Reproducing Kernel Hilbert Space Associated with a Unitary Representation of a Groupoid
aa r X i v : . [ m a t h . F A ] F e b Reproducing Kernel Hilbert Space Associated with a UnitaryRepresentation of a Groupoid
Monika Drewnik , Tomasz Miller , and Zbigniew Pasternak-Winiarski College of Rehabilitation, Department of Rehabilitation, Kasprzaka 49, 01-234 Warsaw, Poland Copernicus Center for Interdisciplinary Studies, Jagiellonian University,Szczepańska 1/5, 31-011 Kraków, Poland Faculty of Natural and Health Sciences, The John Paul II Catholic University of Lublin,Konstantynów 1 H, 20-708 Lublin, Poland
February 22, 2021
Abstract
The aim of the paper is to create a link between the theory of reproducing kernel Hilbertspaces (RKHS) and the notion of a unitary representation of a group or of a groupoid. Morespecifically, it is demonstrated on one hand, how to construct a positive definite kernel and anRKHS for a given unitary representation of a group(oid), and on the other hand how to retrievethe unitary representation of a group or a groupoid from a positive definite kernel defined onthat group(oid) with the help of the Moore–Aronszajn theorem. The kernel constructed fromthe group(oid) representation is inspired by the kernel defined in terms of the convolutionof functions on a locally compact group. Several illustrative examples of reproducing kernelsrelated with unitary representations of groupoids are discussed in detail. The paper is concludedwith the brief overview of the possible applications of the proposed constructions.
Keywords:
Reproducing Kernel Hilbert Spaces, unitary representation, groupoid, group, Haarmeasure, convolution
The theory of reproducing kernel Hilbert spaces (RKHS) provides research tools in such domainsas complex analysis, probability theory and statistics [2], stochastic (Gaussian) processes [14], quan-tum physics [15, 16] or computer science (especially artificial intelligence [9, 11]). Basic propertiesand definitions together with the detailed analysis of RKHS can be found in [18, 23]. In [1] RKHSassociated with the continuous wavelet transform generated by the irreducible representations ofthe Euclidean Motion SE(2) are considered. ∗ Corresponding author: [email protected] C ∗ -algebras [7, 22]. The definition and main properties of(locally compact) groupoids can be found for example in [6, 17]. In [20, 21] unitary representationsof groupoids are considered. The applications of theory of finite groudoids and their representationsare presented in [12, 13]. Considering representations of locally compact groups, the Haar measureand the tensor product of the Hilbert spaces, we refer the Reader to [24].The aim of te present paper is to connect the above two domains. To this end, we study therelationship between unitary representations of groupoids and reproducing kernel Hilbert spaces,proposing how to construct one using the other provided certain conditions are met.The content of the paper is as follows. In Section 2 the fundamental definitions and propertiesof the theory of reproducing kernel Hilbert spaces are introduced. Section 3 covers the concept ofgroupoids (propositions, examples and the notion of their unitary representation). Main results ofthe paper are contained in Section 4. It combines the ideas presented in the two previous sections.The constructions of reproducing kernels associated to a unitary representation of a group and toa unitary representation of a groupoid are described in Subsections 4.1 and 4.2, respectively. Thelatter subsection is concluded with several illustrative examples. Section 5 contains brief discussionon the possible applications of the notions studied in the article. Definition 1.
Let A be a nonempty set. The map K : A × A → C shall be called a kernel on A .We say that the kernel K is positive definite if ∀ n ∈ N ∀ a ,...,a n ∈ A ∀ λ ,...,λ n ∈ C n X k =1 n X l =1 λ k K ( a k , a l )¯ λ l . Let H be an inner product space of complex-valued functions on A equipped with the innerproduct h· , ·i . Definition 2.
The family { K a } a ∈ A ⊂ H is called a reproducing family of H if ∀ a ∈ A ∀ f ∈ H f ( a ) = h f, K a i . (1)Equality (1) is called the reproducing kernel property , and the function K : A × A → C defined as K ( b, a ) := h K a , K b i = K a ( b ) , a, b ∈ A is called a reproducing kernel on the space H . If the latter is a Hilbert space, we call H a reproducingkernel Hilbert space (RKHS).Clearly, by the Riesz representation theorem, any RKHS has a unique reproducing family andthus a unique reproducing kernel, which can be easily shown to be positive definite. The followingseminal result shows that actually the converse is also true: Every positive definite kernel is areproducing kernel on a certain Hilbert space. We adopt the convention that inner products are anti-linear with respect to the second argument. heorem 1 (Moore–Aronszajn) . Let K be a positive definite kernel on a non-empty set A . Thenthere is a unique Hilbert space H ( K ) of complex-valued functions on A with the reproducing kernel K .Proof. (A sketch; for details, see [18, Theorem 2.14]) One considers the vector space H ( K ) :=span { K a } a ∈ A . The map h· , ·i H : H ( K ) × H ( K ) → C given by * m X l =1 λ l K a l , n X k =1 β k K b k + H := n X k =1 m X l =1 λ l ¯ β k K ( b k , a l ) (2)can be demonstrated to be a well-defined inner product. Then one shows that the completion of H ( K ) in the norm induced by that inner product, denoted H ( K ) := ^ H ( K ) = ] span { K a } a ∈ A ,can still be interpreted as a space of complex-valued functions on A , with K as its reproducingkernel.In what follows, we shall be using the following general way of constructing positive definitekernels. Theorem 2.
Let F : A → H be any function from a nonempty set A to a Hilbert space H equippedwith an inner product h· , ·i . Then: • The map K : A × A → C defined as K ( b, a ) := h F ( a ) , F ( b ) i is a positive definite kernel. • By Theorem 1, there exists an RKHS H ( K ) := g span { K ( · , a ) } a ∈ A , for which K is the repro-ducing kernel. • The linear map T : H → H ( K ) defined via ( T v )( a ) := h v, F ( a ) i is a surjective contraction.Moreover, it becomes an isometry when restricted to the closed subspace V := span F ( A ) ofthe space H . • Defining S : H ( K ) → H as S := T | − V , we obtain an isometric embedding of the RKHS H ( K ) into H that satisfies S ( K a ) = F ( a ) for every a ∈ A .Proof. Cf. [23, p. 13].Notice that the third bullet of the above theorem carries a lot of information about the functionsbelonging to H ( K ). For example, if F is weakly continuous, then H ( K ) ⊂ C ( A ).The (unique) reproducing kernel of a given RKHS turns out be tightly related to the so-calledParseval frames, which can be thought of as generalizations of complete orthonormal systems ofvectors. Definition 3 (cf. Definition 2.6 & Proposition 2.8 in [18]) . Let H be a Hilbert space (not necessarilyof functions) with an inner product h· , ·i . The set { w j } j ∈ I ⊂ H is called a Parseval frame for H if k v k = P j ∈ I |h v, w j i| for every v ∈ H or, equivalently, if v = P j ∈ I h v, w j i w j for every v ∈ H . Theorem 3 (Papadakis) . Let H be an RKHS of functions on A with reproducing kernel K . Thenthe family { ϕ j } j ∈ I ⊂ H is a Parseval frame for H iff K ( b, a ) = X j ∈ I ϕ j ( b ) ϕ j ( a ) , a, b ∈ A, (3) where the series converges pointwise.Proof. See [18, Theorem 2.10]. 3 .2 Reproducing kernel defined by the convolution of functions on alocally compact group
Let G be a locally compact group (not necessarily abelian) and let µ be a fixed left Haarmeasure on G . For two continuous compactly supported complex-valued functions f , f ∈ C c ( G ),their convolution f ∗ f is another such function on G defined via( f ∗ f )( g ) = Z G f ( τ ) f ( τ − g ) dµ ( τ ) . The above definition extends to the space L ( G, µ ) in the following sense [10, 444O & 444R]:for any f , f ∈ L ( G, µ ) the convolution f ∗ ˜ f , where ˜ f ( g ) := f ( g − ), is a well-defined elementof C b ( G ) and, moreover, | ( f ∗ ˜ f )( g ) | ¬ k f k L k f k L for every g ∈ G . Example 1.
Let G be a locally compact group and µ be a left Haar measure on G . Fix f ∈ L ( G, µ )and consider the map K : G × G → C defined via K ( h, g ) := ( f ∗ ˜ f )( g − h ) , g, h ∈ G. By the preceding discussion, K is bounded and jointly continuous. Moreover, it is a positive definitekernel, because for any n ∈ N , λ , . . . , λ n ∈ C and g , . . . , g n ∈ G one has that n X i,j =1 λ i K ( g i , g j ) λ j = n X i,j =1 λ i λ j Z G f ( τ ) ˜ f ( τ − g − j g i ) dµ ( τ ) = n X i,j =1 λ i λ j Z G f ( τ ) f ( g − i g j τ ) dµ ( τ )= n X i,j =1 λ i λ j Z G f ( g − j τ ) f ( g − i τ ) dµ ( τ ) = Z G (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X j =1 λ j f ( g − j τ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( τ ) , where in the antepenultimate step we employed the left-invariance of µ .By Theorem 1, K is a reproducing kernel on a certain Hilbert space H ( K ) equipped with theinner product h· , ·i H ( K ) with the reproducing family { K g := K ( · , g ) } g ∈ G . In fact, H ( K ) is a certainspace of bounded continuous functions on G , which can be isometrically embedded into L ( G, µ ).To see why this is the case, let L g : G → G , ξ gξ denote the left shift by the element g ∈ G .Employing the pullback L ∗ g f := f ◦ L g , one can write that K ( h, g ) = Z G f ( τ ) ˜ f ( τ − g − h ) dµ ( τ ) = Z G f ( g − τ ) f ( h − τ ) dµ ( τ ) = D L ∗ g − f, L ∗ h − f E L (4)for any g, h ∈ G . Thus, the above construction is actually a special case of the one described inTheorem 2, with the map F : G → L ( G, µ ) defined as F ( g ) := L ∗ g − f for every g ∈ G . Hence theelements of H ( K ) are mappings of the form g
7→ h v, L ∗ g − f i with v ∈ L ( G, µ ), which are clearlybounded (by k v k L k f k L ) and continuous, whereas the isometric embedding S : H ( K ) → L ( G, µ )satisfies S ( K g ) = L ∗ g − f .As a side remark, observe that the above positive definite kernel K can also be expressed as K ( h, g ) = Z G f ( g − τ ) f ( h − τ ) dµ ( τ ) = Z G ˜ f ( τ − h ) ˜ f ( τ − g ) dµ ( τ )= Z G L ∗ τ − ˜ f ( h ) L ∗ τ − ˜ f ( g ) dµ ( τ ) , g, h ∈ G, { ϕ j } j ∈ I and a countingmeasure we have the family { L ∗ τ − ˜ f } τ ∈ G of the left-translations of the map ˜ f and the left Haarmeasure µ . Groupoids provide both a generalization of groups [3] and of equivalence relations [17].
Definition 4.
Let Γ , X be nonempty sets. A groupoid
Γ over a set X is a septuple (Γ , X, s, r, ε, · , − )with the below described mappings:1. the source mapping s : Γ −→ X , which is surjective.2. the range mapping r : Γ −→ X , which is surjective.3. the multiplication mapping · : Γ (2) −→ Γ, where Γ (2) := { ( γ , γ ) ∈ Γ × Γ | r ( γ ) = s ( γ ) } .The multiplication is associative. For simplicity, instead of γ · γ we write γ γ .
4. the embedding map ε : X −→ Γ fulfilling ε ( r ( γ )) γ = γ = γε ( s ( γ )) .
5. the inversion map − : Γ −→ Γ such that ∀ γ ∈ Γ γ − γ = ε ( s ( γ )) and γγ − = ε ( r ( γ )) . The element γ ∈ Γ can be regarded as an arrow with the starting point at x = s ( γ ) and theending point at y = r ( γ ), where x, y ∈ X . • y w w γ • x Remark . From the above definition it follows that the groupoid (Γ , X, s, r, ε, · , − ) can be identifiedwith a small category in which all morphisms are invertible. In this interpretation X is the set ofobjects, Γ is the set of morphisms, s ( γ ) is the domain of γ , r ( γ ) is the codomain of γ , ε ( x ) is theidentity morphism at x , · is the composition of morphisms and γ − is the inverse of the morphism γ ∈ Γ . Example 2.
A group G is a groupoid over the singleton X := { x } with the mappings definedas ∀ g ∈ G s ( g ) := x, r ( g ) := x and ε ( x ) := e — the unit of the group G . The inversion andmultiplication mappings are given by the respective group operations.Let Γ be a groupoid over the set X . It is easy to show that the relation ∼ on the set X definedas follows: ∀ x,y ∈ X x ∼ y ⇔ ∃ γ ∈ Γ x = s ( γ ) ∧ y = r ( γ )is an equivalence relation on X . The equivalence class of any element x with respect to ∼ is the setOr x := [ x ] ∼ = { y ∈ X | ∃ γ ∈ Γ s ( γ ) = x ∧ r ( γ ) = y } = r (Γ x ) , x := { γ ∈ Γ | s ( γ ) = x } = s − ( x ) . We call Or x the orbit of the element x ∈ X with respect to the groupoid Γ . We say that Γ istransitive if for any x, y ∈ X we have x ∼ y (or, equivalently, if Or x = X for some and hence forevery x ∈ X ).Defining the fibers of the groupoid Γ as Γ x := s − ( x ) and Γ x := r − ( x ) and using the abovedefinition of the equivalence relation ∼ , we have r ( s − ( x )) = s ( r − ( x )).The set of arrows starting at x and ending at y is denoted by Γ yx := Γ x ∩ Γ y . The set Γ xx :=Γ x ∩ Γ x = { γ ∈ Γ | r ( γ ) = s ( γ ) = x } of elements starting and ending at x together with the groupoidmultiplication and inversion map has a group structure, and we call it the isotropy group of theelement x . Remark . If Γ is a transitive groupoid over X then it is easy to show that for any y ∈ X thegroups Γ xx and Γ yy are isomorphic (albeit not canonically!). Without the transitivity, however, itmay happen that for some x = y the fibers Γ xx and Γ yy are not isomorphic as groups. Remark . If (Γ , X, s, r, ε, · , − ) is a groupoid and Γ , X are topological spaces then we call Γ atopological groupoid if all maps s, r, ε, · , − are continuous. In this case the maps ε, − are homeo-morphisms onto their images and all isotropy groups are topological groups. Recall that a unitary representation of a group G is a pair ( U , H ), where H is a Hilbert spaceand U is a map assigning to every g ∈ G a unitary operator U ( g ) : H → H satisfying U ( g g ) = U ( g ) U ( g ) for all g , g ∈ G . A standard example of a unitary representation of a (locally compact)group is the left regular representation U ( g ) := L ∗ g − : L ( G, µ ) → L ( G, µ ), which appeared inExample 1 above.The notion of a unitary representation can be extended onto groupoids.
Definition 5.
Let Γ be a groupoid over the set X . A pair ( U , H ) is called a unitary representation of the groupoid Γ if H := { H x } x ∈ X is a family of Hilbert spaces and U is a mapping assigning toeach γ ∈ Γ a unitary transformation U ( γ ) : H s ( γ ) → H r ( γ ) in such a way that U ( γ γ ) = U ( γ ) U ( γ )for every ( γ , γ ) ∈ Γ (2) .Notice that from the above definition it already follows that• U ( ε ( x )) = id H x for every x ∈ X .• U ( γ − ) = U ( γ ) − = U ( γ ) † for every γ ∈ Γ.From the family H := { H x } x ∈ X one can construct a new Hilbert space H := fL x ∈ X H x definedas the completion of the direct sum L x ∈ X H x with respect to the norm generated by the innerproduct P x ∈ X h· , ·i x , where h· , ·i x denotes the inner product on H x .Many authors [8, 17, 21] use a more general and somewhat more involved definition of a unitaryrepresentation of a groupoid, in which H is a Hilbert bundle or a measurable field of complex Hilbertspaces . This allows for constructing many more interesting Hilbert spaces from the fibers H x thanjust the completed direct sum H above. In the present paper, however, we keep the definition simpleso that the constructions of the positive definite kernels and RKHS become more tractable. Example 3.
Let Γ be a groupoid over X and H be a fixed Hilbert space equipped with an innerproduct h· , ·i . Consider the family { H x := { x } × H } x ∈ X with H x endowed with the inner product6 ( x, h ) , ( x, h ) i x := h h , h i . For any γ ∈ Γ yx , x, y ∈ X the transformation U ( γ ) : H x → H y defined by U ( γ )( x, h ) := ( y, h ) (change of the fiber base point) is of course unitary. Any such arepresentation of the groupoid Γ is called a trivial representation .A less trivial standard examples of groupoid representations arise for locally compact topologicalgroupoids. Observe that for any such a groupoid Γ, the fibers Γ x := r − ( x ) and Γ x := s − ( x ) forany x ∈ X are locally compact spaces, too, and the same concerns the base space X itself. On suchgroupoids one introduces the following generalization of Haar measures [17, 21]. Definition 6.
Let Γ be a locally compact topological groupoid. The family { λ x } x ∈ X of regularBorel measures on Γ x is called a left Haar system for the groupoid Γ if1. For every f ∈ C c (Γ) the function f : X → C given by f ( x ) := R Γ x f dλ x is continuous,2. For every γ ∈ Γ and every f ∈ C c (Γ) Z Γ s ( γ ) f ( γχ ) dλ s ( γ ) ( χ ) = Z Γ r ( γ ) f ( χ ) dλ r ( γ ) ( χ ) . Similarly, the family { λ x } x ∈ X of regular Borel measures on Γ x is called a right Haar system for thegroupoid Γ if1. ′ For every f ∈ C c (Γ) the function f : X → C given by f ( x ) := R Γ x f dλ x is continuous,2. ′ For every γ ∈ Γ and every f ∈ C c (Γ) Z Γ r ( γ ) f ( χγ ) dλ r ( γ ) ( χ ) = Z Γ s ( γ ) f ( χ ) dλ s ( γ ) ( χ ) . Observe that both f and f are automatically compactly supported. Conditions 1. and 1. ′ expressthe demand for the measures λ x and λ x to vary continuously over X , whereas conditions 2. and 2. ′ generalize the notions of, respectively, left and right invariance of Haar measures on locally compactgroups. Example 4.
For any x ∈ X let H x = L (Γ x , λ x ), where { λ x } x ∈ X is a left Haar system on Γ. Forany γ ∈ Γ yx and for f ∈ H x define a unitary transformation U ( γ ) : H x → H y by( U ( γ ) f )( χ ) = f ( γ − χ ) , χ ∈ Γ y . Such a representation ( U , { H x } x ∈ X ) is called the left regular representation of the groupoid Γ. Example 5.
