Contents lists available atScienceDirect
Linear
Algebra
and
its
Applications
www.elsevier.com/locate/laa
Representations
of
∗-semigroups
associated
to
invariant
kernels
with
values
adjointable
operators
Serdar Aya,∗, Aurelian Gheondeaa,b,1
aDepartmentofMathematics,BilkentUniversity,06800Bilkent,Ankara,Turkey bInstitutuldeMatematicăalAcademieiRomâne,C.P.1-764,014700Bucureşti,
Romania
a r t i c l e i n f o a b s t r a c t
Articlehistory:
Received3February2015 Accepted15August2015 Availableonline2September2015 SubmittedbyP.Semrl
MSC:
primary47A20
secondary15B48,43A35,46E22, 46L89
Keywords:
Ordered∗-space
VE-space
Positivesemidefinitekernel
∗-semigroup Invariantkernel Linearisation ∗-representation Reproducingkernel Ordered∗-algebra VE-module
We consider positive semidefinite kernels valued in the ∗-algebra of adjointable operators on a VE-space (Vector Euclidean space) and that are invariant under actions of ∗-semigroups. A rather general dilation theorem is stated and proved: for these kind of kernels, representations of the∗-semigroup on either the VE-spaces of linearisationof the kernels or on their reproducing kernel VE-spaces are obtainable. We point out the reproducing kernel fabric of dilationtheoryandweshowthatthegeneraltheoremunifies manydilationresultsatthenon-topological level.
© 2015ElsevierInc.All rights reserved.
* Correspondingauthor.
E-mailaddresses:serdar@fen.bilkent.edu.tr(S. Ay),aurelian@fen.bilkent.edu.tr(A. Gheondea),
A.Gheondea@imar.ro(A. Gheondea).
1 Theauthor’sworkis supportedbyagrantoftheRomanianNationalAuthorityforScientificResearch, CNCSUEFISCDI,projectnumberPN-II-ID-PCE-2011-3-0119.
http://dx.doi.org/10.1016/j.laa.2015.08.012
0. Introduction
Starting withthecelebratedNaimark’sdilationtheoremsin[22]and[23],apowerful dilationtheoryforoperatorvaluedmapswasobtainedthroughresultsofB. Sz.-Nagy[32], W.F. Stinespring[30],andtheirgeneralisationstoVH-spaces(VectorHilbertspaces)by R.M. Loynes[17],ortoHilbertC∗-modulesbyG.G. Kasparov[15].Takingintoaccount thediversityofdilationtheorems foroperatorvaluedmaps,there isanaturalquestion, whether onecanunify all, or themost,of these dilationtheorems, underonetheorem. Anattempttoapproachthisquestionwasmadein[11]byusingthenotionofVH-space over an admissible space, introduced byR.M. Loynes [17,18]. As asecond step inthis enterprise, aninvestigation at the “ground level”, that is, anon-topological approach, makes perfect sense.Inaddition,animpetusto pursuethisway wasgiventous bythe recentinvestigationoncloselyrelatedproblems,e.g.non-topological theoryforoperator spacesand operatorsystems,cf.[27,5,26,6].
Theaimofthisarticleistopresentageneralnon-topological approachtodilation the-orybasedonpositivesemidefinitekernelsthatareinvariantunderactionsof∗-semigroups and withvaluesadjointableoperators onVE-spaces (Vector Euclidean spaces)over or-dered ∗-spaces. More precisely, we show that at the level of conjunction of order with
∗-spaces or ∗-algebrasand operator valuedmaps, onecanobtaina reasonable dilation theorythatcontainsthefabricofmostofthemoreorlesstopologicalversionsof classi-cal dilationtheorems. Inaddition, weintegrate intonon-topological dilationtheory, on equalfoot,thereproducingkerneltechniqueandshowthatalmosteachdilationtheorem is equivalent to arealisation as a reproducing kernel space with additional properties. Our approach is based onideas already present underdifferent topological versions of dilationtheorems in[24,7,17,3,4,21,8–10,31,12,11] and,probably,manyothers.
Webriefly describethecontentsof thisarticle.InSection1wefix someterminology andfactsonordered∗-spaces,ordered∗-algebras,VE-spacesoverordered∗-spaces,and VE-modules overordered ∗-algebras.Onthese basicobjects,onecanbuildtheordered
∗-algebras ofadjointableoperators onVE-spaces orVE-modules. Weprovidemany ex-amples thatillustrate therichness ofthistheory,evenat thenon-topological level.
Then, inSection2,we considerthemain objectof investigationwhichrefersto pos-itive semidefinite kernels with values adjointable operators on VE-spaces. We make a preparation byshowing that,althoughanalogs ofSchwarz Inequalityis missing atthis level of generality, somebasicresultscanbe obtainedby different techniques.Inorder toachieveasufficientgeneralitythatallows torecoverknowndilationtheoremsforboth
∗-semigroups (B. Sz.-Nagy) and ∗-algebras (Stinespring), in view of [11], we consider positive semidefinitekernels thatare invariantunder actionsof ∗-semigroups and that have values adjointable operators on VE-spaces. In Lemma 2.1, we show that, for a 2-positive kernel,ifboundednessinthesense ofLoynesisassumed foralltheoperators onthediagonal,thentheentirekernelismadeupbyboundedoperators.Inthiswaywe explain howthe investigationofthis articleis situatedwith respect tothatin[11]. We brieflyshowtheconnectionbetweenlinearisationsandreproducingkernelspacesatthis
levelof generality.It isthisstage when weareableto stateandprovethemain result,
Theorem 2.8that, basically, shows that this kind of kernels produce ∗-representations on“dilated”VE-spaces thatlinearise thekernel or,equivalently, onreproducing kernel VE-spacesthatcanbe explicitlydescribed.
Finally,inSection3, as consequencesofTheorem 2.8, we show hownon-topological versionsofmostoftheknowndilationtheorems[32,30,17,15,14]canbeobtained.Onthe other hand,in order to unify theknown dilationtheorems in topological versions, one needscertaintopologicalstructuresonordered∗-spacesandVE-spaces,thatleadclosely to theVH-spaces overadmissible spaces, as consideredin[17]. This way was followed, toacertainextent,in[11]but,inordertoobtainasufficientlylargegeneralityallowing to cover most of the known topological dilation theory, one needs more flexibility by movingbeyondbounded operators.Wewillconsider thisinsubsequentarticles. 1. Preliminaries
Inthissectionwe brieflyreview mostofthedefinitionsandsomebasicfactson VE-spaces over ordered ∗-spaces,inspired bycf. R.M.Loynes, [17–19]. We slightlymodify some definitions in order to match the requirements of this investigation, notably by separatingthe non-topological from the topological cases and by adheringto the con-vention, that is very popular in Hilbert C∗-modules, to let gramians be linear in the secondvariable andconjugatelinearinthefirstvariable,forconsistency.
