) defined abovegives rise to the locally convex groupoid Γ ⇒ A such that the space of arrows Γ isgiven by the relation Γ = { ( x, y ) ∈ A × A : x · y ∈ A} on A . roof. From the above definitions, given that ( A , τ ) is a Hausdorff locally convextopological space; the relation Γ ⊂ A × A has the subspace locally convex topologyinduced from A × A . This follows from the continuity of the partial multiplication ( · ) defining the relation Γ which preserves local convexity. Also, the continuity ofinvolution ∗ is by definition; and the defining maps of the groupoid; the target andsource maps t, s : Γ → A , the inverse i : Γ → Γ, and the composition of arrows m : Γ × Γ → Γ are all continuous maps. Thus, Γ ⇒ A is a locally convex topologicalgroupoid representing the locally convex partial ∗ -algebra ( A , Γ , · , ∗ , τ ).In the following section we show that the locally convex topological groupoidΓ ⇒ A is a Lie groupoid modelled on the locally convex topological vector space A . ( A , τ ) According to Hideki Omokri [22], a Lie group is a group in which the infinitesimalneighbourhood of the identity element generates the connected component of thegroup containing the identity element. Since we are dealing with transformationsof a linear space, the transformation generating the connected component (the Liegroup) can be modified in such a way that any fixed element of the space beingtransformed can give the identity transformation. In the light of this possibility, wecan modify the above description of a Lie group as a group in which the infinitesimalneighbourhood of any element generates the connected component of the group oftransformations giving an identity transformation on the fixed element.We are interested in such (connected) components K of a transformation groupwhich is generated on an open subspace that is dense in the original locally convexlinear space A . The corresponding infinitesimals are properly the unbounded op-erators on the linear space. Their algebras are the partial and partial ∗ -algebras.Hence, we have to consider the open subspace of right multipliers or the target fi-bre Γ( x, − ) (or the source fibre Γ( − , x )) of the groupoid we have constructed fromthe relation Γ on A . As we have noted above, these subspaces are maximal when x ∈ Γ( − , A ) (respectively x ∈ Γ( A , − ). 7urthermore, ’to generate’ a component, as noted by Omokri ([22], p.1), impliesvarious means of ’integrating’ an infinitesimal quantity to give a finite quantity.These various means may be solving an ordinary differential equation, solving apartial differential equation of evolution, product integral, or Feynman path integral.The task therefore is to define the infinitesimal neighbourhood of identity (or of anyelement) in the transformations of the locally convex Hausdorff topological space A defined by the partial multiplication ( · ) giving rise to the relation Γ.Since the space is locally convex, we employ Alain Connes’ technique for a smoothmanifold as in [4]; whereby every point x in the manifold M is identified with aconvex neighbourhood ( x, ε ), where ε ∈ [0 , x, ε ) ∈ M × [0 ,
1) is then used to relate the points to the infinitesimal generators acting at x , as element of the tangent space T x M and the tangent bundle T M .In this case, the pair ( x, ε ) ∈ A × (0 ,
1) can be said to represent an open convexneighbourhood of each point x ∈ A . This follows because the convexity of a set A ⊂ A implies that it is invariant under a homothetic transformation centred atany point a ∈ A with ratio ε ∈ (0 , § A , we have that ( x, ε ) ∈ A × [0 ,
1) encodes open convex neighbourhoods of everypoint x in A .Next, we extended this construction to the groupoid Γ ⇒ A determined by therelation Γ = { ( x, y ) ∈ A × A : x · y ∈ A} . The set of units which is the image of theobjection map o : A → Γ o = { ( x, x ) ∈ ∆ ⊂ A × A} ≃ A is locally convex. The localconvexity of A and the fact that ( A , Γ , · , ∗ , τ ) is a locally convex partial ∗ -algebraassures ( x, x ) ∈ Γ , ∀ x ∈ A , since according to [5], a locally convex topological spaceis an algebra if and only if for any 0-neighbourhood U in ( A , τ ) there is another 0-neighbourhood V satisfying V = { x · y : x, y ∈ V } ⊂ U . Equivalently, thereis a 0-neighbourhood filter in ( A , τ ) with a basis consisting of sets that are stablewith respect to multiplication; that is, for a 0-neighbourhood U , there is stable0-neighbourhood V satisfying V ⊂ V ⊂ U .These imply the compatibility of the algebraic and topological structures in( A , Γ , · , ∗ , τ ), at least in the convex neighbourhood of its points. Thus, the objection8ap x ( x, x ) ∈ Γ is defined since x · x ∈ A and the set of arrows Γ ⊂ A × A is alsolocally convex since it is determined by the (partial) multiplication. Given the par-tial multiplication which presupposes compatibility of the algebraic and topologicalstructures, we define a smooth structure on the algebra as follows.Given the local convexity of A there exists a locally convex neighbourhood ( x, ε )of a point x ∈ A such that we can define a smooth or Lie groupoid by employingConnes’ representation of the locally convex neighbourhoods of the points of A as( x, ε ) , ε ∈ [0 , x, y ) ∈ Γ as ( x, y, ε ) ∈ Γ × [0 , ⊂ A × A × [0 , x, y, ε ) ◦ ( y, z, ε ) = ( x, z, ε ) for ε ∈ [0 , , x, y, z ∈ A . This defines a smooth structure on G := Γ × [0 , ⇒ A × [0 , Proposition 2.1.
The map T A × [0 , ⇒ A → Γ × [0 , ⇒ A × [0 , , defined onthe arrows Γ( − , x ) × [0 , → T x A is a groupoid isomorphism.Proof. First, the tangent bundle T A is a groupoid T A ⇒ A which is a union ofgroups T x A , with x ∈ A as the identity element, such that the objection mapis A ∋ x ( x, ∈ T x A . So the set of arrows of the groupoid T A ⇒ A is T A = { ( x, X ) : x ∈ A} , where X is a vector field or infinitesimal generator at thepoint x ∈ A and X | x is a tangent vector to A at x . The target and source mapsfor an arrow are defined s ( x, X ) = ( x, t ( x, X ) = ( x, x is given as ( x, X ) ◦ ( x, X ) = ( x, X + X ).The equivalence of these groupoids is now established as follows. By definition,the smooth or Lie groupoid Γ × [0 , ⇒ A × [0 ,
1) has an open convex neighbhour-hood of the identity arrow ( x, x ) as ( x, x, ε ) which is constructed from the locallyconvex partial ∗ -algebra. From the locally convex infinitesimal neighbourhood ofthe generators ( x, X, ε ) of the tangent space to A at x , contained in the groupoid T A × [0 , ⇒ A , the exponential map generates the open submanifold Γ( − , x ) of9he Lie groupoid as followsexp : T A × [0 , → G ( − , x ) × [0 , x, X, ε ) ( x, exp x ( − εX ) , ε )where ( x, X, ( x, x ) ∈ ∆ ⊂ A × A ≃ A is an identity. Corollary 2.2.
The locally convex groupoid G := Γ × [0 , ⇒ A × [0 , generatedfrom the locally convex infinitesimal neighbourhoods of T A ⇒ A is a Lie groupoid.Proof. The proof of this follows immediately from the proof of the preceding result.Reference can also be made to Connes’ construction in ([4],Ch.2, Sect.5).
Lemma 2.3.
The connected component K of arrows generated by the locally convexneighbourhood of an identity arrow ( x, x, ε ) , which corresponds to the locally convexneighbourhood ( x, ε ) of the point x ∈ A , has the action of [0 , . It is thereforeisomorphic to the G ( x, x ) -space Γ( x, − ) which is a subspace of Lie groupoid Γ ⇒ A .Proof. Since the smooth locally convex open submanifold Γ( x, − ) of the Lie groupoidΓ ⇒ A is generated by the locally convex neighbourhood ( x, ε ) of x ∈ A (or theneighbourhood ( x, x, ε ) of the identity arrow ( x, x )), it is a connected open subspaceof A × A × [0 ,
1) according to [22]. This is equivalently expressed as
A × [0 , → A .For I = [0 , I → A A , ε γ ε = ( x, y ) which is action of anet on the arrows, such that the arrows { ( x, y ) : y ∈ A} converge to the identityarrows ( x, x ) as ε →
0. This action defines a net of local bisections (to be see in thesubsequent sections). The nature of the open locally convex submanifold of arrowsΓ( x, − ) ⊂ Γ depends on each point x ∈ A . It is maximal when x ∈ Γ( − , A )-a leftmultiplier of A .Now, let K be the connected component of arrows generated by ( x, x, ε ) or ( x, ε ).It is a smooth net resulting from the smoothing action of [0 ,
1) on Γ( x, − ). The par-tial symmetry of Γ( x, − ) is encoded by the net K . Since ( x, εy ) ∈ Γ( x, − ) , ε ∈ [0 , x, − ) × K → Γ( x, − ) encodes the sectional dynamicalsystem of the Lie groupoid Γ. Thus, [0 , ≃ K . When the endpoints of the inter-val [0 ,
1) are identified, the net action is equivalent or similar to the action of theisotropy Lie group G ( x, x ) on Γ( x, − ). 10he inverse operation to the generation of the smooth connected component K isthe limit process on the derived filters {F → x } of the net action of [0 , x ε → x, y ε → x , where ( x ε , y ε ) → ( x, x ) ∈ Γand ( x, y ε ) ∈ Γ( x, − ). This implies that the net of arrows ( x ε , x ) and ( x, y ε ) aregenerated by the smooth net K on a given arrow ( x, y ) making up the locally convexlinear submanifolds Γ( x, − ) , Γ( − , x ).If we now consider the infinitesimal neighbourhood, it will follow that the con-vergence of the two nets gives G ( x, x ) = Γ( x, − ) × [0 , ∩ [0 , × Γ( − , x ); while thefilter F → x is equivalent to Γ( x, − ) × [0 , ∪ [0 , × Γ( − , x ) → G ( x, x ) which isthe isotropy Lie group of the smooth groupoid G = (Γ ⇒ A ). This is equivariant tothe convergence of the quotient( x ε → x, y ε → x, x ε − y ε ε → X, ε → → ( x, X, , for each net of arrows in Γ( x, − ) × [0 , K -action on G ( x, − ).(Cf. [4]). Remark 2.4.
We make the following remarks.First, the vector fields X ∈ T x G ( x, x ) correspond to the universal flows ϕ ( τ, x ) as-sociated with the Lie groupoid Γ ⇒ A , where ϕ (0 , x ) = x for any fixed x ∈ A . Thisimplies X | x ∈ T x (Γ( x, − )) ≃ T x (Γ( − , x )).Second, there are also local fields Y | x ∈ T x (Γ( x, − )) defined for some x ∈ A andnot for all; that is, Y | y = 0 for some y ∈ A ; these are not parallelizable (or glob-alizable through parallel transport). The maximal of these tangent spaces occurswhen x ∈ Γ( − , A ) (or Γ( A , − )). Because the subspace generated by the convexneighbourhood of such x ∈ A is maximal and dense in A , the corresponding flowsare connected to the unbounded infinitesimal generators of the Lie groupoid Γ ⇒ A .They give rise to unbounded operators as described in the introduction of [1].Third, the closed submanifolds Γ( x, y ) ⊂ Γ are G ( x, x )-spaces of the same dimen-sion with G ( x, x ), while the open (maximal) submanifold Γ( − , x ) ≃ Γ( x, − ) areΓ-modules and infinite dimensional. The structure and the symmetry of the open11nd dense submanifolds are now explored with the local bisections B ℓ ( G ) which areLie pseudogroup characterized as follows. Definition 2.5.
Given the isomorphism B (Γ) ≃ Diff( A ), defined by ϕ ↔ t ◦ ϕ ,between the set of diffeomorphisms of the base manifold of a Lie groupoid and theset of bisections of the Lie groupoid in [15], which can be restricted to the collectionof the local diffeomorphisms Diff( U ) of a trivialization U of the base manifold andthe local bisections defined on it B ℓ ( U ), we have the following properties defininga Lie pseudogroup:(1) For any t ◦ ϕ ∈ Diff( U ), t ◦ ϕ : U → A = ⇒ t ◦ ϕ | V ∈ Diff( U ), for all V ⊆ U ;(2) If L ⊆ A is open with L = S α U α , then if t ◦ ϕ : L → A with t ◦ ϕ | Uα ∈ Diff( L ),then t ◦ ϕ ∈ Diff( L );(3) Diff( U ) is closed under composition, since for any two bisections σ and τ , thehomomorphism property of the target map t implies t : σ ⋆ τ t ◦ ( σ ⋆ τ ) =( t ◦ σ ) ◦ ( t ◦ τ ); (closure of composition implies this, since σ ⋆ τ is defined whenevera diffeomorphism φ links σ and τ .)(4) The identity diffeomorphisms are in Diff( U );(5) Each t ◦ ϕ has an inverse ( t ◦ ϕ ) − . This follows because of the localization, sayat the neighbourhood of y , where ( t y ◦ ϕ ) − = ϕ − ◦ t − y ; thus, given any x in theneighbourhood of y , i.e. in U , we have ϕ − ◦ t − y ( x ) = ϕ − (Γ( x, y )) = y . Remark 2.6.
We note the fact that the smoothing action of [0 ,
1) has made thelocally convex space A which is the base space of the Lie groupoid a smooth manifold.Hence the objection map o : x ( x, x ) is an embedding into a smooth manifold.So, the set B ℓ ( U ) of local bisections of a trivialization U is a Lie pseudogroup.We have successfully modelled the locally convex partial ∗ -algebra ( A , Γ , · , ∗ , τ )as a locally convex Lie groupoid Γ ⇒ A which is a smooth locally convex manifoldmodelled on the locally convex topological space A given the relation Γ. This for-mulation gives the isomorphism Γ ⇒ A ≃ C ∞ ( A ) | Γ . The following definitions of asmooth manifold modelled on a locally convex topological space and the diffeomor-phisms arising from the model apply to the formulation.12 efinition 2.7. [9] A smooth manifold M modelled on a locally convex topologicalspace A is a Hausdorff topological space, together with a set G of homeomorphismsfrom open subsets of M onto open subsets of A , such that the domains cover M and the transition maps are smooth. Definition 2.8. [9] Let E and F be real locally convex spaces, U ⊆ E be open,and f : U → F be a map. For x ∈ U and y ∈ E , let ( D y f )( x ) := ddt | t =0 f ( x + ty )be the directional derivative (if it exists). Given k ∈ N ∪ {∞} , the map f iscalled C k if it is continuous, the iterated directional derivatives d j f ( x, y , · · · , y j ) := D yj · · · D y f ( x ) exist for all j ∈ N such that j ≤ k, x ∈ U and y , · · · , y j ∈ E , andall of the maps d j f : U × E j → F are continuous. ( f is smooth if k = ∞ .)As hinted in the introduction, the existence of a Lie group structure on the setof bisections of the Lie groupoid Γ ⇒ A , which is related and derivable from thecanonical smooth structure on the manifold of arrows Γ, is a result of the localconvexity of A . This understanding that the Lie groupoid Γ ⇒ A is a smoothmanifold modelled on a locally convex topological space A carrying a partial ∗ -algebra given by the relation Γ is subsequently employed to define its smooth systemof Haar measures. The diffeomorphims and their differentials constitute differential forms on a smoothmanifold. According to [19], differential forms are used to express various geometricstructures on manofolds.To obtain certain geometric ”invariants”, appropriate oper-ations are applied to differential forms which are integrable on manifolds. We brieflydescribe the operations to be applied to differential forms to obtain Haar measuresfor our Lie groupoid.We first propose that unbounded operators correspond to open and dense locallyconvex subspaces Γ( x, − ) , Γ( − , y ). We shall also linke these to locally convex neigh-bourhoods and their corresponding infinitesimal generators. There is need to con-nect these to differential forms which are integrable objects of a (smooth) manifold.13ince we have already seen that Γ( x, − ) and Γ( − , y ) are locally convex topologicalspaces given the Lie groupoid Γ ⇒ A defined by the relation Γ = { ( x, y ) : x · y ∈ A} on A , we consider the following smooth functions associated with the parameteri-zaton of the open submanifold of arrows Γ( x, − ). φ ( x + ty ) : Γ( x, − ) → IR m (1)where m is the dimension of the maximal closed neighbourhood Γ( x, y ) ⊂ Γ( x, − )containing a trivialization x . These functions constitute the set of homeomorphismsfrom open subsets of Γ( x, − ) of the Lie groupoid onto open subsets of IR m . Fromthis background, we employ the definition of smooth measures on a smooth manifoldas given in [8]. Definition 3.1.
Let M be a smooth manifold of dimension m . Then a smoothmeasure on M is a Borel measure µ which is given in local coordinate x as dµ = φ x dx ,where φ x is a nonnegative smooth function. The change of coordinates for thesmooth measure µ is given as φ x = | det (cid:18) ∂y∂x (cid:19) | φ y = ⇒ dµ = | det (cid:18) ∂y∂x (cid:19) | φ y dy (2)This is also related to density on the smooth manifold Γ ⇒ A , which is asection of the smoothly varying line bundle on the Lie groupoid. From the above,the Borel measures on Γ ⇒ A are differential forms generated or spanned by dx in the coordinate system x of Γ( x, y ) ⊂ Γ( x, − ). It follows that the measures varyaccording to the smooth function φ x ; so that φ x represents a density (or its element)in each coordinate system x , and its variation in coordinate change is governed by(2). This helps us to define both a differential form and a density on Γ ⇒ A asfollows. Definition 3.2.
A differential form or an m -form on an m -dimensional manifoldΓ( x, y ) ⊂ Γ( x, − ) is a section of the line bundle whose transition functions aredet( ∂y∂x ). So, given an m -form ω x for the coordinate system x or ( U, φ x ) in Γ( x, − ),its transformation in the y coordinate system of Γ( y, − ) is ω y ; and the two are14elated by the formula ω x = det (cid:18) ∂y∂x (cid:19) ω y . (3)The usual notation for the differential form is ω = ω x dx ∧ · · · ∧ dx m . Definition 3.3.
A density is a function on coordinate charts of Γ( x, − ) which be-comes multiplied by the absolute value of the Jacobian determinant in the change ofcoordinates φ x = | det( ∂y∂x ) | φ y or between the coordinate charts on the Lie groupoidΓ ⇒ A .Thus, orientation is the only difference between a density φ x and a differentialform ω x ; the former is without an orientation, while the latter is with orienta-tion. The two coincide when there is a restriction to a coordinate system withJocabian matrices of positive determinant. So, the differential forms which aresmooth measures are identified with nonnegative densities on the smooth manifoldof arrows/maps of the Lie groupoid Γ ⇒ A .Based on the treatment of [8], we see that any density φ (a diffeomorphism) onthe manifold of arrows Γ of the Lie groupoid Γ ⇒ A defines, at least locally, asmooth signed (or complex) measure µ on Γ ⇒ A , so that the following integralsare well defined for any compact set K ⊂ Γ; Z K φ = µ ( K ); and Z f φ = Z f dµ ; (4)for any f ∈ C c (Γ ⇒ A ) = C c (Γ). So, the diffeomorphisms defined in (1) above arealso measures supported on compact (closed) subsets Γ( x, y ) of Γ( x, − ).When the densities (as measures) are normalized, they give rise to probabilitymeasures as given in [8]. This is done by having 0 ≤ θ ≤
1, which presents a θ -density on the Lie groupoid Γ ⇒ A to be a section of the line bundle whosetransition functions on Γ( x, − ) ∩ Γ( y, − ) are | det (cid:16) ∂y∂x (cid:17) | θ . Given this normalization,a 1-density becomes a standard density while a 0-density is a smooth function onΓ ⇒ A . When 0 < θ < p = θ − and φ is a θ -density, then | φ | p is well defineddensity and therefore integrable over the Lie groupoid Γ ⇒ A .Alternatively, since densities are also smooth functions on the Lie groupoid Γ ⇒ A , the set of θ -densities φ (usually denoted | Ω | p (Γ), with p = θ − ) satisfying the15orm condition || φ || p = (cid:18)Z | φ | p (cid:19) p < ∞ is a normed linear space with L p (Γ) as its completion. In other words, | Ω | p (Γ) ⊂ L p (Γ). The completion is the intrinsic L p -space of the normed space | Ω | p (Γ) con-nected with the Lie groupoid Γ ⇒ A .By the duality associated to this definition of the space of densities on the Liegroupoid Γ ⇒ A as a dense subset of L p (Γ) which also follows from [8], the productof two -densities is 1-density. Thus, an inner product is defined on the space of -density with compact support as h ω, η i = Z Γ ω ¯ η. (5)This makes the space of -densities | Ω | (Γ) ⊂ L (Γ) a pre-Hilbert space, with L (Γ)as its canonically associated Hilbert space completion. With this understanding, thedefinition of smooth system of Haar measures on a Lie groupoid follows. Since we have shown the groupoid model of the locally convex partial ∗ -algebra( A , Γ , · , ∗ , τ, ) to be a Lie groupoid, and the diffeomorphisms φ ( x + ty ) : Γ( x, − ) → IR m show that Γ( x, − ) is locally compact; it follows that it has a left Haar systemof measures defined by Paterson as follows. Definition 3.4. (cf. [24]) A left Haar system for the Lie groupoid G := Γ ⇒ A isa family { µ x } x ∈A , where each µ x is a positive regular Borel measure on the locallycompact (convex) Hausdorff space Γ( x, − ), such that the following three axioms aresatisfied.(i) the support of each µ x is the t -fibre Γ( x, − );(ii) for any f ∈ C c (Γ), the function f o ( x ) = Z Γ( x, − ) f dµ x belongs to C c ( A );(iii) for any γ ∈ Γ and f ∈ C c (Γ), Z Γ( s ( γ ) , − ) f ( γη ) dµ s ( γ ) ( η ) = Z Γ( t ( γ ) , − ) f ( κ ) dµ t ( γ ) ( κ ).16his is in line with the smoothness condition for a Lie groupoid and our consider-ation above; it is also related to a choice of appropriate local coordinates making theRadon-Nikodym derivatives of the family { µ x } strictly positive and smooth. Hence,the smooth left Haar system of measures on the Lie groupoid Γ ⇒ A is unique upto equivalence due to the smooth net K -action; which means they are in the sameclass of measures. Thus, the system is isomorphic to the strictly positive sections ofthe 1-density line bundle Ω ( T γ (Γ( x, − ) ∗ ). This is established as follows.By the bijection between (cross) sections of a fibre bundle and the set of mapsfrom a base space to the fibres, a section ϕ uniquely defines a function from the basespace to the fibre f : A →
Γ. Thus, a section ϕ is of the form ϕ ( x ) = ( x, f ( x )) , x ∈A , ( x, f ( x )) ∈ Γ ⊂ A × A . (Cf. [11]). By definition of the partial product, thedefinition of the function f : A →
Γ, and subsequently, the section ϕ is localized.Thus, we have ϕ ( x ) = f ( x + εy ) by the smooth action of [0 , ε ∈ [0 , ϕ is a section to the source map s : Γ → A , which means s ◦ φ = I A , it is required to define a diffeomorphism with the target map; that is, t ◦ ϕ : A → A for it to be a bisection of the Lie groupoid Γ ⇒ A . The partialproduct forces it to be a local bisection ϕ ∈ B ℓ ( G ) ⊂ B ( G ) by restricting to an openneighbourhood t ( N x ) = t (( x, ε )) ⊆ A of x ∈ A where the map ϕ t ◦ ϕ is defined,the target map t x : Γ( − , x ) → A is a surjective submersion, and t ◦ ϕ : N x → ( t ◦ ϕ )( N x ) a diffeomorphism. (Cf. [15], Definition 1.4.8).In addition, since the set of (cross) sections of a fibre bundle constitutes a moduleover the ring of continuous functions from the base space to the fibres A →
Γ, whichare manifold-valued functions, it follows that the local bisections B ℓ ( G ) constitutea module over the arrows which are continuous functions uniquely determined bythe local bisections, and a Lie pseudogroup by the smooth action of [0 , ϕ t ◦ ϕ, B ℓ ( G ) → Diff( N x ) which preserves the partition of A underthe relation Γ, we can replace the arrows γ ∈ Γ with the local bisections ϕ ∈ B ℓ ( G ).Thus, the partial product structure of Γ ⇒ A is encoded by the local bisections B ℓ ( G ) ⊂ B ( G ). 17he smooth chart defined by (1) lays bare the structure of this manifold of sec-tions (or arrows), and also defines Borel measures on it as given above. Thus, the useof the open (convex) neighbourhood N x ≃ ( x, x, ε ) ⊂ Γ( x, − ) as a t -fibrewise prod-uct agrees with the definition of the smooth Haar system using the diffeomorphismsof (1), which are also connected to the bisections B ℓ ( G ). Definition 3.5. (cf. [24]) Let N x be an open subset of the Lie groupoid G := Γ ⇒ A . Since t : Γ( x, − ) → A is a submersion, it is open. So, t ( N x ) is open in A . Thepair ( N x , φ ) is called a t -fibrewise product if there exists an open subset W of IR m containing 0, and φ is a diffeomorphism from N x onto t ( N x ) × W preserving t -fibresin the sense that p ( φ ( γ )) = t ( γ ) , ∀ γ ∈ N x , where p is the projection on the firstcoordinate of t ( N x ) × W . Definition 3.6.
A smooth left Haar system for the Lie groupoid Γ ⇒ A is a family { µ x } x ∈A where each µ x is a positive, regular Borel measure on the submanifoldΓ( x, − ) such that:(i) If ( N x , φ ) is a t -fibrewise product open subset of G , N x ≃ t ( N x ) × W , and if µ W = µ | W is Lebesgue measure on IR m , then for each x ∈ t ( N x ), the measure µ x ◦ φ x is equivalent to µ W , since φ x : N x ∩ Γ( x, − ) → IR m is a diffeomorphismand their R-N derivative is the function Φ( x, w ) = d ( µ x ◦ φ x ) /dµ W ( w ) belonging to C ∞ ( t ( N x ) × W ) and is strictly positive.(ii) With Φ | A , we have Φ o ( x ) = Z Γ( x, − ) Φ dµ x , which belongs to C c ( A ).(iii) For any γ ∈ Γ and f ∈ C ∞ c (Γ), we have Z Γ( s ( γ ) , − ) f ( γη ) dµ s ( γ ) ( η ) = Z Γ( t ( γ ) , − ) f ( ξ ) dµ t ( γ ) ( ξ )This definition makes a clear sense in the light of the smooth structure on B ℓ ( G )subsequent on the smooth structure of Γ ⇒ A , for it follows the definition of thesmooth chart ( N x , φ x ) ≃ W ⊂ IR m . So, for each x ∈ N x , µ x is positive and asmooth measure on N x ∩ Γ( x, · ); and since φ x : N x → W ⊂ IR m is a diffeomorphism, µ x ◦ φ x ∼ µ W . Thus, the Radon-Nikodym derivative d ( µx ◦ φx ) dµW varies smoothly onΓ ⇒ A by definition. 18ith these formulations, and given the normalized -densities, where | Ω | / γ isthe fibre over an arrow γ ∈ Γ with t ( γ ) = x, s ( γ ) = y ; a density φ ∈ C ∞ c (Γ , Ω(Γ))determines a functional Ω k T γ (Γ( x, · )) ⊗ Ω k T γ (Γ( · , y )) → IR . The convolution algebraof the Lie groupoid Γ ⇒ A is therefore defined on the space of sections (densities) C ∞ c (Γ , Ω(Γ)) ⊂ L (Γ) of the line bundle, with the convolution product f ∗ g of f, g ∈ C ∞ c (Γ , Ω(Γ)) given as f ∗ g ( γ ) = Z η ◦ ξ = γ f ( η ) g ( ξ ) = Z Γ( t ( γ ) , − ) f ( η ) g ( η − γ ) . (6)The involution is defined as f ∗ ( γ ) = f ( γ − ) . (7)The integral is that of sections on the manifold Γ( t ( γ ) , − ) since f ( η ) g ( η − γ ) is a1-density. Based on Paterson’s reformulation (cf. [24], Appendix F), the above isalternatively given as f ∗ g ( γ ) = ( Z ω )( ω s ( γ ) ⊗ ω t ( γ ) ) , (8)with f, g, f ∗ g ∈ C ∞ c (Γ , Ω(Γ)).As noted above, the definition of 1 / ⇒ A independent of the choice of left Haar system, since theyare intrinsic objects to the Lie groupoid. This also makes the representation of theLie groupoid independent of the choice of smooth left Haar system. As stated above,the choice of left Haar system is made relative by the equivalence established on thespace of densities C ∞ (Γ , Ω(Γ)) by the action of the smooth net K ≃ [0 ,
1) whichwe simply put as (Γ Γ ) I . We give this as a proposition as follows. Proposition 3.7.
The convolution algebra C ∞ (Γ , Ω(Γ)) has a smoothing action of K ≃ [0 , .Proof. Because Ω(Γ) γ are trivial line bundles on Γ, we have Ω(Γ) ∼ = Γ × IR = IR Γ(or Γ × C = C Γ). Therefore, C ∞ c (Γ) can be identified with the space of smoothsections C ∞ c (Γ , Ω(Γ)) which are the smooth functions Γ → Ω(Γ) ≃ IR Γ; i.e. thebisections with the action of [0 , t -fibres. 19he equivalence defined by this action makes the choice of left Haar system ofmeasures defined by the smooth functions on the arrows not to be unique. Thiswas what Paterson [24] meant by positing that Alain Connes’ approach to the con-volution algebra in [4] makes the definition of the convolution algebra, and therepresentations (the unitary and irreducible) of a Lie groupoid G independent of thechoice of smooth left Haar system. The above proposition says that the infinitesimalapproach is equivalent to smoothing net K -action on the space of local bisectionsisomorphic to the arrows in Γ( x, − ). Let ν be a probability measure on A . Then a suitable Hilbert space H for therepresentation of the Lie groupoid Γ ⇒ A arising from the locally convex partial ∗ -algebra ( A , Γ , · , ∗ , τ ) is the Hilbert bundle defined as a triple ( A , H , ν ), where A is thelocally convex Hausdorff space, and ν is an invariant or quasi-invariant probabilitymeasure on A , and H is the collection of Hilbert spaces { H x = L (Γ( x, − ) , µ x ) } indexed by the elements of A , which is a Hilbert bundle over A .The definitions of sections of the Hilbert bundle H and their nets which de-termine the inner product norm on the bundle are done in accordance with theseformulations. The identification of the arrows with the local bisections modifiesthe definition of the unitary operators ℓ ( ϕ ) : H s ( γ ) → H t ( γ ) . Subsequently, a localbisection ϕ ∈ B ℓ ( G ) can also be used to define the C ∗ -representation given by the(densities) map γ f ( γ ) ℓ ( γ ) for each x ∈ A , f ∈ C c (Γ). In this case, the map ϕ f ( γ ) ϕ is a section of the Hilbert bundle (cf. [24]). Thus, given a trivialization U at x ∈ A , the Hilbert space H x = L (Γ( x, − ) , µ x ) can also be given in terms ofthe local bisection H x = L ( B ℓ ( U ) , µ x ).The image of the sections are functions defined on the arrows terminating at x ∈ A , which are square integrable. The net of smooth sections follows on the net oflocal bisections which has a smooth structure (of a Lie pseudogroup) as comparedto the sequence of sections defined in [24]. The inner product h ϕ ( x ) , ϕ ( x ) i is the20iffeomorphism-invariant product of two local bisections defined as Z ( ϕ ⋆ ϕ )( x ) dν ( x ) = Z ϕ ( t ◦ ϕ ( x )) ϕ ( x ) dν ( x ) . This takes care of the convolution of two densities and the diffeomorphism invarianceof integration on a smooth manifold. Following [24], a fundamental net is thereforedefined as follows.
Definition 3.8.
A net ( ϕ n ) of sections is said to be fundamental if for each pairof indices m, n the function x
7→ h ϕ m ( x ) , ϕ n ( x ) i = Z ϕ m ( t ◦ ϕ n ( x )) ϕ n ( x ) dν ( x ) is ν -measurable on A ; and for each x ∈ A , the images ϕ n ( x ) of the net span a densesubspace of H x = L (Γ( x, − ) , µ x ). Proposition 3.9.
A net ( ϕ n ) of local bisection B ℓ ( U ) of Γ ⇒ A is fundamental.Proof. First, bisections ϕ for the Lie groupoids are sections satisfying the definitionabove. Second, since bisections are used to defined left translations L ϕ on a Liegroupoid (see [15]), the image of the net ( ϕ n ) span a dense subspace of the Hilbertspace H x = L (Γ( x, − ) , µ x ) by the openness of the target map t : Γ( x, − ) → A which forms a net of (local) diffeomorphisms t ◦ ϕ n . Finally, the transitive actionof Γ on Γ( x, − ) also points to the denseness of the span of a net of bisections, for arestriction of the left translation by a bisection is open as stated in (1 .
4) of [15].
Remark 3.10. (see [24]) The above result points to the relation between elementsof the base space x ∈ A and the fundamental nets ( ϕ n ), such that:(1) The smooth (fundamental) net ( ϕ n ) can be considered the orthonormal basis ofthe bundle since the Gram-Schmidt process can be used to convert the image of thenet { ϕ n ( x ) } to an orthonormal basis for H x = L (Γ( x, − ) , µ x ) for each x ∈ A .(2) A section ϕ is measurable when the action of a fundamental net ( ϕ n ) on it byinner product is measurable; that is, each function x
7→ h ϕ ( x ) , ϕ n ( x ) i of the net ismeasurable. It follows that the smooth fundamental net ( ϕ n ) defines the notion ofmeasurability for sections. This extends the connection established between localbisections and the sections of the Hilbert bundle H to the sections of line bundledefining densities. 213) Thus, the Hilbert bundle H = L ( A , { H x } , ν ) is the space of measurable sec-tions ϕ with a relation ∼ defined by the convergence of nets ϕ n , which defines a ν -integrable function x
7→ || ϕ ( x ) || , with inner product h ϕ, ψ i = Z A h ϕ ( x ) , ψ ( x ) i dν ( x ).The related norm is || ϕ || = Z A h ϕ ( x ) , ϕ ( x ) i dν ( x ), while the L -norm is || ϕ || = Z A || ϕ ( x ) || dν ( x ). The net convergence constitutes the measurable sections as gen-eralized quantities, as defined in [18]. This gives the following corollary. Corollary 3.11.
The relation ∼ defined by the net ϕ n constitutes a class [ ϕ ] ofsections with an action of the smooth net K .Proof. This follows from the convolution action of K on the local bisection ϕ whichdefines the net of bisections ( ϕ n ). Thus, the class [ ϕ ] of a section ϕ has the innerproduct action h ϕ ( x ) , ϕ n ( x ) i of a fundamental net ( ϕ n ) in terms of measurability.Thus, the fundamental net ( ϕ n ) spans a dense subspace of each L (Γ( x, − ) , µ x )on any x ∈ A , and defines the classes of sections of the Hilbert bundle H = L ( A , { H x } , ν ), with each class a net ϕ n of sections converging to a measurablesection ψ .As was noted earlier, the fibres are not all of same dimension for the Lie groupoid G = Γ ⇒ A ; the right multipliers Γ( x, − ) and left multipliers Γ( − , x ) are not thesame for every x ∈ A . We are interested in the infinite dimensional fibres whichare inductive limit of these fibres. Thus, we have a series of containment for theHilbert space H x = L (Γ( x, − ) , µ x ) : x ∈ A , such that H x ⊃ H x ⊃ · · · ⊃ L (Γ( x, x ) , µ x ), where the right multipliers are also ordered Γ( x , − ) ⊃ Γ( x , − ) ⊃· · · (and respectively the left multipliers Γ( − , x ) ⊃ Γ( − , x ) ⊃ · · · ). We need thefollowing proposition to give the result on the inductive system of locally convex Liegroupoids. Proposition 3.12. [15] Let Γ ⇒ A be a Lie groupoid, and σ : U → V be a diffeo-morphism from U ⊂ Γ open to V ⊆ Γ open, and let f : B → C be a diffeomorphismfrom t ( U ) = B ⊆ A to t ( V ) = C ⊆ A , such that s ◦ σ = s, t ◦ σ = f ◦ σ and ( γη ) = σ ( γ ) η whenever ( γ, η ) ∈ Γ ∗ Γ , γ ∈ U and γη ∈ U . Then σ is the restrictionto U of a unique local translation L ϕ : Γ( B, − ) → Γ( C, − ) where ϕ ∈ B ℓ ( U ) . Proposition 3.13.
The orbits of Γ by the left translation of a bisection L ϕ definesinductive system of Lie groupoids, which reflects as a connected path on A .Proof. Since the partial algebra structure is encoded by the local bisections B ℓ (Γ),given that the target map t restricted to a source fibre t x : Γ( − , x ) → A is ofconstant rank, it gives a diffeomorphism t ◦ ϕ : A → A . This means that the lefttranslation L ϕ is transitive on the fibre. Hence, there exists C ⊂ A such that Γ( C, x )is closed under left multiplication or translation.By definition, Γ defines a relation on A . A conjugate class [ x ] of x ∈ A by thisrelation is the set { y ∈ A : x · y ∈ [ x ] ⊆ A , ( x, y ) ∈ Γ } ; and x · x ∈ [ x ]. Thisdefines an inductive system of locally convex partial ∗ -algebras as follows. Let I bea directed set, with order denoted by ≫ . Given the Lie groupoid Γ ⇒ A , we definea net of subsets of Γ (arrows) { Γ α } α ∈ I of the Lie groupoid such that the followingare satisfied.(1) Γ α ⊂ Γ, for each α , and Γ α ⊆ Γ β for β ≫ α and A α = A β , α = β ;(2) If α, β ∈ I and β ≫ α , then Γ α ⇒ A α is a subgroupoid or a restriction of theLie groupoid for A α ⊂ A .(3) The natural homomorphic embeddings of A α → A and A α → A β is by theextension of a left translation by ϕ ; that is, L ϕ ( γ ) = σ ( γ ), where σ is a restrictionof the left translation L ϕ , ϕ ∈ B ℓ (Γ). Similarly, we can write the left translation L ϕ as σ α and σ β which are the embedding Γ α → Γ and Γ β → Γ respectively.Then σ βα : Γ α → Γ β which satisfies the cocycle condition σ γα = σ γβ ◦ σ βα and σ β ◦ σ βα = σ α for γ ≫ β ≫ α .(4) The linear span of S α ∈ I σ α ( A α ) is A .(5) That the system of Lie subgroupoids { (Γ α ⇒ A α , σ α ) , ( σ βα ) α,β ∈ I : β ≫ α } iscompatible with the algebraic and topological structures of the locally convex partial ∗ -algebra follows from the compatibility of these structures within a topological (Lie)groupoid. 23hus, the system is an inductive system of Lie groupoids with the Lie groupoidΓ ⇒ A as its inductive limit. Corollary 3.14.
The inductive system { Γ α ⇒ A α , σ α , σ βα , α, β ∈ I : β ≫ α } is equivalent to the inductive system { ( A α , ϕ α , τ α ) α ∈ I , ( ϕ βα ) α,β ∈ I : β ≫ α, τ } oflocally convex partial ∗ -algebras defined in [7], whose inductive limit is the locallyconvex ∗ -algebra ( A , Γ , · , ∗ , τ ) .Proof. The proof of this is an immediate consequence of the construction or formu-lation of the Lie groupoid Γ ⇒ A . For by the formulation, as we have seen above,the Lie groupoid Γ α ⇒ A α is always a subgroupoid of the Lie groupoid of pairs G where Γ = A × A . Thus, the Lie groupoid of pairs G ⇒ A on the locally convextopological space A models a locally convex ∗ -algebra ( A , Γ , · , ∗ , τ ). Given the smooth system of Haar measures { µ x } x ∈A supported on the t -fibresΓ( x, − ) as described above, each Haar measure is associated to a Hilbert space H x = L (Γ( x, − ) , µ x ) such that each arrow γ ∈ Γ( t ( γ ) , − ) defines a unitary oper-ator ℓ ( γ ) : H s ( γ ) → H t ( γ ) . This gives rise to a unitary representation of the Liegroupoid Γ ⇒ A on the space U ( H ) of unitary operators on the Hilbert bundle H = { H x } x ∈A . The unitary representation is then used in the definition of the C ∗ -representation of the groupoid convolution algebra C (Γ), which is a representationdefined by the (densities) map γ f ( γ ) ℓ ( γ ) over Γ( x, − ) for each x ∈ A , f ∈ C c (Γ)with respect to the Haar measure µ x .The unitary representation ℓ of the Lie groupoid Γ ⇒ A and the C ∗ -representationof its convolution algebra on the Hilbert space bundle H = L ( A , { H x } , ν ) followdirectly on the above formulations. As we have noted already, given the probabilitymeasure ν which is quasi-invariant on A , it defines the quasi-invariant measures m, m and their inverses m − , ( m ) − on Γ , Γ (2) respectively, and satisfy the re-quirements for measurability of inversion and product. The definitions of these as-24ociated measures m, m − , m , m o to ν and { µ x } x ∈A are given in [24]. So, followingPaterson, we define the representation of the Lie groupoid Γ ⇒ A as follows. Definition 4.1.
A representation of the locally convex groupoid Γ ⇒ A is definedby a Hilbert bundle ( A , { H x } , ν ) where ν is a quasi-invariant measure on A to whichthe measures m, m − , m , m o are associated; and for each γ ∈ Γ, there is a unitaryelement ℓ ( γ ) : H s ( γ ) → H t ( γ ) such that(i) ℓ ( x ) is the identity map on H x for all x ∈ A ;(ii) ℓ ( γ γ ) = ℓ ( γ ) ℓ ( γ ) for m -a.e. ( γ , γ ) ∈ Γ ;(iii) ℓ ( γ ) − = ℓ ( γ − ) for m -a.e. γ ∈ Γ;(iv) for any ξ, η ∈ L ( A , { H x } , ν ), the function γ
7→ h ℓ ( γ ) ξ ( s ( γ )) , η ( t ( γ )) i (9)is m -measurable on Γ.The inner product is defined since ξ ( s ( γ )) ∈ H s ( γ ) and translated by ℓ ( γ ) to ℓ ( γ ) ξ ( s ( γ )) ∈ H t ( γ ) , and η ( t ( γ )) ∈ H t ( γ ) . So the inner product is on the fibre H t ( γ ) .The representation is given as a triple ( ν, H , ℓ ). The identification between an arrowand a local bisection γ ↔ ϕ means that we can use either for the representation.These give rise to the trivial and the left regular representations of the Lie groupoidgiven as follows. (cf. [24], p.93). Working on the field of real numbers, we leave aside the complex trivial represen-tation and focus on the real. We replace the complex numbers C with the orderedspace or connected component [0 ,
1] in the trivial representation ℓ t : C s ( γ ) → C t ( γ ) ,in which ℓ t is an identity map on C . In this case, the natural or trivial represen-tation ℓ t is on the trivial bundle A × [0 ,
1] over A . Thus, each fibre H x is again1-dimensional Hilbert space [0 , x ∈A ≃ A × K . Hence, ℓ t ( γ ) = 1 ∈ [0 ,
1] for all γ ∈ Γ; making ℓ t : [0 , s ( γ ) → [0 , t ( γ ) an identity map on [0 , ℓ r representation whichis built on the trivial. It is defined on the fibres H x = L (Γ( x, − )) (see [24],25.107; [26], p.55). The convolution algebra C c (Γ) is made a space of continuous(smooth) sections of { H x } by identification of each ϕ ∈ C c (Γ) with the section x → ϕ | Γ( x, − ) ∈ C c (Γ( x, − ) ⊂ L (Γ( x, − )). Hence, according to definition, any pairof sections ϕ, ψ ∈ C c (Γ) is required to define a ν -measurable map by inner product γ
7→ h ϕ ( γ ) , ψ ( γ ) i . This is satisfied since ϕψ ∈ B ( H )-bounded operators-and therestriction ( ϕψ ) | A = ( ϕψ ) o ∈ B c ( A ).The generation of the Hilbert bundle from these sections is shown as follows.From our construction and definition of the Lie groupoid Γ ⇒ A , and followingalso from the definition of locally compact groupoid in ([24], Definition 2.2.1), thereis a countable family C of compact (convex) Hausdorff subsets of Γ such that thefamily { C o , C ∈ C} of interiors of C is a basis for the topology of Γ (we have used ordefined this open basis U of locally convex topology of Γ above as N x ⊂ Γ( x, − ).)For C ∈ C , there exists a sup-norm dense countable subset A C (of sections) of C c ( C o ) corresponding to C ∈ C ; a fundamental net of sections is given by the sumsof the functions in A C ⊂ C c ( C o ) as C ranges over the family C (an ultrafilter),which is same as the set of images of all the local bisections B ℓ ( G ).Subsequently, { H x } is a Hilbert bundle, and and the map ℓ r ( γ ) : H s ( γ ) → H t ( γ ) defined by ( ℓ r ( γ ))( f )( γ ) = f ( γ − γ ), with f ∈ L (Γ( s ( γ ) , − )) and γ ∈ Γ( t ( γ ) , − ),is a unitary representation given as follows (see [24]). First, ℓ r ( γ ) is an extension ofa bijective isometry L (Γ( s ( γ ) , − ) , µ s ( γ ) ) → L (Γ( t ( γ ) , − ) , µ t ( γ ) ) , f γ ∗ f ; (10)in the sense that L (Γ( s ( γ ) , − ) , µ s ( γ ) ) ⊂ L (Γ( s ( γ ) , − ) , µ s ( γ ) ); and it defines atranslative (transitive) action of Γ on t -fibres.Second, the restriction of the ℓ r to L (Γ( s ( γ ) , − ) , µ s ( γ ) ) is a representation of γ ∈ Γ as a unitary operator γ ∗ f given as H s ( γ ) → H t ( γ ) , ( γ ∗ f )( γ ) = f ( γ − γ ) = f ( γ − ) f ( γ ) . (11)The restriction holds for 1 < p ≤ ∞ (see [24], p.34). So, ℓ r is a (unitary) represen-tation of Γ ⇒ A on the Hilbert bundle ( A , H , ν ) for it satisfies the other conditionsof definition. 26 emark 4.2. The translative action of Γ on the t -fibres which are open subspacesof the locally convex Lie groupoid Γ can also be given by the action of the localbisections of the Lie groupoid, which are always diffeomorphic to the open subspacesof a Lie groupoid; that is, L ←→ ϕ , where L is an open subspace of Γ and ϕ is alocal bisection of Γ. C ∗ -Representation of the Convolution Algebra The unitary representation of the Lie groupoid Γ ⇒ A on U ( H )-space of unitary op-erators on the Hilbert bundle H comprising of the Hilbert spaces ( L (Γ( t ( γ ) , − ) , µ t ( γ ) )-is connected to ∗ -representation of the convolution algebra C c (Γ) on the space ofoperators on the same bundle space H . The definition of the ∗ -representation pre-supposes the definition of involution or I -norm, the role of which is to keep theinvolution isometric on the convolution algebra C c (Γ). (See also [26], p.51). Definition 4.3. [24] Given the locally convex (compact) groupoid Γ ⇒ A , with C c (Γ) as the space of continuous functions on Γ supported on compact sets, thefollowing norms are defined on C c (Γ) as follows. || f || I,t = sup x ∈A Z Γ( x, − ) | f ( γ ) | dµ x ( γ ); || f || I,s = sup x ∈A Z Γ( − ,x ) | f ( γ ) | dµ x ( γ ) (11) || f || I = max {|| f || I,t , || f || I,s } is the I-norm . We now modify Paterson’s formulations to suit our locally convex Lie groupoidΓ ⇒ A as follows. Proposition 4.4.
Let C be a compact (convex) subset of Γ and { f α } be a net in C c (Γ) such that every f α vanishes outside C ; that is, α < dist (( x, x ) , ∂C ) . Supposethat f α → f uniformly in C c (Γ) . Then f α → f in the I -norm of C c (Γ) .Proof. Using the countable family C of compact (convex) Hausdorff subsets of Γas given above, the compactness of C implies an open covering U , · · · , U n of C byHausdorff subsets of open Hausdorff sets V , · · · , V n (take these to be the imagesof local bisections B ℓ (Γ)) such that the closure of each U i in V i is compact. Let27 i ∈ C c ( V i ) be such that F i ≥ χ Ui . In particular, F i is positive. Let F = n P i =1 F i .(If we use the net of bisections, then F = P α F α .) Then F ∈ C c (Γ) and F ≥ χ C .Hence | f α − f | ≤ | f α − f | F , and we have || f α − f || I,t = sup x ∈A Z Γ( x, − ) | f α ( t ) − f ( t ) | dµ x ( γ ) ≤ sup x ∈A Z Γ( x, − ) | f α ( t ) − f ( t ) | F ( γ ) dµ x ( γ )= || f α − f || ∞ || F o || ∞ → , as α → . Similarly, || f α − f || I,s →
0, and so the same for the I -norm.So C c (Γ) is a normed ∗ -algebra under I -norm, with a I -norm continuous rep-resentation on a Hilbert space. The separable normed ∗ -algebra C c (Γ) generatesseparable C ∗ -algebras on the separable Hilbert bundle H . The bundle space H isseparable because the fibres are separable Hilbert spaces. With this it follows that π ℓ : C c (Γ) → B ( H ) defined as h π ℓ ( f ) ξ, η i = Z Γ f ( γ ) h ℓ ( γ )( ϕ ( s ( γ ))) , ψ ( t ( γ )) i dm o ( γ ) , (12)is a ∗ -representation of the convolution algebra of the Lie groupoid. This followsfrom Paterson’s statement and proof of the results of Renault on ∗ -representationof Lie groupoids in [24]. Proposition 4.5. (cf. [24],Proposition 3.1.1) The equation h π ℓ ( f ) ϕ, ψ i = Z Γ f ( γ ) h ℓ ( γ )( ϕ ( s ( γ ))) , ψ ( t ( γ )) i dm o ( γ ) defines a representation π ℓ of C c (Γ) of norm ≤ on the bundle H = L ( A , { H x } , ν ) . Remark 4.6.
From the result we deduce a relationship between the two repre-sentations; the unitary representation of the Lie groupoid Γ ⇒ A on the Hilbertbundle H = L ( A , { H x } , ν ) and the C ∗ -representation of its convolution algebra C ∞ (Γ , Ω(Γ)) ≃ C ∞ ( A ) | Γ on a dense subspace of the C ∗ -algebra B ( H ) of boundedoperators on the Hilbert bundle H . This relationship, according to Paterson, is basic28o the fundamental theorem of analysis on locally compact groupoids. It has to dowith the fact that every representation of the convolution algebra C c ( G ) of a locallycompact groupoid G is some ∗ -representation π ℓ of the unitary representation ℓ ofthe groupoid.Thus, the C ∗ -representation of the convolution algebra C ∞ (Γ , Ω(Γ)) on a densesubspace of the C ∗ -algebra B ( H ) of bounded operators on the Hilbert bundle H isthe ∗ -representation π ℓ of the unitary representation ℓ of the locally convex groupoidΓ ⇒ A . This is stated in the the following theorem. Theorem 4.7. (Cf. [24]) Given the locally convex Lie groupoid Γ ⇒ A . Therepresentation of C c (Γ , Ω(Γ)) is the ∗ -representation π ℓ of the unitary representation ℓ : Γ → U ( H ) ; and the correspondence ℓ π ℓ preserves the natural equivalencebetween the two representations.Proof. This natural equivalence rests on the idea of groupoid equivalence whichcaptures the partial symmetry encoded by the locally convex Lie groupoid Γ ⇒ A .This partial symmetry is portrayed or captured in the (Ξ , Γ)-equivalence of the t -fibres Γ( x, − ). The isomorphism of the two algebras C c (Γ , Ω(Γ)) ≃ C ∞ ( A ) | Γ brings this partial symmetry to the fore. For it shows that the partial symmetryof the convolution algebra C c (Γ , Ω(Γ)) defined on the arrows of the Lie groupoid isisomorphic or same as the partial symmetry of the smooth algebra defined on A butrestricted to the relation Γ on A .Given that the t -fibre Γ( x, − ) (or the s -fibre Γ( − , x )) is the orbit of the relationΓ through x , it constitutes a representation space of both the smooth algebra andthe convolution algebra of the Lie groupoid Γ ⇒ A . The (Ξ , Γ)-equivalence impliesthat the two actions, Ξ-action and Γ-action commute; while the former defines theunitary representation, the latter defines the ∗ -representation. This gives rise to thenatural equivalence ℓ ↔ π ℓ of the two representations.The following lemma on the groupoid equivalence clarifies this natural equiva-lence and completes the proof of the theorem. It also extends the above result onthe structure of the Lie groupoid we have modelled on the locally convex partial29 -algebra. Lemma 4.8.
Given that
Ξ = F x ∈A Γ( x, x ) is a Lie groupoid. The (Ξ , Γ) -equivalenceof the t -fibres Γ( x, − ) gives rise to a C ∗ -isomorphism.Proof. We have seen above that the sectional transitivity of the Lie groupoid Γ ⇒ A is well reflected in the definition of its convolution algebra C c (Γ) = C ∞ ( A ) Γ , since aconvolution follows the product operation between a pair of arrows γ, η ∈ Γ. Thus,an open submanifold Γ( x, − ) or Γ( − , x ) reflects a (path) connected maximal propersubset B of the locally convex topological vector space A . The maximality of B ⊂ A with respect to path connectedness makes it either open and dense in A or a closedproper subset of A . This is clear from the inductive system defined above (see also[7]).Subsequently, from the definition of a t -fibre Γ( x, − ) as a (Ξ , Γ)-equivalencein [20], the existence of a special equivalence between the C ∗ -algebras C ∗ (Ξ) and C ∗ (Γ) was established, which leads to isomorphism between C ∗ (Γ) and C ∗ (Ξ) ⊗K ( L ( A , ν )), where K is the set of compact operators. On the other hand, fromthe definition of the Hilbert bundle H = F x ∈A H x , where H x = L (Γ( x, − ) , µ x ), it isevident that H is a left Ξ-principal bundle.Finally, from the position of Γ ⇒ A as the limit of the inductive system { (Γ α ⇒ A α , σ α ) , ( σ βα ) α,β ∈ I : β ≫ α } , which implies that Γ is transitive on a dense subsetof A , we conclude that the B ( H ) which has the representation ℓ → π ℓ is isomorphicto C ∗ (Ξ) ⊗ K ( L ( A , ν )). That is, B ( H ) ≃ C ∗ (Ξ) ⊗ K ( L ( A , ν )). Remark 4.9.
This isomorphism depends on the probability measure ν on A whichis unique to each unit x ∈ A , and on the density or local bisection ϕ ∈ B ℓ ( B x )defined on x ∈ A , and related to the system of Haar measures. The implication isthat the inductive system also extends to the C ∗ -algebras, whereby the unboundedoperators are the inductive limit of bounded operators.30 conclusion The groupoid characterizations of the partial algebras characterized in [7] havehelped us to arrive at a clearer understanding of the structures of these partialalgebras. Most important is the Lie groupoid characterization of the locally con-vex partial ∗ -algebras ( A , Γ , · , ∗ , τ ) which clearly demonstrates the facility of (Lie)groupoid framework to handle pathological spaces. This facility is aptly capturedin the correspondence between groupoid equivalence and isomorphism of groupoid C ∗ -algebras established in [20], which is exemplified in this work in a special way.The (Ξ , Γ)-equivalence of the t -fibres { Γ( x, − ) : x ∈ A} relates to the Liegroupoid of pairs G ⇒ A as the inductive limit of the inductive system of Liegroupoids Γ α ⇒ A α given above. Hence, the results show in the case of the locallyconvex Lie groupoid Γ ⇒ A we have formulated from the locally convex partial ∗ -algebra ( A , Γ , · , ∗ , τ ), that both the equivalence of (Lie) groupoids and that oftheir C ∗ -algebras follow from the equivariant actions of the smoothing algebra K .Hence, the fact that they are equivariant K -spaces contributes to the correspon-dence between the two representations and the isomorphism of C ∗ -algebras.In addition, the smooth equivalence also presents the Lie group bundle Ξ = { Γ( x, x ) : x ∈ A} as the deductive limit of a deductive system { Γ( y, x ) , y ε → x : ε → } , which implies the convergence of every closed manifold Γ( y, x ) → Γ( x, x ).This could be considered a deformation of the Lie groupoid Γ( A , x ) to the Lie groupΓ( x, x ) at each unit x ∈ A using ε ∈ [0 ,
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