For any x ∈ X let H x = L (Γ x , λ x ), where { λ x } x ∈ X is a right Haar system on Γ. Forany γ ∈ Γ yx and for f ∈ H x define a unitary transformation U ( γ ) : H x → H y by( U ( γ ) f )( χ ) = f ( χγ ) , χ ∈ Γ y . Such a representation ( U , { H x } x ∈ X ) is called the right regular representation of the groupoid Γ.For more examples of unitary representations of groupoids, the Reader is referred to [12, 21].7 Reproducing kernels and unitary representations
Let G be a group and ( U , H ) be its unitary representation on a Hilbert space H equipped withan inner product h· , ·i . Additionally, let v ∈ H be any fixed vector. Formula (4) in Example 1suggests considering the kernel K : G × G → C defined as K ( h, g ) := hU ( g ) v, U ( h ) v i . (5)Notice that thus defined K is a special case of the construction presented in Theorem 2 with themap F : G → H given by F ( g ) := U ( g ) v , g ∈ G . This means, in particular, that the RKHS H ( K )provided by the Moore–Aronszajn theorem is a space of functions of the form g
7→ h w, U ( g ) v i ,where w ∈ H . Notice that, if U is a weakly continuous unitary representation of a topological group G , then H ( K ) ⊂ C ( G ). Moreover, if { w j } j ∈ I ⊂ H is a Parseval frame for H , then K ( h, g ) = hU ( g ) v, U ( h ) v i = X j ∈ I hU ( g ) v, w j ih w j , U ( h ) v i = X j ∈ I ϕ j ( h ) ϕ j ( g ) , where ϕ j := T w j ∈ H ( K ) (cf. the third bullet in Theorem 2). On the strength of Theorem 3, { ϕ j } j ∈ I is a Parseval frame for H ( K ).By the unitarity of the representation, the kernel K satisfies K ( h, g ) = K ( g − h, e ) for any g, h ∈ G, (6)where e is the unit element of G .Conversely, suppose we are given a positive definite kernel K : G × G → C on a group G satisfying (6). We can employ the Moore–Aronszajn theorem to define its representation ( U , H ( K )).Concretely, for any g ∈ G define U ( g ) P ni =1 λ i K h i := P ni =1 λ i K gh i for any element of H ( K ) :=span { K h } h ∈ G . Since there is no guarantee that the system { K h } h ∈ G is linearly independent, wemust check that such a U ( g ) is well defined. To this end, it suffices to prove that if P ni =1 λ i K h i ≡ G , then also P ni =1 λ i K gh i ≡
0. But thanks to (6) we have that, for any h ∈ G , n X i =1 λ i K gh i ( h ) = n X i =1 λ i K h i ( g − h ) = 0by assumption. Since H ( K ) is dense in H ( K ), thus defined U ( g ) can be uniquely extended to H ( K ). Moreover, also by (6) and by the density argument, one obtains the unitarity of U ( g ) byverifying that for any h, h ′ ∈ G hU ( g ) K h , U ( g ) K h ′ i H ( K ) = h K gh , K gh ′ i H ( K ) = K ( gh ′ , gh ) = K (( gh ) − gh ′ , e )= K ( h − g − gh ′ , e ) = K ( h − h ′ , e ) = K ( h, h ′ ) = h K h ′ , K h i H ( K ) . Observe, finally, that one can retrieve the kernel K from the above ‘Moore–Aronszajn represen-tation’ through formula (5). Indeed, one simply has to take v := K e .8 .2 Reproducing kernel associated to a unitary representation of a groupoid Let us now generalize the above relationship between unitary representations of groups andreproducing kernels onto the groupoid setting. Let thus Γ be a groupoid over X and ( U , H = { H x } x ∈ X ) be its unitary representation as specified by Definition 5. Additionally, let v be a fixed vector field , by which we shall understand a mapping X ∋ x v ( x ) ∈ H x (note: vector fields neednot belong to H := fL x ∈ X H x ). Define the kernel K : Γ × Γ → C via K ( χ, γ ) := (cid:26) hU ( γ ) v ( s ( γ )) , U ( χ ) v ( s ( χ )) i r ( γ ) for r ( γ ) = r ( χ ) , r ( γ ) = r ( χ ) . (7)Observe that for each x ∈ X the restriction K x := K | Γ x × Γ x is a positive definite kernel — itconstitutes a special case of the construction presented in Theorem 2 with F x : Γ x → H x givenby F x ( γ ) := U ( γ ) v ( s ( γ )) for every γ ∈ Γ x . Invoking the Moore–Aronszajn theorem (Theorem 1),we obtain an RKHS given by H ( K x ) := ] span (cid:8) K xγ := K x ( · , γ ) (cid:9) γ ∈ Γ x , whose reproducing kernel is K x . By the third bullet of Theorem 2, every element of H ( K x ) is a function on Γ x of the form γ
7→ h w x , U ( γ ) v ( s ( γ )) i , where w x ∈ H x . Finally, by the fourth bullet of Theorem 2, for any x ∈ X the Hilbert space H ( K x ) can be isometrically embedded into H x , its image being the closed sub-space V x := span {U ( γ ) v ( s ( γ )) } γ ∈ Γ x .Also the kernel K itself is a realization of the general construction described in Theorem 2. Tosee this, consider F : Γ → H given by F ( γ ) := U ( γ ) v ( s ( γ )) and observe that h F ( γ ) , F ( χ ) i H := P x ∈ X h F ( γ ) , F ( χ ) i x indeed equals K ( χ, γ ) for all γ, χ ∈ Γ, because H x ⊥ H y as subspaces of H for x = y . Therefore, K is positive definite and the Moore–Aronszajn theorem yields an RKHS definedas H ( K ) := ] span { K γ := K ( · , γ ) } γ ∈ Γ .Notice that the two above constructions are of RKHS compatible in the sense that H ( K ) = gM x ∈ X H ( K x ) . (8)Indeed, observe that both spaces have the same dense subspaces, namely (cf. the sketch of the proofof Theorem 1 above) H ( K ) := span { K γ } γ ∈ Γ = M x ∈ X span { K γ } γ ∈ Γ x = M x ∈ X span (cid:8) K xγ (cid:9) γ ∈ Γ x =: M x ∈ X H ( K x ) , where the maps K γ : Γ → C and K xγ : Γ x → C have been identified (the former being the extensionby zero of the latter). Since both above spaces are equipped with the same inner product (given by(2)), they yield the same Hilbert spaces after completion.Similarly as before, by the third bullet of Theorem 2, every element of H ( K ) is a function on Γof the form γ
7→ h w, U ( γ ) v ( s ( γ )) i , where w ∈ H . Notice that such functions must vanish on all butcountably many fibers Γ x . Moreover, if { w j } j ∈ I ⊂ H is a Parseval frame for H , then K ( χ, γ ) = hU ( γ ) v ( s ( γ )) , U ( χ ) v ( s ( χ )) i H = X j ∈ I hU ( γ ) v ( s ( γ )) , w j i H h w j , U ( χ ) v ( s ( χ )) i H = X j ∈ I ϕ j ( χ ) ϕ j ( γ ) , ϕ j := h w j , U ( · ) v ( s ( · )) i H . Similarly as in the group case, by Theorem 3 we obtain that { ϕ j } j ∈ I is a Parseval frame for H ( K ).Finally, by the fourth bullet of Theorem 2, the Hilbert space H ( K ) can be isometrically embed-ded into H , its image being the closed subspace V := span {U ( γ ) v ( s ( γ )) } γ ∈ Γ = L x ∈ X V x , wherethe last equality can be proven completely analogously to (8).Additionally, the unitarity of the representation implies that for γ, χ ∈ Γ such that r ( γ ) = r ( χ ) K ( χ, γ ) = K ( γ − χ, ε ( r ( γ − χ ))) , (9)which is nothing but a straightforward generalization of (6). Indeed, one has K ( χ, γ ) = hU ( γ ) v ( s ( γ )) , U ( χ ) v ( s ( χ )) i r ( γ ) = (cid:10) v ( s ( γ )) , U ( γ − χ ) v ( s ( χ )) (cid:11) s ( γ ) = K ( γ − χ, ε ( s ( γ ))) = K ( γ − χ, ε ( r ( γ − χ ))) . Let us now consider the converse problem. That is, given a groupoid Γ over X and a pos-itive definite kernel K : Γ × Γ → C such that K ( γ, χ ) = 0 if r ( γ ) = r ( χ ) and (9) holds, weconstruct the ‘Moore–Aronszajn representation’ ( U , { H ( K x ) } x ∈ X ) of Γ. To this end, define each U ( γ ) : H ( K s ( γ ) ) → H ( K r ( γ ) ) first on the dense subspace H ( K s ( γ ) ) := span { K χ } χ ∈ Γ s ( γ ) by U ( γ ) n X i =1 λ i K χ i := n X i =1 λ i K γχ i , where, similarly as for groups, one can easily check that this definition is sound (analogously as inthe group case, it is here where property (9) steps in). Observe that r ( χ i ) = s ( γ ) for all i = 1 , . . . , n ,so the products γχ i are all well defined. It is also straightforward to prove (again, thanks to (9))that thus defined U ( γ ) preserves inner products. Extending it onto the entire H ( K s ( γ ) ), by thearbitrariness of γ we obtain the desired unitary representation of Γ.Finally, notice that the kernel K can be retrieved from the ‘Moore–Aronszajn representation’through formula (7), where one has to take the vector field v ( x ) := K ε ( x ) , x ∈ X .Before moving to examples, let us remark that formula (7) ‘works well’ with the basic algebraicaloperations on the groupoid representations. Remark . Let Γ be a groupoid over X and let ( U ( j ) , H ( j ) := { H ( j ) x } x ∈ X ), j = 1 , , . . . , l , be itsunitary representations. Fixing l vector fields v , v , . . . , v l on X such that v j ( x ) ∈ H ( j ) x for x ∈ X and j = 1 , , . . . , l we can construct positive definite kernels K , K , . . . , K l on Γ, respectively,using formula (7). Consider now the direct sum ( L lj =1 U ( j ) , L lj =1 H ( j ) := { L lj =1 H ( j ) x } x ∈ X ) ofthe above representations, and, taking the vector field v ⊕ := L lj =1 v j , define a kernel K ⊕ on Γvia (7). Similarly, consider the tensor product ( N lj =1 U ( j ) , N lj =1 H ( j ) = { N lj =1 H ( j ) x } x ∈ X ) of therepresentations and, taking this time the vector field v ⊗ := N lj =1 v j , define another kernel K ⊗ onΓ, again employing formula (7). It follows from [23, p. 16,17] that K ⊕ = K + K + . . . + K l and K ⊗ = K · K · . . . · K l . Remark . Suppose now we have a family of groupoids { (Γ i , X i , s i , r i , ε i , · i , − i ) } indexed by i ∈ I .Let also ( U ( i ) , H ( i ) := { H ( i ) x } x ∈ X i ) be a unitary representation of the groupoid Γ i for every i ∈ I .The disjoint union Γ := F i ∈ I Γ i has a natural groupoid structure with X := F i ∈ I X i as a base10pace, Γ (2) := F i ∈ I Γ (2) i , γ · γ := γ · i γ for γ , γ ∈ Γ (2) i , ε ( x ) := ε i ( x ) for x ∈ X i and, moreover, s ( γ ) := s i ( γ ), r ( γ ) := r i ( γ ), γ − := γ − i for γ ∈ Γ i .Consider now the family of Hilbert spaces H := { H x } x ∈ X , where we put H x := H ( i ) x for x ∈ X i and define a mapping U assigning to each γ ∈ Γ a unitary transformation U ( γ ) : H s ( γ ) → H r ( γ ) by U ( γ ) := U ( i ) ( γ ) for γ ∈ Γ i . Then ( U , H ) is a unitary representation of Γ.For any i ∈ I let us fix a vector field X i ∋ x → v i ( x ) ∈ H ( i ) x and use formula (7) to obtain apositive definite kernel K i from the unitary representation ( U ( i ) , H ( i ) ) of Γ i . Additionally, introducea vector field v on X by setting v ( x ) := v i ( x ) if x ∈ X i and use it together with the representation( U , H ) to obtain, again via (7), another positive definite kernel K on Γ. It is not difficult to observethat K ( χ, γ ) = (cid:26) K i ( χ, γ ) for χ, γ ∈ Γ i , χ ∈ Γ i , γ ∈ Γ i ′ with i = i ′ . (10)What is more, reasoning analogously as when proving formula (8), one can show that the RKHS H ( K ) obtained from K by means of the Moore–Aronszajn theorem satisfies H ( K ) = fL i ∈ I H ( K i ). Example 6.
Fix a Hilbert space H endowed with the inner product h· , ·i and let ( U , { H x := { x } × H } x ∈ X ) be the trivial representation of the groupoid Γ. For any fixed vector field v , whichhere can be regarded as an element of H X , we have U ( γ ) v ( s ( γ )) = v ( r ( γ )) and formula (7) yieldsthe kernel K ( χ, γ ) = (cid:26) h v ( r ( γ )) , v ( r ( χ )) i = k v ( r ( γ )) k for r ( γ ) = r ( χ ) , r ( γ ) = r ( χ ) . In other words, for every x ∈ X the kernel’s restriction K x is a constant map. Every H ( K x )is thus either one-dimensional (if v ( x ) = 0) or zero-dimensional (if v ( x ) = 0). Moreover, if v isa nowhere-vanishing vector field, the set {k v ( x ) k Γ x } x ∈ X (where Γ x : Γ → { , } denotes theindicator function of the fiber Γ x ) constitutes an orthonormal basis of H ( K ), and hence the latterHilbert space is isometrically isomorphic to l ( X ). Example 7.
As a simple nontrivial example, consider the four-element groupoid Γ := { ε (+) , ε ( − ) ,α, α − } over a two-element set X := { + , −} , visualized in Figure 1. Such a groupoid (the pairgroupoid over a two-element set) is studied e.g. in the context of quantum information [13]. • − ε ( − ) w w αα − • + ε (+) h h Figure 1: Structure of the groupoid Γ in Example 7.To further simplify the example, let us consider a one-dimensional representation of Γ. Con-cretely, let us take H + := { + }× C , H − := {−}× C and let the unitary representation ( U , { H + , H − } )be given by U ( α ) : H + → H − , U ( α )(+ , z ) := ( − , λz )11 ( · , · ) ε (+) α − α ε ( − ) ε (+) | v + | ¯ λv − ¯ v + α − λ ¯ v − v + | v − | α | v + | ¯ λv − ¯ v + ε ( − ) 0 0 λ ¯ v − v + | v − | Table 1: The kernel obtained from the representation studied in Example 7.and hence U ( α − ) : H − → H + , U ( α − )( − , z ) := (+ , ¯ λz ), where λ is a fixed complex number ofmodulus 1. Choosing a generic vector field v (+) := (+ , v + ), v ( − ) := ( − , v − ), where v ± ∈ C , formula(7) yields a kernel whose respective values are presented in Table 1.Notice that the columns of the above table contain the values of the functions K ε (+) , K α − , K α , K ε ( − ) , respectively. These four functions span the space H ( K ), but they are not linearlyindependent. In fact, one can write that H ( K ) = span { ϕ + , ϕ − } , where the functions ϕ ± : Γ → C are defined as ϕ + ( ε (+)) := ¯ v + , ϕ + ( α − ) := λ ¯ v − , ϕ + ( α ) := 0 , ϕ + ( ε ( − )) := 0 ,ϕ − ( ε (+)) := 0 , ϕ − ( α − ) := 0 , ϕ − ( α ) := ¯ λ ¯ v + , ϕ − ( ε ( − )) := ¯ v − . Unless v + = v − = 0, the functions ϕ ± can be shown to be orthonormal: h ϕ + , ϕ + i H ( K ) = h ϕ − , ϕ − i H ( K ) = 1 and h ϕ + , ϕ − i H ( K ) = h ϕ − , ϕ + i H ( K ) = 0 . Therefore, we have H ( K ) ∼ = C as Hilbert spaces, and so in this case H ( K ) is isomorphic to H := H + ⊕ H − and not just isometrically embedded in the latter (cf. Theorem 2). In addition,observe that the (restricted) functions ϕ + | Γ + , ϕ − | Γ − span the RKHS’s H ( K + ), H ( K − ), respectively,built from the restricted kernels. All in all, we have that H ( K ) = span { ϕ + , ϕ − } = span { ϕ + } ⊕ span { ϕ − } = H ( K + ) ⊕ H ( K − ) , where again we have identified the restricted functions with its extensions by zero, in full agreementwith formula (8).Finally, the set { ϕ + , ϕ − } , being an orthonormal basis of H ( K ), is a Parseval frame for H ( K ),and hence by Theorem 3 we must have, for every χ, γ ∈ Γ, K ( χ, γ ) = ϕ + ( χ ) ϕ + ( γ ) + ϕ − ( χ ) ϕ − ( γ ) , As one can check directly, the above equality indeed holds in the considered case.
Example 8.
Let Γ be a locally compact groupoid endowed with a left Haar system { λ x } x ∈ X , andlet ( U , { H x := L (Γ x , λ x ) } x ∈ X ) be its left regular representation, i.e. ( U ( γ ) f )( ξ ) := f ( γ − ξ ) for any f ∈ H r ( γ ) .For any fixed vector field X ∋ x v ( x ) ∈ H x the reproducing kernel reads, in the case when12 ( χ ) = r ( γ ), K ( χ, γ ) = Z Γ r ( γ ) v ( s ( γ ))( γ − ξ ) v ( s ( χ ))( χ − ξ ) dλ r ( γ ) ( ξ )= Z Γ s ( γ ) v ( s ( γ ))( ξ ) v ( s ( χ ))( χ − γξ ) dλ s ( γ ) ( ξ )= Z Γ s ( γ ) v ( s ( γ ))( ξ ) ^ v ( s ( χ ))( ξ − γ − χ ) dλ s ( γ ) ( ξ ) . We note that this is an analogue of the kernel defined by the convolution on a group withrespect to a left Haar measure (cf. Example 1). Although there exists a standard definition of aconvolution on a groupoid (see, e.g., [17, p. 38]), the above expression does not entirely fit into it,because the convoluted functions v ( s ( γ )) and ^ v ( s ( χ )) are not defined over entire Γ. In fact, v ( s ( γ )) ∈ L (Γ s ( γ ) , λ s ( γ ) ), whereas ^ v ( s ( χ )) ∈ L (Γ s ( χ ) , inv ∗ λ s ( χ ) ) (where inv denotes here the inversion map,inv( γ ) := γ − ) and as such their convolution is well defined only on Γ s ( γ ) s ( χ ) .Of course, when r ( γ ) = r ( χ ) the kernel is by definition K ( χ, γ ) = 0. Example 9.
Let Γ be a locally compact groupoid endowed this time with a right Haar system { λ x } x ∈ X , and let ( U , { H x := L (Γ x , λ x ) } x ∈ X ) be its right regular representation, i.e. ( U ( γ ) f )( ξ ) := f ( ξγ ) for any f ∈ H r ( γ ) .For any fixed vector field X ∋ x v ( x ) ∈ H x the reproducing kernel reads, in the case when r ( χ ) = r ( γ ), K ( χ, γ ) = Z Γ r ( γ ) v ( s ( γ ))( ξγ ) v ( s ( χ ))( ξχ ) dλ r ( γ ) ( ξ )= Z Γ s ( γ ) v ( s ( γ ))( ξ ) v ( s ( χ ))( ξγ − χ ) dλ s ( γ ) ( ξ )= Z Γ s ( γ ) v ( s ( γ ))( ξ ) ^ v ( s ( χ ))( χ − γξ − ) dλ s ( γ ) ( ξ ) . This also can be seen as something analogous to the convolution (only this time, related to theright Haar measure in the group setting).Of course, when r ( γ ) = r ( χ ) the kernel K ( χ, γ ) is defined to vanish, just as in the previousexamples. Example 10.
This example is inspired by [4]. Let X be the set of all pairs (Ω , z ), where Ω is adomain (nonempty, open and connected set) in C n and z ∈ Ω. Let also Γ be the set of all pairs(Φ , z ), where Φ : Ω → Ω is a biholomorphism between open domains in C n and z ∈ Ω . We definethe groupoid (Γ , X, s, r, ε, · , − ) as follows.If (Φ , z ) ∈ Γ, where Φ : Ω → Ω , then s (Φ , z ) := (Ω , z ) and r (Φ , z ) := (Ω , Φ( z )). The embed-ding map is given by ε (Ω , z ) := (id Ω , z ), whereas the inversion map reads (Φ , z ) − := (Φ − , Φ( z )).Finally, the multiplication · of pairs (Φ : Ω → Ω , z ) and (Ψ : Ω → Ω , ζ ) is defined if Ω = Ω and ζ = Φ( z ), in which case (Ψ , ζ )(Φ , z ) := (Ψ ◦ Φ , z ), where ◦ is an ordinary composition of mappings.In order to introduce a kernel on Γ, define a map k : Γ → C via k (Φ , z ) := J Φ( z ) | J Φ( z ) | , (Φ , z ) ∈ Γ , J Φ( z ) denotes the complex Jacobian determinant of Φ at the point z . Notice that the map k is multiplicative, i.e., k ( γχ ) = k ( γ ) k ( χ ) for any ( γ, χ ) ∈ Γ (2) . Indeed, using elementary propertiesof the Jacobian, we can write k ((Ψ , ζ )(Φ , z )) = k (Ψ ◦ Φ , z ) = J (Ψ ◦ Φ)( z ) | J (Ψ ◦ Φ)( z ) | = J (Ψ)( ζ ) | J (Ψ)( ζ ) | · J (Φ)( z ) | J (Φ)( z ) | = k (Ψ , ζ ) k (Φ , z ) , (11)where ζ = Φ( z ) by the multiplicability of (Ψ , ζ ), (Φ , z ). What is more, for any (Ω , z ) ∈ X onetrivially has that k ( ε (Ω , z )) = k (id Ω , z ) = 1 and hence, moreover, k ( γ − ) = k ( γ ) − k ( γ ) k ( γ − ) = k ( γ ) − k ( γγ − ) = k ( γ ) − k ( ε ( r ( γ ))) = k ( γ ) − = k ( γ ) (12)for every γ ∈ Γ.Let now K : Γ × Γ → C be a kernel given by the formula K ( χ, γ ) := (cid:26) k ( γ ) k ( χ ) for r ( γ ) = r ( χ ) , r ( γ ) = r ( χ ) , (13)that is K ((Ψ , ζ ) , (Φ , z )) := ( J Φ( z ) | J Φ( z ) | J Ψ( ζ ) | J Ψ( ζ ) | for r (Φ , z ) = r (Ψ , ζ ) , r (Φ , z ) = r (Ψ , ζ ) . One can easily show that the above kernel is positive definite. In fact, it is a realization of thegeneral construction described in Theorem 2 with the function F : Γ → l ( X ) defined as F (Φ , z ) := k (Φ , z ) δ r (Φ ,z ) , where δ x : X → C denotes the Kronecker delta concentrated at x ∈ X .Moreover, the above kernel satisfies (9). Indeed, by (11,12) one has that, whenever r ( χ ) = r ( γ ), K ( χ, γ ) = k ( γ ) k ( χ ) = k ( γ − ) k ( χ ) = k ( γ − χ ) = k ( ε ( γ − χ )) k ( γ − χ ) = K ( γ − χ, ε ( r ( γ − χ ))) . On the strength the discussion following formula (9), the just proven properties of the ker-nel K mean that the latter can be used to construct a unitary ‘Moore–Aronszajn representation’( U , { H ( K x ) } x ∈ X ) of Γ, with each U ( γ ) : H ( K s ( γ ) ) → H ( K r ( γ ) ) satisfying U ( γ ) K χ = K γχ , χ ∈ Γ s ( γ ) . (14)In order to better understand this representation, notice first that for any χ ∈ Γ K χ = k ( χ )¯ k · Γ r ( χ ) , where Γ r ( χ ) denotes the indicator function of the fiber Γ r ( χ ) (cf. Example 6). This in particularmeans that for every x ∈ X the space H ( K x ) is spanned by the function ¯ k · Γ x , which can beeasily shown to be of norm 1. Observe now that, for any chosen χ ∈ Γ s ( γ ) U ( γ )(¯ k · Γ s ( γ ) ) = k ( χ ) − U ( γ ) K χ = k ( χ ) − K γχ = k ( χ ) − k ( γχ )¯ k · Γ r ( γχ ) = k ( γ )¯ k · Γ r ( γ ) , where we have used (14) and (11). In other words, under the above choice of the orthonormalbases of the one-dimensional spaces H ( K x ), the transformation U ( γ ) can be regarded simply as themultiplication by the complex number k ( γ ). 14he above groupoid Γ together with the simple kernel given by (13) is by no means the onlyone worth investigating in the context of biholomorphisms. The construction can be generalized,e.g., by considering, for some fixed natural N >
1, the sets X N := { (Ω , z ) | z ∈ Ω N } and Γ N := { (Φ , z ) | z ∈ Ω N } , where z := ( z , . . . , z N ) and the symbols Ω and Φ have the same meaning asbefore. For any biholomorphism Φ : Ω → Ω we put s (Φ , z ) := (Ω , z ) , r (Φ , z ) := (Ω , Φ( z )) , (Φ , z ) − := (Φ − , Φ( z )) , where Φ( z ) := (Φ( z ) , . . . , Φ( z N )), and we define the multiplication(Ψ , ζ )(Φ , z ) := (Ψ ◦ Φ , z )provided Φ( ζ ) = z and Ψ : Ω → Ω is another biholomorphism. Finally, the embedding map isgiven by ε (Ω , z ) := (id Ω , z ) for any (Ω , z ) ∈ X N . The septuple (Γ N , X N , s, r, ε, · , − ) is a groupoidand we can define N positive definite kernels K , . . . , K N : Γ N × Γ N → C via K j ((Ψ , ζ ) , (Φ , z )) := (cid:26) k (Φ , z j ) k (Ψ , ζ j ) for r (Φ , z ) = r (Ψ , ζ ) , r (Φ , z ) = r (Ψ , ζ ) , (15) j = 1 , . . . , N . Since all K j ’s satisfy condition (9), then for any α , . . . , α N > m , . . . , m N the function K := α K m + α K m + . . . + α N K m N N is a positive definitekernel on Γ N also satisfying condition (9). Therefore, K defines a unitary ‘Moore–Aronszajn repre-sentation’ of Γ N , which no longer offers such a straightforward interpretation as the one presentedabove. This, however, goes beyond the scope of the current article and will be addressed in thefuture work. Let us briefly discuss possible applications of the presented relationship between unitary repre-sentations of groupoids and reproducing kernels.Kernel methods are practically utilized, e.g., in the machine learning field. Currently, theirtypical implementation is the classification or regression task, where the kernel-based method canbe used to process the feature vector (representing the analyzed object) and produce the desiredoutput, ensuring the minimum error, even if the data are difficult to distinguish. The most popularmethod is the Support Vector Machines Classifier (SVC), used to identify linearly inseparableobjects. Their original features, based on which the decision is made, are transformed using thekernel function to the new space, where separation of examples belonging to various categories iseasier [9]. However, one of the requirements for the kernel function K ( x, y ) = h τ ( y ) , τ ( x ) i is that itsinput arguments are real numbers. This function can be substituted by K ( h, g ) = hU ( g ) v, U ( h ) v i .As a result, the unitary representation U on the group (instead of object transformation τ ) is used.Another domain in which kernel-based methods prove their usefulness is optimization. Theproblem, often encountered in data processing modules (implemented in such fields as electronicsor control engineering) is the selection of the optimal kernel regarding the distance between featurevectors in the multidimensional space. According to [19], such a measure (on the groupoids) can It is well known that the finite product of positive definite kernels on A is itself a positive definite kernel (see,e.g., [23, p. 6,17]). The same, of course, concerns linear combinations with positive coefficients of positive definitekernels defined on the same set A .
15e used to solve the generalized version of the Traveling Salesman Problem. The overall distanceto minimize is given as: L = P Ni =1 d ( c i , c i +1 ) , where c i and c i +1 are two subsequent nodes fromthe graph in the optimized cycle. Using the kernel to describe distances between nodes in the newspace allows for estimating the distance components as: d ( x, y ) = p K ( x, x ) − K ( x, y ) + K ( y, y ) , where kernel K is a real-valued function. This type of distance is described in [11, p. 78]. Inour proposed application, selection of the optimal kernel among various candidates (ensuring theminimal or maximal distance between the points) can be done without the actual transformationof the original space to the new one (which is the core of the kernel applicability in the machinelearning).Yet another possible application concerns quantum physics, where both reproducing kernels [15]and groupoids and their representations [5] have been used in the description of quantum systems.For instance, the simple representation of the pair groupoid of a two-element set, considered inExample 7 above, can be used in the description of a qubit — the central notion of the theoryof quantum information [13]. In the future work we shall investigate the physical meaning andsignificance of the kernels and RKHS associated to the quantum-mechanically relevant unitaryrepresentations of groupoids.On the mathematical side, let us add that the simple definition of a unitary representation of agroupoid employed in the paper can be generalized onto the setting of measurable fields of Hilbertspaces [8, 21, 22]. The natural question whether the above-studied relationship between RKHS andunitary groupoid representations still holds in this more general setting will also be addressed inthe future work. This line of research might in the end offer some new tools and insights in thefields of complex analysis, quantum physics and computer science. References [1] Barbieri, D., Citti, G.: Reproducing kernel Hilbert spaces of CR functions for the EuclideanMotion group. Anal. Appl. 13(03), 331–346 (2015)[2] Berlinet, A., Thomas-Agnan, C.: Reproducing kernel Hilbert spaces in probability and statistics.Kluwer Academic Publishers, Boston/Dordrecht/London (2004)[3] Brown, R.: From groups to groupoids. Bull. Lond. Math. Soc. 19, 113–134 (1987)[4] Burzyńska M., Pasternak-Winiarski Z.: Differential Groupoids, J. Math. Sys. Sci. 5, 39–45 (2015)[5] Ciaglia, F.M., Ibort, A., Marmo, G.: A gentle introduction to Schwinger’s formulation of quan-tum mechanics: The groupoid picture. Mod. Phys. Lett. A 33(20), 1850122 (2018)[6] da Silva, A., Weinstein, A.: Geometric models for noncommutative algebras. Berkeley Mathe-matics Lecture Notes, AMS, Providence (1999)[7] Debord, C., Lescure, J.M.: Index Theory and Groupoids. Geometric and topological methodsfor quantum field theory. Cambridge Univ. Press, 86–158 (2010)[8] Dixmier, J.: Von Neumann Algebras. North Holland Publ. Comp., Amsterdam (1981)169] Drewnik, M., Pasternak-Winiarski, Z.: SVM Kernel Configuration and Optimization for theHandwritten Digit Recognition. In: Saeed, K., Homenda, W., Chaki, R. (eds.) Proceedings ofthe CISIM, pp. 87–98. Springer, Białystok (2017)[10] Fremlin, D.H.: Measure Theory Vol.4. Topological Measure Spaces. Torres Fremlin, Colchester(2003)[11] G¨artner, T., Lloyd J.W., Flach, P.A.: Kernels for Structured Data. In: Matwin, S., Sammut,C. (eds.) Proceedings of ILP, pp. 66–83. Springer-Verlag, Sydney (2002)[12] Ibort, A., Rodr´ıguez M.A.: An Introduction to Groups, Groupoids and Their Representations.CRC Press, Boca Raton (2019)[13] Ibort, A., Rodr´ıguez M.A.: On the Structure of Finite Groupoids and Their Representations.Symmetry, 11, 414 (2019)[14] Kallianpur, G.: The role of reproducing kernel Hilbert spaces in the study of Gaussian processes.Advances in Prob. 2, 49–83 (1970)[15] Odzijewicz, A.: On reproducing kernels and quantization of states. Commun. Math. Phys., 114,577–597 (1988)[16] Pasternak-Winiarski, Z.: On the dependence of the reproducing kernel on the weight of inte-gration. J. Funct. Anal. 94, 110–134 (1990)[17] Paterson, A.L.T.: Groupoids, Inverse Semigroups, and Their Operator Algebras. Birkh¨auser,Boston (1999)[18] Paulsen, V.I., Raghupathi, M.: An Introduction to the Theory of Reproducing Kernel HilbertSpaces. Cambridge Univ. Press (2016)[19] Pissanetzky, S.: Causal Groupoid Symmetries and Big Data. J. Appl. Math. 5, 3489–3510(2014)[20] Pysiak, L.: Groupoids, their representations and imprimitivity systems. Demonstr. Math. 37,661–670 (2004)[21] Pysiak, L.: Imprimitivity theorem for groupoid representations. Demonstr. Math. 44, 29–48(2011)[22] Renault, J.: The ideal structure of groupoid crossed product C ∗∗