1.1. Ordered∗-spaces
AcomplexvectorspaceZ iscalled ordered∗-space,cf.[27],if:
(a1) Z has an involution ∗, that is, a map Z z → z∗ ∈ Z that is conjugate linear
((sx+ ty)∗ = sx∗+ ty∗ foralls,t∈ C andallx,y ∈ Z)and involutive ((z∗)∗ = z forallz∈ Z).
(a2) InZ thereisacone Z+ (sx+ ty∈ Z+ forallnumberss,t≥ 0 andallx,y∈ Z+),
thatisstrict (Z+∩−Z+={0}),andconsistingofself-adjointelements only(z∗= z forallz∈ Z+).Thisconeisusedtodefineapartialorder ontherealvectorspace
ofallselfadjointelementsin Z:z1≥ z2 ifz1− z2∈ Z+.
Recallthata∗-algebra A isacomplexalgebraontowhichthereisdefinedaninvolution A a→ a∗∈ A, thatis, (λa+ μb)∗ = λa∗+ μb∗, (ab)∗ = b∗a∗, and (a∗)∗ = a,for all
a,b∈ A andallλ,μ∈ C.
Anordered∗-algebra A isa∗-algebrasuchthatitisanordered∗-space,moreprecisely, ithasthefollowing property.
Clearly, any ordered ∗-algebra is an ordered ∗-space. In particular, given a ∈ A, we denotea≥ 0 ifa∈ A+and,fora= a∗∈ A andb= b∗∈ A,wedenotea≥ b ifa−b≥ 0.
Remark 1.1. Inanalogy with thecase of C∗-algebras, givena∗-algebra A, onedefines an element a ∈ A to be ∗-positive if a =k=1n a∗kak for some natural number n and someelements a1,. . . ,an∈ A. Thecollectionof all∗-positive elementsina∗-algebrais acone,butitmayfailtobestrictandhence,associatedisonlyaquasi-order,e.g.see[5]
for arecentinvestigation.Thus, ourdefinitionofanordered ∗-algebraspecifies astrict cone A+ and, ingeneral,it doesnotreferto theconeof ∗-positiveelements as defined
above,exceptspecialcasesas,forexample,preC∗-algebrasor prelocallyC∗-algebras. Examples 1.2. (1) Any C∗-algebra, e.g. see [2], A is an ordered ∗-algebra and any
∗-subspace S of a C∗-algebra A, with the positive cone S+ = A+ ∩ S and all other
operations (addition,multiplicationwithscalars,andinvolution) inheritedfrom A,isa
∗-space.
(2)Anypre-C∗-algebraisanordered∗-algebra.Any∗-subspaceS ofapre-C∗-algebra
A is anordered ∗-space,with thepositivecone S+ =A+∩ S andall otheroperations
inherited fromA.
(3) Any locally C∗-algebra, see [13,28], is an ordered ∗-algebra. In particular, any
∗-subspace S of a locally C∗-algebra A, with the cone S+ = A+ ∩ S and all other operations inheritedfromA, isanordered ∗-space.
(4)Anylocallypre-C∗-algebraisanordered∗-algebra.Any∗-subspaceS ofalocally pre-C∗-algebraisanordered∗-space,withS+=A+∩S andallotheroperationsinherited
from A.
(5) LetV beacomplexvector spaceand letV be its conjugatedual space.On the vector space L(V,V) of all linear operators T : V → V, anatural notion of positive operatorcanbe defined:T ispositive if(T v)(v)≥ 0 for allv∈ V . LetL+(V,V) be the
collection ofallpositiveoperators andnote thatitis astrictcone. Theinvolution ∗ in
L(V,V) is defined inthefollowing way:for any T ∈ L(V,V), T∗ = T|V , thatis, the restrictionto V ofthedual operatorT: V→ V. Withrespectto theconeL+(V,V)
and the involution ∗ just defined,L(V,V) becomes an ordered ∗-space.See A. Weron
[33],aswell asD. GaşparandP. Gaşpar[8].
(6)LetX beanonemptysetanddenotebyK(X) thecollectionofallcomplexvalued kernels on X, thatis, K(X) ={k | k : X × X → C}, consideredas acomplex vector space withtheoperationsofadditionandmultiplicationofscalarsdefinedelementwise. An involution ∗ canbe definedonK(X) as follows: k∗(x,y)= k(y, x),for allx,y ∈ X
and allk∈ K(X).TheconeK(X)+ consistsonallpositivesemidefinite kernels,thatis, those kernelsk∈ K(X) withtheproperty that,foranyn∈ N andanyx1,. . . ,xn ∈ X, thecomplexmatrix[k(xi,xj)]ni,j=1 ispositivesemidefinite.
OnK(X) amultiplicationcanbedefinedelementwise:ifk,l∈ K(X) then(kl)(x,y)=
k(x,y)l(x,y) forallx,y∈ X.Withrespecttothismultiplicationandtheotheroperations described before,K(X) isanordered∗-algebra.
Using the notion of Schur product, e.g. see [25], it can be proven that the ordered
∗-algebraK(X) hasthefollowing property:ifk,l∈ K(X) arepositivesemidefinite ker-nels,thenkl ispositivesemidefinite.However,thisisacasethatillustratesRemark 1.1: itisnottrue,ingeneral,thatkernelsof typek∗k are positivesemidefinite.
(7)Let A andB be two ordered ∗-spaces.Inaddition, we assumethatthespecified strict cone A+ linearly generates A. On L(A,B), the vector space of all linear maps
ϕ :A → B, we define an involution: ϕ∗(a) = ϕ(a∗)∗, for all a ∈ A. A linear map
ϕ ∈ L(A,B) is called positive if ϕ(A+) ⊆ B+. It is easy to see that L(A,B)+, the
collectionofallpositivemaps fromL(A,B), isacone, andthatitis strictbecauseA+ linearlygeneratesA.Inaddition,anyϕ∈ L(A,B)+ isselfadjoint, againduetothefact
thatA+ linearlygeneratesA.Consequently,L(A,B) hasanaturalstructure ofordered
∗-space.
(8)Let{Zα}α∈Abeafamilyofordered∗-spaces suchthat,foreachα∈ A,Zα+isthe specifiedstrict coneofpositiveelementsinZα.Ontheproduct spaceZ =α∈AZαlet
Z+ =
α∈AZα+ and observe thatZ+ isastrictcone. Lettingtheinvolution ∗ on Z be definedelementwise,itfollowsthatZ+consistsonselfadjointelementsonly.Inthisway,
Z isanordered∗-space.
(9) LetZ be an ordered ∗-spacewith thespecified strict cone Z+. A subspaceJ of
Z iscalled an order ideal if itis selfadjoint, thatis, J = J∗ ={z∗ | z ∈ J},and solid,
thatis,forany z∈ J ∩ Z+ andany y∈ Z+ suchthaty≤ z it followsy ∈ J.Then, on
thequotientvectorspaceZ/J ,aninvolution∗ canbedefinedby:(z + J )∗= z∗+ J ,for
z∈ Z.Also, letting(Z/J )+={z + J | z ∈ Z+},itfollows that(Z/J )+ isastrict cone inZ/J consistingofselfadjointelementsonlyand,hence,Z/J isanordered∗-space.See
[27].
1.2. VectorEuclidean spaces andtheir linearoperators
Given acomplex linearspace E and an ordered ∗-spacespace Z, a Z-gramian, also calledaZ-valuedinnerproduct,isamappingE × E (x,y)→ [x,y]∈ Z subject tothe followingproperties:
(ve1) [x,x]≥ 0 forallx∈ E,and[x,x]= 0 ifandonlyifx= 0. (ve2) [x,y]= [y,x]∗ forallx,y∈ E.
(ve3) [x,αy1+ βy2]= α[x,y1]+ β[x,y2] forallα,β∈ C andallx1,x2∈ E.
AcomplexlinearspaceE ontowhichaZ-gramian[·,·] isspecified,foracertainordered
∗-spaceZ, iscalledaVE-space (VectorEuclidean space)overZ,cf. [17].
Remark 1.3. In any VE-space E over an ordered ∗-space Z, the familiar polarisation formula
4[x, y] =
3
k=0
ik[(x + iky, x + iky], x, y∈ E, (1.1)
holds,whichshowsthattheZ-valuedinnerproductisperfectlydefinedbytheZ-valued
quadratic mapE x→ [x,x]∈ Z.
Actually, theformula(1.1)ismoregeneral:givenapairing[·,·]: E × E → Z,whereE issomevectorspaceandZ isa∗-space,andassumingthat[·,·] satisfiesonlytheaxioms (ve2) and(ve3),then(1.1)isstillvalid.
The conceptofVE-spacesisomorphism is alsonaturallydefined:this isjustalinear bijection U : E → F, for two VE-spaces over the same ordered ∗-space Z, which is
isometric,thatis,[U x,U y]F = [x,y]E forallx,y∈ E.
IngeneralVE-spaces,ananalogoftheSchwarzInequalitymaynothold butsomeof its consequencescanbe proven usingslightly differenttechniques. Onesuchmethodis providedbythefollowinglemma.
Lemma 1.4. (See Loynes [17].) Let Z be an ordered ∗-space, E a complex vector space and [·,·]: E × E → Z a positive semidefinite sesquilinear map, that is, [·,·] is linear in thesecond variable, conjugatelinear inthefirstvariable, and[x,x]≥ 0 forallx∈ E.If f ∈ E issuchthat[f,f ]= 0,then[f,f]= [f,f ]= 0 for allf ∈ E.
GiventwoVE-spacesE andF,overthesameordered∗-space Z,onecanconsiderthe vectorspaceL(E,F) ofalllinearoperatorsT :E → F.A linearoperatorT ∈ L(E,F) is
called adjointable ifthere existsT∗∈ L(F,E) suchthat
[T e, f ]F= [e, T∗f ]E, e∈ E, f ∈ F. (1.2) The operatorT∗,if itexists, isuniquely determined byT andcalled its adjoint.Since ananalogoftheRieszRepresentationTheoremforVE-spacesmaynotexist,ingeneral, theremaybenotsomanyadjointableoperators.DenotebyL∗(E,F) thevectorspaceof all adjointableoperators from L(E,F).Note thatL∗(E)=L∗(E,E) isa∗-algebrawith respect totheinvolution∗ determinedbytheoperationoftaking theadjoint.
AnoperatorA∈ L(E) iscalledselfadjoint if[Ae,f ]= [e,Af ],foralle,f ∈ E.Clearly, anyselfadjointoperatorA isadjointableandA= A∗.Bythepolarisationformula(1.1),
A is selfadjointifandonlyif[Ae,e]= [e,Ae],e∈ E.AnoperatorA∈ L(E) is positive if
[Ae, e]≥ 0, e ∈ E. (1.3)
Since thecone Z+ consistsof selfadjointelements only,any positiveoperatoris selfad-jointandhenceadjointable.Ontheotherhand,note thatanyVE-spaceisomorphismis adjointableandhence,itmakessense tocallitunitary.
Examples 1.5. (1) If E is some VE-space over an ordered ∗-space Z, then L∗(E) is an ordered∗-algebra,wheretheconeofpositiveelementsisdefinedby(1.3).Notethatthis
cone is strict. In connection with Remark 1.1, note thatany operatorA ∈ L∗(E) that canberepresentedA=Nj=1A∗jAj ispositive,buttheconverse,ingeneral,isnottrue. (2) Let {Eα}α∈A be a family of VE-spaces such that, for each α ∈ A, Eα is a VE-space over the ordered ∗-spaceZα. Consider the ordered ∗-spaceZ = α∈AZα as in
Example 1.2.ConsiderthevectorspaceE =α∈AEαonwhichwedefine [(eα)α∈A, (fα)α∈A] = ([eα, fα])α∈A∈ Z, (eα)α∈A, (fα)α∈A∈ E. ThenE isaVE-spaceover Z.
(3)LetH beapre-HilbertspacehavinganorthonormalbasisandE aVE-spaceover theordered∗-spaceZ.Onthealgebraictensorproduct H ⊗ E defineagramianby
[h⊗ e, l ⊗ f]H⊗E = h, l H[e, f ]E ∈ Z, h, l ∈ H, e, f ∈ E,
andthen extendit toH ⊗ E bylinearity.Byastandardbutratherlongargument,e.g. similarto[16, p. 6],itcanbeproventhat,inthisway, H ⊗ E becomesaVE-spaceover
Z aswell.
Remark 1.6. Given a finite collection of VE-spaces E1,. . . ,EN, over the same ordered
∗-spaceZ,onecandefinenaturallytheVE-spaceE1⊕ · · · ⊕ EN overZ where,forej,fj ∈
Ej,j = 1,. . . ,N wedefine [e1⊕ · · · ⊕ eN, f1⊕ · · · ⊕ fN] = N j=1 [ej, fj].
We use the notation EN for E
1⊕ · · · ⊕ EN when E = Ej for all j = 1,. . . ,N . Then observe that L∗(EN) can be naturallyidentified with M
N(E), the space of all N× N matriceswithentriesinL∗(E).Thisidentificationprovidesanaturalstructureofordered
∗-algebraofL∗(EN) overZ,withanevenricher structure,seeRemarks 3.5.
An operator A ∈ L(E,F), for two VE-spaces over the same ordered ∗-space Z, is calledbounded if,forsomeμ≥ 0,
[Ah, Ah]F ≤ μ[h, h]E, h∈ E. (1.4) We denote the class of bounded operators by B(E,F). For a bounded operator A ∈ B(E,F),itsoperatornorm isdenotedbyA anditisdefinedbysquarerootoftheleast
μ≥ 0 satisfying(1.4),thatis,
It iseasytoseethattheinfimumisactuallyaminimumandhence,thatwehave [Ah, Ah]≤ A2[h, h], x∈ H. (1.6)
B(E) =B(E,E) isanormed algebrawithrespect to theusual algebraic operationsand theoperatornorm,cf. Theorem 1in[18].
Let B∗(E,F) denote the collection of all bounded and adjointable linear operators
A : E → F.A contraction isalinearoperatorT :E → F suchthat[T x,T x]≤ [x,x] for
allx∈ H.ByTheorem 2in[18],ifT ∈ B∗(E,F) isacontractionthenT∗isacontraction as well,hence,forallT ∈ B∗(E,F) wehaveT∗=T .
IfA∈ B∗(E) isselfadjoint, then,byTheorem 3in[18],
−A[h, h] ≤ [Ah, h] ≤ A[h, h], h ∈ E. (1.7)
Moreover, ifA is alinearoperatorinE and,forsomerealnumbersm,M ,wehave
m[h, h]≤ [Ah, h] ≤ M[h, h], h ∈ E, (1.8)
thenA∈ B∗(E) andA= A∗.If,inaddition, m istheminimumandM isthemaximum with theseproperties,thenA= min{|m|,|M|},seeTheorem 3in[18].
Accordingto Theorem4in[18],thealgebraB∗(E) of boundedandadjointable oper-atorsonE isapreC∗-algebraandwehaveA∗A=A2 forallA∈ B∗(E).
1.3. VE-modules overordered∗-algebras
A VE-module E over an ordered ∗-algebra A is an ordered ∗-space over A, thatis, (ve1)–(ve3) hold,subjecttothefollowingadditionalproperties
(vem1) E is a right module over A, compatible with the multiplication with scalars:
λ(ea)= (λe)a= e(λa) forallλ∈ C,e∈ E,and a∈ A.
(vem2) [e,f a+ gb]E = [e,f ]Ea+ [e,g]Eb foralle,f,g∈ E andalla,b∈ A.
Given an ordered ∗-algebra A and two VE-modules E and F over A, an operator
T ∈ L(E,F) iscalledamodulemap if
T (ea) = T (e)a, e∈ E, a ∈ A.
Any operatorT ∈ L∗(E,F) isamodule map:given arbitrarye∈ E, f ∈ F anda∈ A
we have
[T (ea), f ]F= [ea, T∗(f )]E = a∗[e, T∗(f )]E = a∗[T (e), f ]F = [T (e)a, f ]F,
hence T isa modulemap. See[16,20,29], forthe morespecial caseof Hilbert modules over C∗-algebras.
Examples1.7.LetE andF betwoVE-spaces overthesameordered∗-spaceZ.
(1)ThevectorspaceL∗(E,F) hasanaturalstructureofVE-moduleovertheordered
∗-algebraL∗(E),seeExample 1.5,moreprecisely,
[T, S] = T∗S, T, S∈ L∗(E, F). (1.9) (2) LetS be a∗-subspace of L∗(E,F) and define agramian [·,·] on S by(1.9). Let
Z bethe∗-subspaceofL∗(E) generatedbyalloperatorsT∗S, whereT,S∈ S.Z hasa naturalstructure ofordered ∗-space,where positivityof T ∈ S isinthesense of (1.3). Thus,S isaVE-spaceoverZ that,ingeneral,isnotaVE-module.
2. Linearisationsforinvariantkernels
Inthissectionwepresent themaindilationtheorem forkernels.Westartwithsome preliminaryresults.
2.1. Kernels withvaluesadjointableoperators
LetX beanonemptysetandletH beaVE-spaceovertheordered∗-space Z.A map k : X× X → L(H) is called a kernel on X and valued inL(H). In case the kernel k hasallitsvaluesinL∗(H),anadjoint kernelk∗: X× X → L∗(H) canbe associatedby k∗(x,y)= k(y,x)∗ forallx,y∈ X.Thekernelk iscalled Hermitian ifk∗= k.
LetF = F(X;H) denote thecomplex vector space of all functionsf : X → H and
letF0 =F0(X;H) beits subspace of those functions having finite support. A pairing
[·,·]F0:F0× F0→ Z canbedefinedby
[g, h]F0 =
y∈X
[g(y), h(y)]H, g, h∈ F0. (2.1)
ThispairingisclearlyaZ-gramian onF0,hence(F0;[·,·]F0) isaVE-space.
Letusobservethatthesumin(2.1)makessenseevenwhenonlyoneofthefunctions
g orh has finite support,theother canbe arbitrary inF. Thus, anotherpairing [·,·]k canbe definedonF0 by
[g, h]k= x,y∈X
[k(y, x)g(x), h(y)]H, g, h∈ F0. (2.2)
In general, the pairing [·,·]k is linear in the first variable and conjugate linear in the secondvariable.If,inaddition,k = k∗ then thepairing[·,·]k isHermitianas well,that is,
A convolutionoperator K :F0→ F canbeassociated tothekernelk by
(Kg)(y) = x∈X
k(y, x)g(x), g∈ F0, (2.3)
anditiseasytoseethatK isalinearoperator.Thereisanaturalrelationbetweenthe pairing[·,·]k andtheconvolutionoperatorK givenby
[g, h]k= [Kg, h]F0, g, h∈ F0.
Given n ∈ N, the kernel k is called n-positive if for any x1,x2,. . . ,xn ∈ X and any
h1,h2,. . . ,hn∈ H wehave n i,j=1
[k(xi, xj)hj, hi]H≥ 0. (2.4)
The kernel k is called positive semidefinite (or of positive type) if it is n-positive for all natural numbers n. The proof of the following lemma is the same as the proof of Lemma 3.1from [11].
Thethirdassertioninthenextresultmakestheconnectionwiththekernelsmadeup of boundedoperatorsonlyas in[11].
Lemma 2.1.Assume thatthekernel k : X× X → L∗(H) is2-positive. Then:
(1) k is Hermitian.
(2) If,forsome x∈ X, wehave k(x,x)= 0,thenk(x,y)= 0 forally∈ X.
(3) Assume that, for x,y ∈ X the operators k(x,x) and k(y,y) are bounded. Then
k(x,y) andk(y,x)= k(x,y)∗ are boundedand
k(x, y)2≤ k(x, x) k(y, y). (2.5)
In particular,if k(x,x)∈ B∗(E) forallx∈ X, thenk(y,x)∈ B∗(E) for allx,y∈ X.
Proof. The proofof(1)and(2)is thesameastheproofof Lemma 3.1from [11]. (3) Assumethatboth operatorsk(x,x) andk(y,y) arebounded, seeSubsection1.2, hence k(x,x),k(y,y) ∈ B∗(E). If k(y,y) = 0 then, by (2), k(x,y) = 0 and k(y,x) = k(x,y)∗ = 0,hencebounded,andtheinequality(2.5)holdstrivially.
Assumethatk(y,y)= 0,hencek(y,y)> 0.Sincek is2-positive, foranyh,g∈ H
we have
[k(x, x)h, h] + [k(x, y)g, h] + [k(y, x)h, g] + [k(y, y)g, g]≥ 0. (2.6) Weletg =−k(x,y)∗h/k(y,y) in(2.6),takeintoaccount (1.8)and get
2
k(y, y)[k(y, x)h, k(y, x)h]≤ [k(x, x)h, h] +
1
k(y, y)2[k(y, y)k(y, x)h, k(y, x)h]
≤ [k(x, x)h, h] +k(y, y)k(y, y)2[k(y, x)h, k(y, x)h] = [k(x, x)h, h] + 1
k(y, y)[k(y, x)h, k(y, x)h],
hence
[k(y, x)h, k(y, x)h]≤ k(y, y)[k(x, x)h, h] ≤ k(x, x) k(y, y)[h, h], whichprovesthatk(y,x) is abounded operatorandtheinequality(2.5). 2
Example2.2.ThisexampleisageneralisationofExample 1.2.(6).LetX beanonempty set, E be aVE-space over the ordered ∗-space Z. Let K(X;E) be the vector space of allkernelsk : X× X → L∗(E), andletK(X;E)+ be theset ofallpositivesemidefinite
kernels. Then K(X;E)+ is a cone and, by Lemma 2.1, it consists only of selfadjoint elements.If k ∈ (K(X;E)+∩ −K(X;E)+), we obtain [k(x,x)h,h]
E = 0 for all x∈ X and h ∈ E by strictness of the cone of Z. Since k(x,x) is a positive operator, hence selfadjoint,bymeansoftheanalogofthepolarisationformula(1.1),seethesecondpart of Remark 1.3, it follows thatk(x,x) = 0 for any x∈ X. Then, byLemma 2.1 again, k(x,y) = 0 for allx,y ∈ X, i.e. k = 0. Therefore K(X;E) is an ordered ∗-space with cone K(X;E)+. A multiplication can be defined onK(X;E): for k,l ∈ K(X;E) we let
(kl)(x,y)= k(x,y)l(x,y) for allx,y ∈ X.Withrespect tothis multiplication, K(X;E) isanordered∗-algebra.
Given anL∗(H)-valuedkernel k ona nonemptyset X,for someVE-spaceH on an ordered ∗-space Z, a VE-space linearisation or, equivalently, a VE-space Kolmogorov decomposition ofk is,bydefinition,a pair(K;V ),subjecttothefollowing conditions: (kd1) K isaVE-spaceoverthesameordered∗-spaceZ.
(kd2) V : X→ L∗(H,K) satisfiesk(x,y)= V (x)∗V (y) forallx,y∈ X.
TheVE-spacelinearisation(K;V ) iscalled minimal if
(kd3) Lin V (X)H = K.
TwoVE-spacelinearisations(V ;K) and(V;K) ofthesamekernelk arecalledunitary equivalent ifthereexistsaunitaryoperatorU :K → K suchthatU V (x)= V(x) forall
x∈ X.
TheuniquenessofaminimalVE-spacelinearisation(K;V ) ofapositivesemidefinite kernelk,modulounitaryequivalence,followsintheusualway:if(K;V) isanother
min-imal VE-spacelinearisationof k,for arbitraryx1,. . . ,xm,y1,. . . ,yn ∈ X andarbitrary h1,. . . ,hm,g1,. . . ,gn ∈ H,wehave [ m j=1 V (xj)hj, n i=1 V (yi)gi]K= m j=1 n i=1 [V (xj)hj, V (yi)gi]K = n i=1 m j=1 [k(yi, xj)hj, gi]K = m j=1 n i=1 [V(xj)hj, V(yi)gi]K = [ m j=1 V(xj)hj, n i=1 V(yi)gi]E,
henceU : Lin V (X)→ Lin V(X) definedby m j=1 V (xj)hj → m j=1 V(xj)hj (2.7)
is acorrectlyeverywheredefinedlinearoperator,isometric andonto.Thus, U isa VE-space isomorphismU :K → K andU V (x)= V(x) for allx∈ X,byconstruction.
2.2. Reproducingkernel VE-spaces
Let H be a VE-space over the ordered ∗-space Z, and let X be a nonempty set. A VE-space R, over the same ordered ∗-space Z, is called an H-reproducing kernel VE-space on X if there exists a Hermitian kernel k : X × X → L∗(H) such that the following axiomsaresatisfied:
(rk1) R isasubspaceofF(X;H),withallalgebraic operations.
(rk2) For allx∈ X andallh∈ H,theH-valuedfunctionkxh= k(·,x)h∈ R. (rk3) For allf ∈ R wehave[f (x),h]H= [f,kxh]R,forallx∈ X andh∈ H.
As a consequence of (rk2), Lin{kxh | x ∈ X, h ∈ H} ⊆ R. The reproducing kernel VE-spaceR iscalledminimal ifthefollowingpropertyholds aswell:
(rk4) Lin{kxh| x∈ X, h∈ H}=R.
Observe that if R is an H-reproducing kernel VE-space on X, with kernel k, then k is positive semidefiniteand uniquely determined by R hence we cantalk about the H-reproducingkernelk correspondingtoR.Ontheotherhand,a minimalreproducing kernel VE-spaceR isuniquelydeterminedbyitsreproducingkernelk.
Theclassicalreproducingkernel Hilbertspaces, e.g.see[1],arecharacterised,within theHilbert functionspaces, by thecontinuity ofthe evaluation functionals.Inthe fol-lowing, we generalise this by showing that,in theabsence of an analogueof the Riesz Representation Theorem,it istheadjointabilitywhich makesthedifference. LettingH be a VE-space over an ordered ∗-space Z, for X a nonempty set, an evaluation oper-ator Ex: F(X;H) → H can be defined for each x∈ X by letting Exf = f (x) for all
f ∈ F(X;H).Clearly,Ex islinear.
Proposition 2.3. Let X be a nonempty set, H a VE-space over an ordered ∗-space Z, and letR⊆ F(X;H), with all algebraic operations, be aVE-space over Z.Then R is an H-reproducing kernel VE-space if and only if, for all x∈ X, the restriction of the evaluationoperatorEx toR isadjointableas alinearoperatorR→ H.
Proof. AssumefirstthatR isanH-reproducingkernelVE-spaceonX andletk beits reproducingkernel.Forany h∈ H andany f ∈ R
[Exf, h]H= [f (x), h]H= [f, kxh]R. (2.8)
Sincekx ∈ L(H,R),itfollows thatExis adjointableand,inaddition, Ex∗= kx, forall
x∈ X.
Conversely, assume that, for all x ∈ X, the evaluation operator Ex ∈ L∗(R,H). Equation(2.8) shows that,inorder to show thatR is areproducing kernel VE-space, weshoulddefinethekernelk inthefollowing way:
k(y, x)h = (Ex∗h)(y), x, y∈ X, h ∈ H. (2.9) Itis clearthatk(y,x) :H → H is alinearoperatorand observethatkxh= Ex∗h forall
x∈ X andallh∈ H.Thereproducingproperty (rk3)holds:
[f (x), h]H= [Exf, h]H = [f, E∗xh]R= [f, kxh]R, f ∈ R, h ∈ H, x ∈ X. Theaxioms(rk1) and(rk2)areclearlysatisfied,so itonlyremainstoprovethatk isa Hermitiankernel. Toseethis,fix x,y∈ X andh,l∈ H.Then
[k(y, x)h, l]H= [(kxh)(y), l]H= [kxh, kyl]R
= [kyl, kxh]∗R= [k(x, y)l, h]∗R= [h, k(x, y)l]R.
Therefore, k(y,x) is adjointableand k(y,x)∗ = k(x,y), hencek isaHermitian kernel. Wehaveproventhatk isthereproducingkernelofR. 2
There is a very close connection between VE-space linearisations and reproducing kernelVE-spaces.
Proposition2.4.LetX beanonemptyset,H aVE-space overanordered∗-spaceZ,and let k : X× X → L∗(H) beaHermitiankernel.
(1) Any H-reproducing kernel VE-space R withkernel k is aVE-space linearisation
(R;V ) ofk,withV (x)= kx forallx∈ X.
(2) Forany minimalVE-space linearisation(K;V ) ofk,letting
R = {V (·)∗f | f ∈ K}, (2.10)
we obtaintheminimalH-reproducing kernel VE-space withreproducing kernelk. Proof. (2)⇒(1).Let(K;π;V ) beaminimalVE-spacelinearisationofthekernelk on X. Let R be the set of all functionsX x → V (x)∗f ∈ H,in particular R ⊆ F(X;H), and weendowR withthealgebraicoperationsinherited fromthecomplexvectorspace
F(X;H).
Thecorrespondence
K f → Uf = V (·)∗f ∈ R (2.11)
is bijective.Bythe definitionofR, thiscorrespondence issurjective. Inorder to verify thatitisinjectiveaswell,letf,g∈ K besuchthatV∗(·)f = V∗(·)g.Then,forallx∈ X
and allh∈ H wehave
[V (x)∗f, h]H= [V (x)∗g, h]H,
equivalently,
[f− g, V (x)h]K= 0, x∈ X, h ∈ H.
BytheminimalityoftheVE-spacelinearisation(K;V ) itfollowsthatg = f .Thus,U is
abijection.
Clearly,thebijectivemapU definedat(2.11)islinear,hencealinearisomorphismof complexvectorspacesK → R.OnR weintroduceaZ-valued pairing
[U f, U g]R= [V (·)∗f, V (·)∗g]R = [f, g]K, f, g∈ K. (2.12) Then (R;[·,·]R) is a VE-spaceover Z since, by (2.12), we transported the Z-gramian
from K toR or, inother words, wehave definedon R the Z-gramianthat makes the linearisomorphismU aunitaryoperatorbetweentheVE-spacesK andR.
We show that (R;[·,·]R) is an H-reproducing kernel VE-space with corresponding reproducing kernelk.Bydefinition,R⊆ F(X;H).Ontheotherhand,since
taking into account thatV (x)h ∈ K, by (2.10) it follows that kx ∈ R for all x ∈ X. Further,forallf ∈ R,x∈ X,andh∈ H,wehave
[f, kxh]R= [V (·)∗g, kxh]R= [V (·)∗g, V (·)∗V (x)h]R = [g, V (x)h]K= [V (x)∗g, h]H= [f (x), h]H,
whereg∈ K istheuniquevectorsuchthatV (x)∗g = f (x),whichshowsthatR satisfies thereproducingaxiomas well.
(1)⇒(2). Assume that(R;[·,·]R) is an H-reproducing kernel VE-space on X, with reproducingkernelk.WeletK = R anddefine
V (x)h = kxh, x∈ X, h ∈ H. (2.13)
NotethatV (x) :H → K islinearforallx∈ X.
We show thatV (x) ∈ L∗(H,K) for all x ∈ X. To see this, first note that, by the reproducingproperty,
[f, V (x)h]K= [f, kxh]R= [f (x), h]H, x∈ X, h ∈ H. (2.14) Letus then, for fixed x∈ X, consider thelinear operatorW (x) :R=K → H defined byW (x)f = f (x) forallf ∈ R=K.From (2.14)we concludethatV (x) isadjointable andV (x)∗= W (x) for allx∈ X.
Finally,bythereproducingaxiom,forallx,y∈ X andallh,g∈ H wehave [V (y)∗V (x)h, g]H= [V (x)h, V (y)g]R= [kxh, kyg]R= [k(y, x)h, g]H,
hence V (y)∗V (x) = k(y,x) for all x,y ∈ X. Thus, (K;V ) is a VE-space linearisation of k (actually,aminimal one). 2
Remark2.5. Theproof ofProposition 2.4 provides anexplicit correspondencebetween theclassofminimalVE-spacelinearisationsofk,identifiedbyunitaryequivalence, and theminimalH-reproducingkernelVE-spaceassociatedtok.
2.3. ∗-representations onVE-spacesassociatedto invariantkernels
Leta (multiplicative)semigroup Γ act onX, denotedby ξ· x,for allξ ∈ Γ andall
x∈ X.Bydefinition,wehave
α· (β · x) = (αβ) · x for all α, β ∈ Γ and all x ∈ X. (2.15) Equivalently,thismeansthatwehaveasemigroupmorphismΓ ξ → ξ·∈ G(X),where
the semigroup Γ has a unit , the action is called unital if · x = x for all x ∈ X,
equivalently, ·= IdX.
WeassumefurtherthatΓ isa∗-semigroup,thatis,thereisaninvolution ∗ onΓ;this meansthat(ξη)∗= η∗ξ∗and(ξ∗)∗= ξ forallξ,η ∈ Γ.Notethat,incaseΓ hasaunit
then ∗= .
GivenaVE-spaceH weareinterestedinthoseHermitiankernelsk : X× X → L∗(H) thatareinvariant undertheactionofΓ on X,thatis,
k(y, ξ· x) = k(ξ∗· y, x) for all x, y ∈ X and all ξ ∈ Γ. (2.16) A triple (K;π;V ) iscalled aninvariant VE-space linearisation of thekernel k and the actionof Γ on X,shortlya Γ-invariantVE-space linearisation of k,if:
(ikd1) (K;V ) isaVE-spacelinearisationofthekernelk.
(ikd2) π : Γ→ L∗(K) isa∗-representation,thatis,a multiplicative∗-morphism. (ikd3) V and π arerelatedbytheformula:V (ξ· x)= π(ξ)V (x), forallx∈ X,ξ∈ Γ.
Remarks 2.6. (1) Let(K;π;V ) beaΓ-invariant VE-spacelinearisation ofthekernel k. Since (K;V ) isa VE-spacelinearisation and taking into account theaxiom (ikd3),we have
k(y, ξ· x) = V (y)∗V (ξ· x) = V (y)∗π(ξ)V (x)
= (π(ξ∗)V (y))∗V (x) = k(ξ∗· y, x), x, y ∈ X, ξ ∈ Γ, (2.17) hencek isinvariantundertheactionofΓ onX.
(2) Observe that,ifthe action of Γ onX is unital then, for aΓ-invariant VE-space linearisation(K;π;V ),thetwoconditionsk(x,y)= V (x)∗V (y),x,y∈ X,andV (ξ· x)=
π(ξ)V (x), ξ ∈ Γ and x ∈ X, can be equivalently combined into two slightly different conditions,namely,π unital andk(x,ξ· y)= V (x)∗π(ξ)V (y),ξ∈ Γ andx,y∈ X.
If, inadditionto theaxioms (ikd1)–(ikd3),thetriple(K;π;V ) hastheproperty (ikd4) Lin V (X)H = K,
thatis,theVE-spacelinearisation(K;V ) isminimal, then(K;π;V ) iscalledaminimal
Γ-invariant VE-space linearisation ofk andtheactionof Γ on X.
MinimalinvariantVE-spacelinearisationshaveabuilt-inlinearityproperty;theproof is thesamewiththatofProposition 4.1in[11].
Proposition 2.7. Assume that, given a VE-space adjointable operator valued kernel k,
invariantundertheactionofthe∗-semigroupΓ on X,forsomefixedα,β,γ∈ Γ wehave
k(y,α· x)+ k(y,β· x)= k(y,γ· x) for allx,y ∈ X. Then, for any minimal invariant VE-space linearisation (K;π;V ) ofk,therepresentationsatisfies π(α)+ π(β)= π(γ).
TwoΓ-invariantVE-spacelinearisations(K;π;V ) and(K;π;V),ofthesame Hermi-tiankernelk, arecalled unitaryequivalent ifthereexistsaunitaryoperatorU :K → K
suchthatU π(ξ)= π(ξ)U forallξ∈ Γ, andU V (x)= V(x) forallx∈ X. Letusnote that, incase both of these invariant VE-space linearisations are minimal, then this is equivalentwiththerequirementthattheVE-spacelinearisations(K;V ) and(K;V) are unitaryequivalent.
Themain resultofthisarticle isthefollowing theorem.It isstatedintermsof both linearisationsandreproducingkernelsandtheproofpointsoutessentiallyareproducing kernelandoperatorrangeconstruction.
Theorem2.8.LetΓ bea∗-semigroupthatactsonthenonemptysetX andletk : X×X →
L∗(H) be a kernel, for some VE-space H over an ordered ∗-space Z. The following
assertionsare equivalent:
(1) k ispositivesemidefinite,in thesense of (2.4),andinvariantunder theaction ofΓ
on X,that is, (2.16) holds.
(2) k hasaΓ-invariant VE-spacelinearisation (K;π;V ).
(3) k admitsan H-reproducing kernel VE-space R and there exists a∗-representation ρ : Γ→ L∗(R) suchthat ρ(ξ)kxh= kξ·xh forallξ∈ Γ,x∈ X, h∈ H.
In addition, in case any of the assertions (1), (2), or (3) holds, then a minimal
Γ-invariant VE-space linearisation can be constructed, any minimal Γ-invariant VE-spacelinearisationis uniqueuptounitary equivalence,apair(R;ρ) asin assertion (3) withR minimalcanbealwaysobtainedand, inthis case,itisuniquelydeterminedby k
aswell.
Proof. (1)⇒(2). Assumingthat k ispositive semidefinite,by Lemma 2.1.(1) itfollows thatk isHermitian, thatis,k(x,y)∗ = k(y,x) forall x,y∈ X. Weconsider the convo-lutionoperatorK definedat (2.3)andletG = G(X;H) beitsrange,moreprecisely,
G = {f ∈ F | f = Kg for some g ∈ F0}
={f ∈ F | f(y) = x∈X
k(y, x)g(x) for some g∈ F0 and all x∈ X}. (2.18)
Apairing[·,·]G:G × G → Z canbedefinedby [e, f ]G= [Kg, h]F0 = y∈X [e(y), h(y)]H = x,y∈X [k(y, x)g(x), h(y)]H, (2.19)
[e, f ]G = y∈X [e(y), h(y)]H= x,y∈X [k(y, x)g(x), h(y)]H = x,y∈X [g(x), k(x, y)h(y)]H= x∈X [g(x), f (x)]H,
whichshowsthatthedefinitionin(2.19)iscorrect,thatis,independentofg andh such
thate= Kg andf = Kh.
We claim that [·,·]G is aZ-valued gramian,that is, it satisfies all the requirements (ve1)–(ve3).Theonlyfactthatneedsaproofis[f,f ]G = 0 impliesf = 0 andthisfollows byLemma 1.4.
Thus, (G;[·,·]G) is a VE-space that we denote by K. For each x ∈ X we define
V (x) : H → G by
V (x)h = Khx, h∈ H, (2.20)
where hx= δxh∈ F0 isthe functionthattakes thevalueh at x and isnullelsewhere.
Equivalently,
(V (x)h)(y) = (Khx)(y) = z∈X
k(y, z)(hx)(z) = k(y, x)h, y∈ X. (2.21)
Note thatV (x) is an operatorfrom theVE-spaceH to theVE-spaceG = K and it remainstoshowthatV (x) isadjointableforallx∈ X.Toseethis,letusfixx∈ X and
takeh∈ H andf ∈ G arbitrary.Then, [V (x)h, f ]G =
y∈X
[(hx)(y), f (y)]H= [h, f (x)]H, (2.22)
whichshowsthatV (x) isadjointableandthatitsadjointV (x)∗istheoperatorG f →
f (x)∈ H ofevaluationat x.
Ontheother hand,forany x,y∈ X,by(2.21),wehave
V (y)∗V (x)h = (V (x)h)(y) = k(y, x)h, h∈ H,
hence (V ;K) is aVE-spacelinearisation of k. Weprove thatit is minimal as well.To seethis,notethatatypicalelementinthelinearspanofV (X)H is,forarbitraryn∈ N, x1,. . . ,xn∈ X,andh1,. . . ,hn ∈ H, n j=1 V (xj)hj = n j=1 Khj,xj = n j=1 y∈X k(·, y)hj,xj(y) = n j=1 k(·, xj)hj,
and then takeinto account that G isthe range of the convolution operatorK defined
at(2.3). Theuniquenessoftheminimal VE-spacelinearisation(V ;K) just constructed followsasin(2.7).
Foreachξ∈ Γ weletπ(ξ) :F → F bedefinedby
(π(ξ)f )(y) = f (ξ∗· y), y ∈ X, ξ ∈ Γ. (2.23) Weprovethatπ(ξ) leavesG invariant. Tosee this,letf ∈ G,thatis,f = Kg forsome
g∈ F0or,evenmoreexplicitly,by(2.18),
f (y) = x∈X k(y, x)g(x), y∈ X. (2.24) Then, f (ξ∗· y) = x∈X k(ξ∗· y, x)g(x) = x∈X k(y, ξ· x)g(x) = z∈X k(y, z)gξ(z), (2.25) where, gξ(z) = ⎧ ⎨ ⎩
0, if ξ· x = z has no solution x ∈ supp g,
ξ·x=zg(x), otherwise.
(2.26)
Sincegξ ∈ F
0,it followsthatπ(ξ) leavesG invariant.Inthefollowing wedenotebythe
samesymbol π(ξ) themapπ(ξ) :G → G.
Weprovethatπ isarepresentationofthesemigroupΓ onthecomplexvectorspaceG, thatis,
π(αβ)f = π(α)π(β)f, α, β∈ Γ, f ∈ G. (2.27)
To see this, let f ∈ G be fixed and denote h = π(β)f , that is, h(y) = f (β∗ · y) for all y ∈ X. Then π(α)π(β)f = π(α)h,that is, (π(α)h)(y) = h(α∗· y) = h(β∗α∗· y) =
h((αβ)∗· y)= (π(αβ))(y),forally∈ X,whichproves (2.27)
Weshowthatπ isactuallya∗-representation,thatis,
[π(ξ)f, f]G = [f, π(ξ∗)f]G, f, f∈ G. (2.28) To see this, let f = Kg and f = Kg for some g,g ∈ F0. Then, recalling (2.19) and (2.25),
[π(ξ)f, f]G = y∈X [f (ξ∗y), g(y)]H= x,y∈X [k(ξ∗· y, x)g(x), g(y)]H = x,y∈X [k(y, ξ· x)g(x), g(y)]H= x,y∈X [g(x), k(ξ· x, y)g(y)]H = x∈X [g(x), f(ξ· x)]H= [f, π(ξ∗)f]H,
and hencetheformula(2.28) isproven.
Inordertoshowthattheaxiom(ikd3)holdsaswell,weuse(2.21).Thus,forallξ∈ Γ, x,y∈ X,h∈ H,andtakingintoaccountthatk isinvariantundertheactionofΓ on X, we have
(V (ξ· x)h)(y) = k(y, ξ · x)h = k(ξ∗· y, x)h
= (V (x)h)(ξ∗· y) = (π(ξ)V (x)h)(y), (2.29) which proves (ikd3). Thus, (K;π;V ), here constructed, is a Γ-invariant VE-space lin-earisation oftheHermitian kernelk. Notethat(K;π;V ) isminimal, thatis, theaxiom (ikd4)holds,sincetheVE-spacelinearisation(K;V ) isminimal.
Let(K;π;V) beanotherminimalinvariantVE-spacelinearisationof K.Weconsider theunitaryoperatorU :K → K definedasin(2.7)andwealreadyknowthatU V (x)=
V(x) forallx∈ X.Since, foranyξ∈ Γ,x∈ X,andh∈ H,wehave
U π(ξ)V (x)h = U V (ξ· x)h = V(ξ· x)h = π(ξ)V(x)h = π(ξ)U V (x)h, and takingintoaccount theminimality,itfollows thatU π(ξ)= π(ξ)U forallξ∈ Γ.
(2)⇒(1).Let(K;π;V ) beaