Locally Convex Bialgebroid of an Action Lie Groupoid

Action Lie groupoids are used to model spaces of orbits of actions of Lie groups on manifolds. For each such action groupoid M⋊H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M\rtimes H$$\end{document}, we construct a locally convex bialgebroid Dirac(M⋊H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathord {\textrm{Dirac}}(M\rtimes H)$$\end{document} with an antipode over Cc∞(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathord {{\mathcal {C}}^{\infty }_{c}}(M)$$\end{document}, from which the groupoid M⋊H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M\rtimes H$$\end{document} can be reconstructed as its spectral action Lie groupoid AGsp(Dirac(M⋊H))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathord {\mathcal{A}\mathcal{G}_{ sp }}(\mathord {\textrm{Dirac}}(M\rtimes H))$$\end{document}.


Introduction
Our motivation for this paper originates from the Gelfand-Naimark theorem.To any locally compact, Hausdorff topological space X one assigns the C ˚-algebra C 0 pXq of all continuous functions on X that vanish at infinity.The set SpecpC 0 pXqq of all characters on C 0 pXq can be equipped with the weak-˚topology so that it becomes a locally compact Hausdorff space.The Gelfand-Naimark theorem then says that the map Φ lc X : X Ñ SpecpC 0 pXqq, which assigns to a point x P X the evaluation δ x at x, is a homeomorphism.
We wish to obtain a similar result for the class of geometric spaces which can be represented by Lie groupoids [4,17,20,21].These spaces include orbifolds, spaces of leaves of foliations and spaces of orbits of Lie groups actions.In [22] a result in the spirit of the Gelfand-Naimark theorem was constructed for the class of étale Lie groupoids, which can be used to model orbifolds and spaces of leaves of foliations.To any étale Lie groupoid G one assigns the Hopf algebroid C 8 c pG q of smooth compactly supported functions on G (if G is not Hausdorff, one needs to be careful with the definition).As an algebra C 8 c pG q coincides with the Connes convolution algebra in the Hausdorff case [5], while the coalgebra structure is basically induced from the sheaf [23], corresponding to the target map t : G Ñ M of G .Finally, the antipode on C 8 c pG q is induced by the inverse map of G .For each such Hopf algebroid C 8 c pG q one can construct the spectral étale Lie groupoid G sp pC 8 c pG qq so that there is a natural isomorphism Φ egr G : G Ñ G sp pC 8  c pG qq of Lie groupoids.Similar ideas were used in [10] to extend these results to the semi-direct products of étale Lie groupoids and bundles of Lie groups.
Structure maps of an étale Lie groupoid G are local diffeomorphisms so, in particular, the fibres of the target map are discrete.It is therefore enough to reconstruct the fibres of such a groupoid just as sets, which can be done by utilizing the coalgebra structure on C 8 c pG q.However, in the case of a general groupoid one needs some additional information, which enables us to recover the topology along the fibres.Let us explain the main idea on a simple example.If Γ is a discrete group, then C 8 c pΓq is just the group Hopf algebra of Γ. Elements of Γ correspond to grouplike elements of C 8 c pΓq, while the multiplication and inverse of Γ are encoded in multiplication and antipode of C 8 c pΓq.If we now replace Γ with a non-discrete Lie group H, one can still define its group Hopf algebra and reconstruct H, but only as a group.To recover the topology and the smooth structure of H we need some additional structure.One way to solve this problem is to identify the group Hopf algebra of H with the space DiracpHq of distributions on H which is spanned by Dirac distributions.The space DiracpHq is a subspace of the space E 1 pHq of compactly supported distributions on H.If we equip DiracpHq with the induced strong topology from E 1 pHq, the following two things happen.The group H is naturally homeomorphic to the space of Dirac distributions, which are precisely the grouplike elements of the Hopf algebra DiracpHq.On the other hand, since DiracpHq is dense in E 1 pHq, the strong dual DiracpHq 1 is isomorphic to C 8 pHq.Now observe that the space C 8 pHq is a Fréchet algebra, from which H can be reconstructed as a manifold.Judging by the above example, we are led to consider Hopf algebroids not only as purely algebraic objects, but with some additional topological structure.The main idea consists of two parts.First of all we assign to a Lie groupoid G a certain Hopf algebroid, from which the algebraic structure of G can be reconstructed.We then equip this Hopf algebroid with a suitable locally convex structure, which enables us to recover the topology and smooth structure of G .
In this paper we use this idea on the class of action Lie groupoids, which are used to describe spaces of orbits of Lie groups actions on manifolds.Each such action Lie groupoid M ¸H is isomorphic as a groupoid to an étale Lie groupoid M ¸H# , where H # is the group H with discrete topology.We use this identification to define the Dirac bialgebroid DiracpM ¸Hq of M ¸H as a certain subspace of the space E 1 t pM ¸Hq of t-transversal distributions on M ¸H.Transversal distributions on Lie groupoids were studied in [1,2,6,11,15] and, crucially for our problem, it was shown in [15] that the space E 1 t pM ¸Hq is a locally convex algebra, if we equip it with the strong topology of uniform convergence on bounded subsets.With the induced topology DiracpM ¸Hq becomes a locally convex bialgebroid with an antipode.
The paper is organized as follows.In Section 2 we recall the basic definitions and known results that are used in the rest of the paper.In Section 3 we construct for every trivial bundle π : M ˆN Ñ M the space Dirac π pM ˆN q of transversal distributions of constant Dirac type.These are families of Dirac distributions, supported on constant sections of π.If the fiber N is discrete, Dirac π pM ˆN q coincides with the LF -space C 8 c pM ˆN q, while in general we show that it is a dense subspace of the space of π-transversal distributions E 1 π pM ˆN q.In Section 4 we define on Dirac π pM ˆN q a structure of a locally convex coalgebra over C 8 c pM q and show that its strong C 8 c pM q-dual is naturally isomorphic to the Fréchet algebra C 8 pM ˆN q.The combination of the coalgebra structure and locally convex topology enables us to reconstruct from Dirac π pM ˆN q the bundle π : M ˆN Ñ M as the spectral bundle B sp pDirac π pM ˆN qq.Finally, in Section 5 we use these results to assign to each action Lie groupoid M ¸H its Dirac bialgebroid DiracpM ¸Hq.The space DiracpM ¸Hq is a locally convex bialgebroid with an antipode, which coincides with the locally convex Hopf algebra DiracpHq in the case when M is a point.We then show that the groupoid M ¸H can be reconstructed from DiracpM ¸Hq as its spectral action Lie groupoid AG sp pDiracpM ¸Hqq

Preliminaries
In this subsection we will review basic definitions and results that will be needed in the rest of the paper.More details concerning locally convex vector spaces and Lie groupoids can be found for example in [7,14,26] respectively [17,20,21].
We will assume all our manifolds to be smooth, Hausdorff and paracompact, but not necessarily second-countable.For any such manifold M we will denote by C 8 pM q the vector space of smooth C-valued functions on M .The subspaces of compactly supported and R-valued functions on M will be denoted by C 8 c pM q respectively C 8 pM, Rq.

2.1.
Locally convex spaces.All our locally convex vector spaces will be complex and Hausdorff.A subset B of a locally convex space E is bounded if and only if the set ppBq is a bounded subset of R for any continuous seminorm p on E. For locally convex vector spaces E and F we will denote by HompE, F q the space of all continuous linear maps from E to F , equipped with the strong topology of uniform convergence on bounded subsets.The basis of neighbourhoods of zero in HompE, F q consists of sets of the form where B is a bounded subset of E and V is a neighbourhood of zero in F .If E and F are modules over an C-algebra A, we will denote by Hom A pE, F q the corresponding space of continuous A-module homomorphisms and equip it with the induced topology from HompE, F q.
The space C 8 pR l q has a structure of a Fréchet algebra for any l P N. Topology on C 8 pR l q is generated by a family of seminorms tp L,m u, indexed by compact subsets L of R l and m P N 0 , given by p L,m pF q " sup xPL,|α|ďm |D α pF qpxq| for F P C 8 pR l q.Here we denoted , where α " pα 1 , . . ., α l q P N l 0 is a multi-index and |α| " α 1 `α2 `. . .`αl .If M is a second-countable manifold, one can choose similar seminorms with respect to some open cover of M with local coordinate charts to define the Fréchet topology on C 8 pM q.This topology coincides with the topology of uniform convergence of all derivatives on compact subsets of M .The strong dual of the space C 8 pM q is the space E 1 pM q " HompC 8 pM q, Cq of compactly supported distributions on M .
If M is not compact, the subspace C 8 c pM q of C 8 pM q is not complete in the Fréchet topology, so we consider a finer LF-topology on C 8 c pM q.For any compact subset L of M we denote by C 8 c pLq the subspace of functions with support contained in L. The space C 8 c pLq is a closed subspace of C 8 pM q and hence a Fréchet space itself.The LF-topology on C 8 c pM q is now defined as the inductive limit topology with respect to the family of all subspaces of the form C 8 c pLq for L Ă M compact.The space C 8 c pM q with LF-topology is a complete locally convex space, which is not metrizable, if M is not compact.
If M is a smooth manifold and E is a locally convex vector space, a vector valued function u : M Ñ E is smooth if in local coordinates all partial derivatives exist and are continuous.We will denote by C 8 pM, Eq the space of smooth functions on M with values in E and by C 8 c pM, Eq its subspace, consisting of compactly supported functions.To make a distinction between scalar functions and vector valued functions, we will denote by f pxq P C the value of a function f P C 8 pM q at x and by u x P E the value of a function u P C 8 pM, Eq at x.

Lie groupoids.
A Lie groupoid is given by a manifold M of objects and a manifold G of arrows together with smooth structure maps: target t : G Ñ M , source s : G Ñ M , multiplication mlt : G ˆs,t M G Ñ G , inverse inv : G Ñ G and unit uni : M Ñ G .We assume that the source and the target maps are submersions to ensure that G ˆs,t M G is a smooth manifold.A Lie groupoid is étale if all its structure maps are local diffeomorphisms.Note that there exist more general definitions of Lie groupoids, which we will not need.
Example 2.1.We will be mostly interested in action Lie groupoids.Suppose H is a Lie group which acts from the right on the manifold M .The associated action Lie groupoid G " M ¸H is then a Lie groupoid over M with the manifold of arrows M ˆH and with the following structure maps: tpx, hq " x, spx, hq " xh, mltppx, hq, pxh, h 1 qq " px, hqpxh, h 1 q " px, hh 1 q, invpx, hq " px, hq ´1 " pxh, h ´1q, unipxq " px, eq.
Here x P M and h, h 1 P H are arbitrary, while e is the unit of the Lie group H.The action groupoid G is étale if and only if the group H is discrete.

Real commutative algebras.
Let A be an R-algebra.A real character on A is a nontrivial multiplicative homomorphism from A to R. We will denote by SpecpAq the space of all real characters on A, equipped with the Gelfand topology (i.e. the relative weak-˚topology).If the algebra A satisfies the conditions of the Theorem in [19], the space SpecpAq also has a natural smooth structure.
If Q is a smooth manifold, we have SpecpC 8 pQqq " tδ q | q P Qu, where δ q is the Dirac functional, concentrated at the point q.In this case we can equip the set SpecpC 8 pQqq with a topology and a smooth structure such that the map Φ man Q : Q Ñ SpecpC 8 pQqq, defined by Φ man Q pqq " δ q , is a diffeomorphism.

2.4.
Coalgebras.Let R be a commutative ring.We say that R has local identities if for any r 1 , . . ., r n P R there exists r P R such that rr i " r i for i " 1, . . ., n.
Similarly, a left R-module C is locally unitary if for any c 1 , . . ., c n P C there exists r P R such that rc i " c i for i " 1, . . ., n. Suppose now that R is an associative, commutative algebra with local identities over the field of complex numbers C and let C be a locally unitary left R-module.A coalgebra structure on C over R consists of two R-linear maps: called comultiplication and counit, which satisfy the conditions pid b ǫq ˝∆ " id and pǫ b idq ˝∆ " id.Note that these conditions make sense because we can identify C with R b R C and C b R R since C is locally unitary.A coalgebra over R is a locally unitary left R-module C, equipped with a coalgebra structure p∆, ǫq, which is coassociative in the sense that pid b ∆q ˝∆ " p∆ b idq ˝∆.A coalgebra C over R is cocommutative if σ ˝∆ " ∆, where the flip isomorphism σ : Our main examples of coalgebras will be coalgebras associated to sheaves, which were introduced in [23].
Example 2.2.Let P and M be manifolds and let π : P Ñ M be a local diffeomorphism (i.e.P is a sheaf over M ).The ring C 8 c pM q always has local units, but it is unital if and only if M is compact.It will be convenient to denote for any One can show that the definition of ∆ is independent of the various choices that we have made and that we obtain in this way a cocommutative coalgebra C 8 c pP q over C 8 c pM q.For our purposes we will be mostly interested in trivial sheaves.If Γ is a discrete topological space, then the projection π : M ˆΓ Ñ M is a local diffeomorphism.We can decompose the vector space C 8 c pM ˆΓq as a direct sum C 8 c pM ˆΓq " Using this decomposition we can write every element a P C 8 c pM ˆΓq uniquely in the form a " n ÿ i"1 f i ¨δyi for some f 1 , . . ., f n P C 8 c pM q and some y 1 , . . ., y n P Γ.Here we have denoted for any f P C 8 c pM q and any y P Γ by f ¨δy P C 8 c pM ˆΓq the function, given by pf ¨δy qpx, y 1 q " " f pxq ; y 1 " y, 0 ; y 1 ‰ y for px, y 1 q P M ˆΓ.The comultiplication and counit are then given on the generators of C 8 c pM ˆΓq by the formulas: ∆pf ¨δy q " pf ¨δy q b p1 f ¨δy q, ǫpf ¨δy q " f.
In the rest of this subsection we will focus on coalgebras over the algebra C 8 c pM q for some manifold M and recall the main results from [23].For any x P M we denote by c pM q, consisting of all functions with trivial germ at x.The quotient algebra of C 8 c pM q with respect to this ideal will be denoted by then inherits a structure p∆ x , ǫ x q of a coalgebra over C 8 c pM q x which is called the local coalgebra of C at x.The image of c P C in the quotient C x will be denoted by c| x P C x .
An element c P C is weakly grouplike if ∆pcq " c b c 1 for some c 1 P C. A weakly grouplike element c P C is normalized on an open subset U Ă M if ǫ x pc| x q " 1 for all x P U .Weakly grouplike elements of the sheaf coalgebra C 8 c pP q are precisely elements of the form f ˝π| W for some π-elementary open subset W of P and some c pM q x is unital for any x P M , we can also define the set of grouplike elements of C x by An element ξ P C x is grouplike if and only if ξ " c| x for some weakly grouplike element c P C, which is normalized on some open neighbourhood of x.
The spectral sheaf E sp pCq of a C 8 c pM q-coalgebra C is the sheaf π sp pCq : E sp pCq Ñ M with the stalk E sp pCq x " GpC x q.The topology on E sp pCq is defined by the basis, consisting of π sp pCq-elementary subsets of E sp pCq of the form where c P C is a weakly grouplike element, normalized on an open subset U of M .Now let π : P Ñ M be a sheaf over M .By Theorem 2.4 in [23] we have a natural isomorphism of sheaves Φ shv P : P Ñ E sp pC 8 c pP qq defined by Φ shv P ppq " pf ˝π| W q| πppq , where p P P , W is a π-elementary neighbourhood of p in P and f P C 8 c pπpW qq is such that f | πppq " 1 P C 8 c pM q πppq .Moreover, by Theorem 2.10 in [23], a coalgebra C is isomorphic to some sheaf coalgebra C 8 c pP q if and only if C is locally grouplike, which by definition means that for every x P M the C 8 c pM q x -module C x is free with the basis GpC x q.
Let R be a commutative C-algebra with local units.We say that a C-algebra A extends R if R is a subalgebra of A and A has local units in R. We do not assume that R is a central subalgebra of A. Any C-algebra A which extends R, is naturally an R-R-bimodule.We will denote by A b R A " A b ll R A the tensor product of left R-modules, which has two natural right R-module structures.
A bialgebroid over R is a C-algebra A which extends R, together with structure maps ∆ : A Ñ A b R A and ǫ : A Ñ R for which pA, ∆, ǫq is a cocommutative coalgebra and such that: (i) ∆pAq Ă Ab R A, where Ab R A is the algebra consisting of those elements of A b R A, on which both right R-actions coincide, (ii) ǫ| R " id and ∆| R is the canonical embedding R Ă A b R A, (iii) ǫpabq " ǫpaǫpbqq and ∆pabq " ∆paq∆pbq for any a, b P A. Antipode on a bialgebroid A is a C-linear involution S : A Ñ A which satisfies the conditions: (i) S| R " id and Spabq " SpbqSpaq for any a, b P A, (ii) µ A ˝pS b idq ˝∆ " ǫ ˝S, where µ A denotes the multiplication in A. A Hopf algebroid over R is a bialgebroid A over R with an antipode.Note that in the case when R is a central subalgebra of A the notions of bialgebroid and Hopf algebroid coincide with the more familiar notions of bialgebra and Hopf algebra.
In the next example we will recall from [22] the Hopf algebroid associated to an étale Lie groupoid.
Example 2.3.Let G be an étale Lie groupoid over M .Multiplication on G induces a convolution product [5] on C 8 c pG q, given by the formula pa 1 ˚a2 qpgq " ÿ g"g1g2 for a 1 , a 2 P C 8 c pG q.Note that this sum is always finite as a 1 and a 2 are compactly supported.Since t : G Ñ M is a sheaf, we can also consider the space C 8 c pG q as a locally grouplike coalgebra over C 8 c pM q.Finally, the antipode S : C 8 c pG q Ñ C 8 c pG q is defined by the formula Spaq " a ˝inv for a P C 8 c pG q.In this way C 8 c pG q becomes a Hopf algebroid over C 8 c pM q.Suppose now that Γ is a discrete group which acts from the right on the manifold M and denote by M ¸Γ the associated action groupoid.We then have the decomposition C 8 c pM ¸Γq " For any g P Γ and f P C 8 c pM q let us denote by gf P C 8 c pM q the function, given by pgf qpxq " f pxgq for x P M .If we use the notation from the Example 2.2, the convolution product and the antipode on C 8 c pM ¸Γq can be described on the set of generators by the formulas: pf 1 ¨δg1 q ˚pf 2 ¨δg2 q " pf 1 pg 1 f 2 qq ¨δg1g2 , Spf ¨δg q " pg ´1f q ¨δg ´1 .Now let A be a Hopf algebroid over C 8 c pM q.We will next recall from [22] the construction of the spectral étale Lie groupoid G sp pAq, associated to A. Note that A is a coalgebra over C 8 c pM q, so we have the notion of weakly grouplike elements.We say that a weakly grouplike element a P A is S-invariant if there exists a 1 P A such that ∆paq " a b a 1 and ∆pSpaqq " Spa 1 q b Spaq.In the case of the Hopf algebroid C 8 c pG q of an étale Lie groupoid G , an element a P C 8 c pG q is S-invariant weakly grouplike if and only if it is of the form f ˝t| W for some bisection W of G and some f P C 8 c ptpW qq (a bisection of an étale Lie groupoid G is an open subset W of G which is both t-elementary and s-elementary).
An arrow of A with target y P M is an element g P GpA y q, for which there exists an S-invariant weakly grouplike element a P A such that g " a| y .The set of all arrows of A with target y will be denoted by G sp pAq y .All arrows of A form a subsheaf G sp pAq of E sp pAq, whose projection will be denoted by t " π sp pAq| Gsp pAq : G sp pAq Ñ M.
To describe the source map of G sp pAq, we first recall that each S-invariant weakly grouplike element a P A induces a C-linear map T a : C 8 c pM q Ñ C 8 c pM q, given by T a pf q " ǫpSpf aqq.If a is normalized on an open subset U of M , one can find an open subset U 1 of M and a unique diffeomorphism τ U,a : U 1 Ñ U such that T a pC 8 c pU qq Ă C 8 c pU 1 q and T a pf q " f ˝τU,a for any f P C 8 c pU q.Furthermore, the element Spaq is S-invariant weakly grouplike, normalized on U 1 and we have that τ U 1 ,Spaq " τ ´1 U,a .The source map s : G sp pAq Ñ M is now defined by spa| y q " τ ´1 U,a pyq, where a P A is an S-invariant weakly grouplike element, normalized on U .Now choose elements a, b P A which represent an arrow a| y P G sp pAq y and an arrow b| x P G sp pAq x such that spa| y q " x.The product of a| y and b| x is then defined by a| y b| x " pabq| y .
The unit unipxq P G sp pAq at the point x P M is given by unipxq where f P C 8 c pM q Ă A is any function with germ f | x " 1 P C 8 c pM q x .Finally, the inverse of an arrow a| y P G sp pAq y with spa| y q " x is defined by The groupoid G sp pAq is an étale Lie groupoid over M , called the spectral étale Lie groupoid of the Hopf algebroid A. For any étale Lie groupoid G over M we have a natural isomorphism of Lie groupoids Φ egr G pgq " pf ˝t| W q| tpgq , where W is any bisection of G which contains g and f P C 8 c ptpW qq is any function with f | tpgq " 1 P C 8 c pM q tpgq .Hopf algebroid A is isomorphic to a Hopf algebroid of the form C 8 c pG q for an étale Lie groupoid G if and only if it is locally grouplike, which means that for every y P M the C 8 c pM q y -coalgebra A y is a free C 8 c pM q y -module with the basis consisting of arrows of A at the point y.

Transversal distributions of constant Dirac type
Let π : M ˆN Ñ M be a trivial bundle over M with fiber N .In the spirit of the Gelfand-Naimark theorem we will assign to it a locally convex C 8 c pM q-module Dirac π pM ˆN q of distributions of constant Dirac type on M ˆN and show that its strong C 8 c pM q-dual is isomorphic to the space C 8 pM ˆN q.In general, the space Dirac π pM ˆN q is a dense subspace of the space E 1 π pM ˆN q of compactly supported transversal distributions.However, in the case when N " Γ is discrete, the space Dirac π pM ˆΓq is complete and isomorphic to the LF-space C 8 c pM ˆΓq.We start with the definition of transversal distributions on a trivial bundle.Definition 3.1.Let M and N be second-countable manifolds and let M ˆN be the trivial bundle over M with fibre N and projection π : M ˆN Ñ M .The space of π-transversal distributions with compact support is the space π pM ˆN q " Hom C 8 c pMq pC 8 pM ˆN q, C 8 c pM qq.In other words, E 1 π pM ˆN q is the space of continuous C 8 c pM q-linear maps from C 8 pM ˆN q to C 8 c pM q, where the C 8 c pM q-module structure on C 8 pM ˆN q is given by pf ¨F qpx, yq " f pxqF px, yq for f P C 8 c pM q, F P C 8 pM ˆN q and px, yq P M ˆN .The space E 1 π pM ˆN q is a C 8 c pM q-module as well, with module structure given by pf ¨T qpF q " T pf ¨F q for f P C 8 c pM q, F P C 8 pM ˆN q and T P E 1 π pM ˆN q.If we equip the space E 1 π pM ˆN q with the strong topology of uniform convergence on bounded subsets, it becomes a complete locally convex space.Remark 3.2.p1q It was shown in [15] that there is an isomorphism π pM ˆN q -C 8 c pM, E 1 pN qq, which enables us to identify a π-transversal distribution T P E 1 π pM ˆN q with a smooth, compactly supported family u " upT q P C 8 c pM, E 1 pN qq.We will denote the value of u at x P M by u x P E 1 pN q.If we denote by π N : M ˆN Ñ N the projection to N , the distribution u x is given by the formula u x pF q " T pF ˝πN qpxq for any F P C 8 pN q.We can also view any u P C 8 c pM, E 1 pN qq as a π-transversal distribution T " T puq, if we define T pF qpxq " u x pF ˝ιx q for F P C 8 pM ˆN q.Here ι x : N Ñ txu ˆN is given by ι x pyq " px, yq for x P M .Different letters T and u are used intentionally to make a slight distinction between transversal distributions and smooth families of distributions along the fibres.
p2q We can define a support of a family u P C 8 c pM, E 1 pN qq either as a subspace of M or a subspace of M ˆN .The support of u is the subset supppuq of M , defined by supppuq " tx P M | u x ‰ 0u.On the other hand, the total support of u is the subset supp MˆN puq of M ˆN , consisting of all points px, yq P M ˆN , which satisfy the condition that for every open neighbourhood U of px, yq there exists F P C 8 c pU q such that upF q ‰ 0. For u P C 8 c pM, E 1 pN qq both supports are compact and we have πpsupp MˆN puqq " supppuq.We can also define the support of a π-transversal distribution T P E 1 π pM ˆN q as the subset supppT q " supp MˆN pupT qq of M ˆN .For more details about supports we refer the reader to [9].
p3q If dimpM q ą 0, the space Hom C 8 c pMq pC 8 pM ˆN q, C 8 c pM qq in fact coincides with the space Lin C 8 c pMq pC 8 pM ˆN q, C 8 c pM qq of C 8 c pM q-linear maps, without any assumption on continuity (see [8]).
Let us take a look at some important examples of transversal distributions that will be used throughout the paper.
Example 3.3.(1) Let π : M ˆN Ñ M be a trivial bundle and denote for any y P N by E y " M ˆtyu the horizontal subspace of M ˆN .For any f P C 8 c pM q we define a π-transversal distribution E y , f P E 1 π pM ˆN q by E y , f pF qpxq " f pxqF px, yq, for F P C 8 pM ˆN q.We think of E y , f as a smooth family of Dirac distributions, supported on the constant section E y and weighted by the function f .In particular, we have E y , f x " f pxqδ y .(2) Let M " R l , N " R k and let π : R l ˆRk Ñ R l be the projection onto R l .For φ P C 8 c pR l ˆRk q we define a π-transversal distribution T φ P E 1 π pR l ˆRk q by φpx, yqF px, yq dy for F P C 8 pR l ˆRk q.The distribution T φ corresponds to the family of smooth densities on R k , parametrized by R l and explicitly given by where dV is the Lebesgue measure on R k .
The map φ Þ Ñ T φ defines a continuous, injective C 8 c pR l q-linear map C 8 c pR l ˆRk q ãÑ E 1 π pR l ˆRk q.Note that the LF -topology on C 8 c pR l ˆRk q is strictly finer than the subspace topology that is induced from E 1 π pR l ˆRk q via the above map.In particular, if M " R 0 is a point, the above construction enables us to consider the space C 8 c pR k q as a subspace of the space E 1 pR k q.
Let us now denote for a manifold N by N # the set N with the discrete topology.The projection π # : M ˆN # Ñ M is then a local diffeomorphism.Note that the manifold M ˆN # is paracompact, but not second-countable if dimpN q ą 0.
Using the notation from the Example 2.2 we have a decomposition which enables us to write every element a P C 8 c pM ˆN # q uniquely in the form a " for some f 1 , . . ., f n P C 8 c pM q and some y 1 , . . ., y n P N .Now define an injective Definition 3.4.Let M ˆN be a trivial bundle with projection π : M ˆN Ñ M .The space of π-transversal distributions of constant Dirac type is the space Dirac π pM ˆN q " Ψ MˆN pC 8 c pM ˆN # qq Ă E 1 π pM ˆN q.The space Dirac π pM ˆN q is equipped with the induced topology from E 1 π pM ˆN q.If M is a point, we denote by DiracpN q the subspace of E 1 pN q, spanned by Dirac distributions.
We will show in the sequel that Dirac π pM ˆN q is a proper, dense subspace of E 1 π pM ˆN q if dimpN q ą 0. However, in the case when N " Γ is discrete, we have the following description of the space Dirac π pM ˆΓq.Proposition 3.5.Let M ˆΓ be a trivial bundle over a second-countable manifold M with a countable discrete fiber Γ and bundle projection π : M ˆΓ Ñ M .
(a) The map Ψ MˆΓ : C 8 c pM ˆΓq Ñ E 1 π pM ˆΓq is an isomorphism of C 8 c pM qmodules, so we have Dirac π pM ˆΓq " E 1 π pM ˆΓq.(b) The map Ψ MˆΓ is an isomorphism of locally convex spaces with respect to the LF -topology on C 8 c pM ˆΓq and the strong topology on E 1 π pM ˆΓq.Proof.(a) First recall that we have an isomorphism E 1 π pM ˆΓq -C 8 c pM, E 1 pΓqq of C 8 c pM q-modules.It is therefore enough to show that every u P C 8 c pM, E 1 pΓqq is of the form u " Ψ MˆΓ paq for some a P C 8 c pM ˆΓq.Since Γ is discrete, the space E 1 pΓq is isomorphic to the locally convex direct sum À yPΓ Spanpδ y q of one-dimensional subspaces, spanned by Dirac distributions.Any u P C 8 c pM, E 1 pΓqq has compact support, so its image upM q Ă E 1 pΓq is compact and hence bounded.This implies that upM q lies in some finite-dimensional subspace À n i"1 Spanpδ yi q Ă E 1 pΓq for some y 1 , . . ., y n P Γ.We can therefore find functions f 1 , . . ., f n : M Ñ C such that u x " n ÿ i"1 f i pxqδ yi for every x P M .If we denote by 1 yi P C 8 pM ˆΓq the function, which is equal to 1 on M ˆty i u and zero elsewhere, we have f i " up1 yi q P C 8 c pM q and therefore (b) To see that Ψ MˆΓ is continuous, first choose a basic neighbourhood KpB, V q of zero in E 1 π pM ˆΓq, where B is a bounded subset of C 8 pM ˆΓq and V is a neighbourhood of zero in C 8 c pM q.From the definition of LF -topology on C 8 c pM ˆΓq it follows that we only have to show that the restrictions of Ψ MˆΓ onto subspaces of the form C 8 c pL ˆtyuq are continuous for all y P Γ and all compact subsets L of M .Define a multiplication map µ : C 8 c pL ˆtyuq ˆC8 pM ˆΓq Ñ C 8 c pM q by µpf ¨δy , F qpxq " f pxqF px, yq for f ¨δy P C 8 c pL ˆtyuq and F P C 8 pM ˆΓq.Note that µ is continuous, so we can find neighbourhoods V 1 and V 2 of zero in C 8 c pL ˆtyuq respectively C 8 pM ˆΓq such that µpV 1 , V 2 q Ă V .Since B Ă C 8 pM ˆΓq is bounded, we can assume that B Ă V 2 (if necessary, rescale V 1 and V 2 by appropriate inverse factors).Now observe that µpf ¨δy , F q " Ψ MˆΓ pf ¨δy qpF q.
For f ¨δy P V 1 and F P B we thus have Ψ MˆΓ pf ¨δy qpF q " µpf ¨δy , F q P V , which shows that Ψ MˆΓ pV 1 q Ă KpB, V q.
Finally, we have to show that the map Ψ ´1 MˆΓ is continuous.Let us choose a net pu α q αPA that converges to zero in E 1 π pM ˆΓq.The set tu α | α P Au is then a bounded subset of E 1 π pM ˆΓq, so there exists a compact subset of M ˆΓ which contains all supports supppu α q for α P A. In particular, we can find a compact subset L of M and elements y 1 , y 2 , . . ., y n P Γ, such that for every α P A we can write for some f α,1 , . . ., f α,n P C 8 c pLq.If we evaluate u α at 1 yi (see part (a) of the proof for the definition of 1 yi ), we get that u α p1 yi q " f α,i converges to zero in C 8 c pM q for 1 ď i ď n, which implies that Ψ ´1 MˆΓ pu α q " ř n i"1 f α,i ¨δyi converges to zero in C 8 c pM ˆΓq.
We will now move on to the study of the space Dirac π pM ˆN q in the case of non-discrete fibre and show that it is a dense subspace of the space E 1 π pM ˆN q.To do that, we first recall some facts about the convolution of distributions on euclidean spaces.We will use the definition of the convolution product on E 1 pR k q that is easily generalized to arbitrary Lie groupoids (see Section 5).For any F P C 8 pR k q we can define a smooth map R k Ñ C 8 pR k q by y Ñ F ˝Ly , where the left translation L y : R k Ñ R k is defined by L y py 1 q " y`y 1 for y P R k .If we compose this map with an arbitrary distribution w P E 1 pR k q, we thus get a smooth map R k Ñ C, given by y Ñ wpF ˝Ly q.The convolution product ˚: E 1 pR k q ˆE1 pR k q Ñ E 1 pR k q can be then described by the formula pv ˚wqpF q " vpy Ñ wpF ˝Ly qq for F P C 8 pR k q and v, w P E 1 pR k q.The convolution product is a bilinear, jointly continuous map and it turns E 1 pR k q into a commutative, locally convex algebra.
As we have seen in the Example 3.3, we can consider C 8 c pR k q as a subspace of E 1 pR k q.We can explicitly describe the convolution of an arbitrary distribution with a smooth function as follows.Choose any ρ P C 8 c pR k q and consider it as an element T ρ P E 1 pR k q, which we will for simplicity denote just by ρ.Let ρ P C 8 pR k q be defined by ρpzq " ρp´zq.For any v P E 1 pR k q we then have that v ˚ρ P C 8 c pR k q Ă E 1 pR k q is a smooth function, given by pv ˚ρqpyq " vpρ ˝L´y q for y P R k .This shows that C 8 c pR k q is actually an ideal of E 1 pR k q.We will now use these results in the setting of transversal distributions.For any u " pu x q xPR l P C 8 c pR l , E 1 pR k qq and any ρ P C 8 c pR k q we define u ˚ρ P C 8 c pR l , E 1 pR k qq pointwise by pu ˚ρq x " u x ˚ρ.Smoothness of u ˚ρ follows from bilinearity and continuity of the convolution product.One can moreover show that u ˚ρ is actually of the form u ˚ρ " T φ for the smooth function φ P C 8 c pR l ˆRk q, given by φpx, yq " u x pρ ˝L´y q.
We will next show that the image of the map C 8 c pR l ˆRk q ãÑ C 8 c pR l , E 1 pR k qq, given by φ Þ Ñ T φ , is a dense subspace of C 8 c pR l , E 1 pR k qq.To do that, we first recall from [9] an explicit description of a neighbourhood basis of zero in the space C 8 c pR l , E 1 pR k qq.Denote K 0 " H and let K n " tx P R l | |x| ď nu be the ball with centre at zero and radius n P N. Choose an increasing sequence of natural numbers m " pm 1 , m 2 , . ..q, a decreasing sequence of positive real numbers e " pǫ 1 , ǫ 2 , . ..q and let B " pB 1 , B 2 , . ..q be an increasing sequence of bounded subsets of C 8 pR k q.Now define a subset V B,m,e Ă C 8 c pR l , E 1 pR k qq by V B,m,e " tu P C 8 c pR l , E 1 pR k qq | p Bn ppD α uq x q ă ǫ n for x P K c n´1 and |α| ď m n u, where the seminorm p Bn on E 1 pR k q is given by p Bn pvq " sup F PBn |vpF q| for v P E 1 pR k q.The family of all such sets V B,m,e , with B, m and e as defined above, then forms a basis of neighbourhoods of zero for a topology on C 8 c pR l , E 1 pR k qq, for which the natural identification C 8 c pR l , E 1 pR k qq -E 1 π pR l ˆRk q becomes an isomorphism of locally convex vector spaces (see [9]).
Proposition 3.6.The image of the map C 8 c pR l ˆRk q ãÑ C 8 c pR l , E 1 pR k qq, given by φ Þ Ñ T φ , is a dense subspace of C 8 c pR l , E 1 pR k qq.Proof.Let us choose a one-parameter family pρ t q P C 8 c pR k q Ă E 1 pR k q, for t P p0, 1q, which converges to the Dirac distribution δ 0 P E 1 pR k q as t Ñ 0. Now choose any u P C 8 c pR l , E 1 pR k qq and define u t " u ˚ρt P C 8 c pR l ˆRk q for t P p0, 1q.We will show that u t Ñ u as t Ñ 0. To do that, take an arbitrary basic neighbourhood of zero in C 8 c pR l , E 1 pR k qq of the form V B,m,e .Since u is compactly supported, we have supppuq " supppu t q Ă K n for some n P N and all t P p0, 1q.We need to show that for t small enough, we have u ´ut P V B,m,e , which by the above observation means that p Bn pD α pu ´ut q x q ă ǫ n for all x P K n and all α with |α| ď m n .Equivalently, if we denote Dpǫ n q " tz P C | |z| ă ǫ n u, then for all such x and α we have D α pu ´ut q x P KpB n , Dpǫ n qq Ă E 1 pR k q.Now note that the set A " tpD α uq x | x P K n , |α| ď m n u is compact and hence a bounded subset of E 1 pR k q.Since the convolution ˚: E 1 pR k q ˆE1 pR k q Ñ E 1 pR k q is bilinear and continuous, we can find a neighbourhood V of zero in E 1 pR k q such that V ˚A Ă KpB n , Dpǫ n qq.Since ρ t Ñ δ 0 as t Ñ 0, we have that δ 0 ´ρt P V for t small enough and hence D α pu ´ut q x " pD α uq x ´pD α u ˚ρt q x " pδ 0 ´ρt q ˚pD α uq x P KpB n , Dpǫ n qq.
We will next show, by using ideas from Riemannian integration, that arbitrary transversal distribution of the form T φ P C 8 c pR l , E 1 pR k qq can be approximated by elements of Dirac π pR l ˆRk q.
Choose any L ą 0 and any n P N and denote t j " ´L 2 `jL n for 0 ď j ď n ´1.The set I " tt 0 , t 1 , . . ., t n´1 u k is then a finite subset of the cube D " r´L 2 , L 2 s k .If we denote for t P I by D t " t `r0, L n s k the cube with volume volpD t q " `L n ˘k, we can express D " Ť tPI D t as a union of n k such small cubes.Now define a distribution ∆ n P DiracpR k q Ă E 1 pR k q by Using the fundamental theorem of calculus one can show that for any F P C 8 pR k q we have the following bound ˇˇˇż D F pyq dy ´∆n pF q ˇˇˇď kL k`1 n p D,1 pF q, where p D,1 measures the size of the gradient of F and is defined as in Subsection 2.1.This bound basically says that the sequence p∆ n q nPN converges to ş D in E 1 pR k q.Proposition 3.7.The space Dirac π pR l ˆRk q is a dense subspace of C 8 c pR l , E 1 pR k qq.Proof.By Proposition 3.6 it is enough to show that for every φ P C 8 c pR l ˆRk q the distribution T φ P C 8 c pR l , E 1 pR k qq can be approximated arbitrarily well by elements of Dirac π pR l ˆRk q.
Choose any φ P C 8 c pR l ˆRk q and suppose π R k psupppφqq Ă D Ă R k for some L ą 0 as above.For n P N we define a π-transversal distribution ∆ φ,n P Dirac π pR l ˆRk q by the formula ∆ φ,n pF qpxq " `L n ˘k ÿ tPI φpx, tqF px, tq.
We will show that ∆ φ,n Ñ T φ in C 8 c pR l , E 1 pR k qq as n Ñ 8.This means that for every set of the form V B,m,e Ă C 8 c pR l , E 1 pR k qq we have T φ ´∆φ,n P V B,m,e for n P N big enough.Both T φ and ∆ φ,n have supports contained in πpsupppφqq Ă K j for some j P N, so we have to show that |D α x pT φ pF q ´∆φ,n pF qqpxq| ă ǫ j for x P K j , F P B j and |α| ď m j .Since B j is a bounded subset of C 8 pR l ˆRk q, the set Bj " φB j " t F " φF | F P B j u is bounded in C 8 pR l ˆRk q as well, so there exists a constant C ă 8 such that suptp Kj ˆD,mj`1 p F q | F P Bj u ă C. For F P B j , x P K j and |α| ď m j we now compute: x pT φ pF q ´∆φ,n pF qqpxq| " We conclude that ∆ φ,n Ñ T φ in C 8 c pR l , E 1 pR k qq as n Ñ 8.As a corollary we get the following result.Proposition 3.8.Let M ˆN be a trivial bundle over M with fiber N and projection π : M ˆN Ñ M .The space Dirac π pM ˆN q is a dense subspace of E 1 π pM ˆN q.
Proof.Every transversal distribution T P E 1 π pM ˆN q has compact support, so we can write it as a sum T " T 1 `T2 `. . .`Tn , where each T i P E 1 π pM ˆN q has support contained in the set of the form U i ˆU 1 i for some domains of coordinate charts U i « R l on M and U 1 i « R k on N .By Proposition 3.7 we can find for each neighbourhood of zero V Ă E 1 π pM ˆN q elements u i P Dirac πi pU i ˆU 1 i q Ă Dirac π pM ˆN q, such that T i ´ui P 1 n V for 1 ď i ď n.If we define u " u 1 `u2 `. . .`un P Dirac π pM ˆN q, we then have T ´u P V .
Let us now denote for simplicity by: c pM q-duals of the C 8 c pM q-modules Dirac π pM ˆN q and E 1 π pM ˆN q.Define a C 8 c pM q-linear map ˆ: C 8 pM ˆN q Ñ Dirac π pM ˆN q 1 , by F puq " upF q for F P C 8 pM ˆN q and u P Dirac π pM ˆN q.
Theorem 3.9.Let M ˆN be a trivial bundle over M with fiber N and bundle projection π : M ˆN Ñ M .The map ˆ: C 8 pM ˆN q Ñ Dirac π pM ˆN q 1 is an isomorphism of locally convex C 8 c pM q-modules.Proof.We first show that the mapˆ: C 8 pM ˆN q Ñ Dirac π pM ˆN q 1 is a C 8 c pM qlinear isomorphism.It is injective since Dirac π pM ˆN q separates the points of C 8 pM ˆN q.To see that it is surjective, choose any φ P Dirac π pM ˆN q 1 .Since Dirac π pM ˆN q is a dense subspace of E 1 π pM ˆN q and since C 8 c pM q is complete, there exists a unique continuous extension φ : E 1 π pM ˆN q Ñ C 8 c pM q of φ to E 1 π pM ˆN q.From Theorem 4.5 in [9] it now follows that φ " F for some F P C 8 pM ˆN q.
It remains to be shown that the map ˆ: C 8 pM ˆN q Ñ Dirac π pM ˆN q 1 is a homeomorphism.It is continuous as it can be written as a composition where the left map is continuous by Theorem 4.5 in [9] and the right map is the continuous restriction of functionals from E 1 π pM ˆN q to Dirac π pM ˆN q.In the remainder of the proof we will show that the above map is open.Let us choose an arbitrary subbasic neighbourhood of zero in C 8 pM ˆN q of the form x D β y F px, yq| ă ǫ for px, yq P LˆK, |α|`|β| ď mu, where m P N, ǫ ą 0, L is a compact subset of M which lies in some chart U M « R l and K is a compact subset of N which lies in some chart U N « R k .Our goal is to find a bounded subset B Ă Dirac π pM ˆN q and a neighbourhood V of zero in C 8 c pM q such that KpB, V q Ă { V LˆK,m,ǫ .For n P N, t P p0, 8q and y P R we define a distribution ∆ n t pyq P DiracpRq by ∆ n t pyq " Using the Taylor's theorem one can show that ∆ n t pyq converges in E 1 pRq to D n y | y as t Ñ 0, where D n y | y is the distribution which computes the n-th derivative at the point y.More generally, denote β " pβ 1 , β 2 , . . ., β k q P N k 0 , y " py 1 , y 2 , . . ., y k q P R k and define ∆ β t pyq " ∆ β1 t py 1 q b ∆ β2 t py 2 q b ¨¨¨b ∆ β k t py k q P E 1 pR k q.
Again we have that ∆ β t pyq P DiracpR k q converges to D β y | y in E 1 pR k q as t Ñ 0. Using K and m from the definition of V LˆK,m,ǫ we now define the subset Using estimates from the Taylor's theorem one can show that B K,m is a bounded subset of E 1 pR k q.Now note that the bilinear map C 8 c pM qˆE 1 pN q Ñ C 8 c pM, E 1 pN qq, given by pf, vq Þ Ñ f v for pf vq x " f pxqv, is continuous.If we choose a function η P C 8 c pU M q Ă C 8 c pM q, such that η " 1 on some neighbourhood of L, it now follows from the above observation that is a bounded subset of Dirac π pM ˆN q.Finally, let us define an open neighbourhood V of zero in C 8 c pM q by V " tf P C 8 c pM q | |D α x f pxq| ă ǫ 2 for x P L, |α| ď mu.
Now choose any φ " F P KpB, V q so that φpuq " upF q P V for u P B. If we write u " ηv " η ř n i"1 a i δ yi for some a 1 , . . ., a n P C and some y 1 , . . ., y n P U N , we have for x P L and |α| ď m the following estimate Here we have used the fact that η " 1 on some neighbourhood of L and denoted by D α x F the α-partial derivative of F in the horizontal direction.For any y P K and any β with |β| ď m the net ∆ β t pyq P B K,m converges to D β y | y in E 1 pR k q as t Ñ 0. If we define u t " η∆ β t pyq P B, we then have for x P L the estimate To sum it up, for px, yq P L ˆK and |α|, |β| ď m we have |D β y D α x F px, yq| ă ǫ, which implies that F P V LˆK,m,ǫ and consequently φ " F P { V LˆK,m,ǫ .

Spectral bundle of the coalgebra of transversal distributions of constant Dirac type
From the Theorem 3.9 it follows that C 8 pM ˆN q is isomorphic to the strong C 8 c pM q-dual of Dirac π pM ˆN q.We will now equip the space Dirac π pM ˆN q with a structure of a locally convex coalgebra over C 8 c pM q, such that its strong C 8 c pM q-dual Dirac π pM ˆN q 1 is a Fréchet algebra, isomorphic to C 8 pM ˆN q. will use the isomorphism Ψ MˆN : C 8 c pM ˆN # q Ñ Dirac π pM ˆN q to transfer coalgebra structure from C 8 c pM ˆN # q to Dirac π pM ˆN q.Explicitly, using the notation from the Example 2.2, we define on Dirac π pM ˆN q a structure of a coalgebra over C 8 c pM q with structure maps: ∆ : Dirac π pM ˆN q Ñ Dirac π pM ˆN q b C 8 c pMq Dirac π pM ˆN q, ǫ : Dirac π pM ˆN q Ñ C 8 c pM q, explicitly given by: ∆p Since C 8 pM ˆN q is isomorphic to the strong dual of Dirac π pM ˆN q, we can use it to define the C 8 c pM q-injective topology on Dirac π pM ˆN qb C 8 c pMq Dirac π pM ˆN q.For any pair of functions F, G P C 8 pM ˆN q we define a C 8 c pM q-linear map The C 8 c pM q-injective topology on Dirac π pM ˆN q b C 8 c pMq Dirac π pM ˆN q is now defined by specifying basic neighbourhoods of zero of the form KpA, B, V q " tũ P Dirac π pM ˆN q b2 | pF b Gqpũq P V, for F P A, G P Bu, where A, B Ă C 8 pM ˆN q -Dirac π pM ˆN q 1 are bounded subsets and V is a neighbourhood of zero in C 8 c pM q.Proposition 4.1.The triple pDirac π pM ˆN q, ∆, ǫq is a cocommutative, locally convex coalgebra over C 8 c pM q, in the sense that ∆ and ǫ are continuous maps.Proof.Let us denote by 1 the unit of the algebra C 8 pM ˆN q.We then have ǫ " 1, which shows that ǫ is a continuous map.
To see that ∆ is continuous, we choose any basic neighbourhood of zero in Dirac π pM ˆN q b C 8 c pMq Dirac π pM ˆN q of the form KpA, B, V q as above.The set A ¨B " tF G | F P A, G P Bu is then a bounded subset of C 8 pM ˆN q.For any u " ř n i"1 E yi , f i P KpA ¨B, V q any F P A and any G P B we now have This implies that ∆pKpA ¨B, V qq Ă KpA, B, V q hence ∆ is continuous.
Since Dirac π pM ˆN q is a cocommutative, counital coalgebra over C 8 c pM q, its dual Dirac π pM ˆN q 1 naturally becomes a commutative algebra with unit ǫ over C 8 c pM q, if we define pφ ¨ψqpuq " pφ b ψqp∆puqq for φ, ψ P Dirac π pM ˆN q 1 and u P Dirac π pM ˆN q.Continuity of φ ¨ψ follows from continuity of φ b ψ and ∆.On both Dirac π pM ˆN q and Dirac π pM ˆN q 1 we can naturally define conjugation as follows.For any u P Dirac π pM ˆN q we define conjugation by u " Using the above formula and complex conjugation on C 8 c pM q we now define for any φ P Dirac π pM ˆN q 1 the element φ P Dirac π pM ˆN q 1 by φpuq " φpuq for u P Dirac π pM ˆN q.It is now a straightforward calculation to extend the Theorem 3.9 in the following way.Proposition 4.2.Let M ˆN be a trivial bundle over M with fiber N and bundle projection π : M ˆN Ñ M .The map ˆ: C 8 pM ˆN q Ñ Dirac π pM ˆN q 1 is an isomorphism of locally convex algebras with involutions.
Using the definitions and notations from the Subsection 2.4 we now define for any x P M the local C 8 c pM q x -coalgebra Dirac π pM ˆN q x " Dirac π pM ˆN q{I x Dirac π pM ˆN q.
It follows from [23] that the space Dirac π pM ˆN q x is a free C 8 c pM q x -module, generated by the set GpDirac π pM ˆN q x q of grouplike elements.The spectral sheaf of the C 8 c pM q-coalgebra Dirac π pM ˆN q is the sheaf π sp : E sp pDirac π pM ˆN qq Ñ M with the stalk at the point x P M given by E sp pDirac π pM ˆN qq x " GpDirac π pM ˆN q x q.
Note that the sheaves M ˆN # and E sp pDirac π pM ˆN qq over M are isomorphic via the map px where f P C 8 c pM q is any function with f | x " 1 P C 8 c pM q x .Let us now define the real part of Dirac π pM ˆN q 1 by Dirac π pM ˆN q 1 R " tφ P Dirac π pM ˆN q 1 | φ " φu and note that it corresponds to the algebra C 8 pM ˆN, Rq via the isomorphism from Proposition 4.2.This implies that Dirac π pM ˆN q 1 R satisfies the conditions of the main theorem in [19], so it can be used to define a smooth structure on the space SpecpDirac π pM ˆN q 1 R q.Furthermore, we have a natural bijection Θ MˆN : E sp pDirac π pM ˆN qq Ñ SpecpDirac π pM ˆN q 1 R q, defined by Θ MˆN p E y , f | x qpφq " φp E y , f qpxq for φ P Dirac π pM ˆN q 1 R .We will now use this bijection to transfer the smooth structure from SpecpDirac π pM ˆN q 1 R q to E sp pDirac π pM ˆN qq.Proof.Let us denote by Sp : SpecpC 8 pM ˆN, Rqq Ñ SpecpDirac π pM ˆN q 1 R q the diffeomorphism, induced by the inverse of ˆ: C 8 pM ˆN, Rq Ñ Dirac π pM ˆN q 1 R .We then have the commutative diagram Let us now take a look at this construction in the case of a single manifold.
Example 4.5.Let M be a single point and consider the manifold N as a trivial bundle over a point.In this case we have Dirac π pM ˆN q " DiracpN q " Spantδ y | y P N u.
Every element u P DiracpN q can be expressed as a finite sum u " ř n i"1 λ i δ yi for unique λ 1 , . . ., λ n P C and y 1 , . . ., y n P N .The space DiracpN q is a coalgebra over C with structure maps: Grouplike elements of DiracpN q are precisely Dirac distributions, so we have GpDiracpN qq " tδ y | y P N u.
The spectral sheaf E sp pDiracpN qq is the set GpDiracpN qq with the discrete topology and projection onto the point.The map Θ N : E sp pDiracpN qq Ñ SpecpDiracpN q 1 R q is defined by Θ N pδ y qpφq " φpδ y q, which means that Θ N pδ y q " δy .The topology on B sp pDiracpN qq " GpDiracpN qq, which is induced by Θ N , coincides with the subspace topology on GpDiracpN qq, induced from E 1 pN q.Finally, the diffeomorphism Φ bun N : N Ñ B sp pDiracpN qq is given by Φ bun N pyq " δ y .

Locally convex bialgebroid of an action Lie groupoid
In this section we will assign to each action groupoid M ¸H a locally convex bialgebroid with antipode DiracpM ¸Hq over C 8 c pM q, from which the Lie groupoid M ¸H can be reconstructed.
Let M be a second-countable manifold and let H be a second-countable Lie group, which acts on M from the right.If we denote by H # the group H with the discrete topology, the group H # acts on M from the right as well, so we obtain two action groupoids M ¸H and M ¸H# .These two groupoids are isomorphic as groupoids but not as Lie groupoids if dimpHq ą 0.
Groupoid M ¸H# is étale, so we can construct its Hopf algebroid Moreover, since M ¸H is a Lie groupoid, we also have a convolution product, as defined in [15], on the space E 1 t pM ¸Hq " Hom C 8 c pMq pC 8 pM ˆHq, C 8 c pM qq of t-transversal distributions on M ¸H.It can be described explicitly as follows.The left translation by g P M ¸H is the diffeomorphism L g : t ´1pspgqq Ñ t ´1ptpgqq, defined by L g phq " gh.For any F P C 8 pM ˆHq and any g P M ¸H it follows that F ˝Lg P C 8 pt ´1pspgqqq and one can show that the function M ˆH Ñ R, g Þ Ñ T spgq pF ˝Lg q, is smooth for any T P E 1 t pM ¸Hq.For any T 1 , T E hi , f i .
Proposition 5.1.The map Ψ M¸H : C 8 c pM ¸H# q Ñ E 1 t pM ¸Hq is multiplicative.
Multiplicativity of the map Ψ M¸H now follows from linearity of Ψ M¸H and bilinearity of both convolution products.Definition 5.2.Let M ¸H be an action groupoid of an action of a second-countable Lie group H on a second-countable manifold M and let M ¸H# be the assoicated étale groupoid.The Dirac bialgebroid of M ¸H is the space DiracpM ¸Hq " Ψ M¸H pC 8 c pM ¸H# qq.The Dirac bialgebroid DiracpM ¸Hq inherits from C 8 c pM ¸H# q a structure of a locally grouplike Hopf algebroid over C 8 c pM q.Moreover, by Proposition 4.1 we obtain on DiracpM ¸Hq a structure of a locally convex coalgebra.Finally, as shown in [15], the multiplication on E 1 t pM ¸Hq and hence on DiracpM ¸Hq is separately continuous.We sum up these observations in the following proposition.Remark 5.4.A locally convex bialgebroid is a bialgebroid pA, ∆, ǫ, µq, equipped with a locally convex structure such that ∆ and ǫ are continuous maps and µ is separately continuous.We do not know if the antipode S on DiracpM ¸Hq is continuous in general, which would mean that it is a locally convex Hopf algebroid.
Example 5.5.Let us take a look at the case when the group H acts trivially on M .The associated action groupoid M ¸H is in this case just the trivial bundle of Lie groups over M with fiber H, which will be denoted by M ˆH.The multiplication and antipode can be expressed on generators by the formulas: Sp E h , f q " E h ´1 , f .
In this case C 8 c pM q is a central subalgebra of DiracpM ˆHq.Moreover, from the equality t ˝inv " t it follows that S is continuous.As a result we see that DiracpM ˆHq is a locally convex Hopf algebra over C 8 c pM q.
Now take any action Lie groupoid M ¸H.The spectral étale Lie groupoid G sp pDiracpM ¸Hqq of DiracpM ¸Hq is then isomorphic to the étale groupoid M ¸H# .Moreover, since DiracpM ¸Hq is a locally convex coalgebra over C 8 c pM q, we also have the bijection Θ M¸H : G sp pDiracpM ¸Hqq Ñ SpecpDiracpM ¸Hqq 1 R q. Proof.Since M ¸H is isomorphic to M ¸H# and AG sp pDiracpM ¸Hqq is isomorphic to G sp pDiracpM ¸Hqq, the map Φ agr M¸H is an isomorphism of groupoids.By Theorem it is also a diffeomorphism, which implies that it is an isomorphism of Lie groupoids.
Example 5.8.Let H be a Lie group so that DiracpHq is a locally convex Hopf algebra over C. Example 4.5 shows that AG sp pDiracpHqq " tδ h | h P Hu is naturally diffeomorphic to H.The multiplication and inverse maps on AG sp pDiracpHqq are induced by the multiplication and the antipode on DiracpHq.Namely, for any h, h 1 P H we have:

Proposition 5 . 3 .
The Dirac bialgebroid DiracpM ¸Hq of any action Lie groupoid M ¸H is a locally convex bialgebroid with an antipode over C 8 c pM q.

Definition 5 . 6 .Theorem 5 . 7 .
The spectral action Lie groupoid AG sp pDiracpM ¸Hqq of the Dirac bialgebroid DiracpM ¸Hq is the groupoid G sp pDiracpM ¸Hqq, equipped with the smooth structure such that the map Θ M¸H is a diffeomorphism.Define a map Φ agr M¸H : M ¸H Ñ AG sp pDiracpM ¸Hqq, by Φ agr M¸H px, hq " E h , f | x , where f P C 8 c pM q is such that f | x " 1 P C 8 c pM q x .Let M ¸H be an action groupoid of an action of a second-countable Lie group H on a second-countable manifold M .The map Φ agr M¸H : M ¸H Ñ AG sp pDiracpM ¸Hqq is an isomorphism of Lie groupoids.
Definition 4.3.Let π : M ˆN Ñ M be a trivial bundle over M with fiber N .The spectral bundle B sp pDirac π pM ˆN qq of the coalgebra Dirac π pM ˆN q is the set E sp pDirac π pM ˆN qq, equipped with the bundle projection π sp : B sp pDirac π pM ˆN qq Ñ M and the topology and smooth structure such that Θ MˆN is a diffeomorphism.We will show in the next theorem that B sp pDirac π pM ˆN qq is a trivial bundle over M , naturally isomorphic to the bundle M ˆN .Define a map Φ bun Theorem 4.4.Let π : M ˆN Ñ M be a trivial bundle over M with fiber N .The map Φ bun MˆN : M ˆN Ñ B sp pDirac π pM ˆN qq is an isomorphism of trivial bundles over M .
2P E 1 t pM ¸Hq the convolution T 1 ˚T 2 P E 1 t pM ¸Hq is then defined bypT 1 ˚T 2 qpF qpxq " T 1 ´g Þ Ñ T 2 spgq pF ˝Lg q ¯pxq,for any F P C 8 pM ˆHq and any x P M .Using the notation from the Example 2.2 we define an injective C 8 c pM q-linear map Ψ M¸H : C 8 c pM ¸H# q Ñ E 1 t pM ¸Hq by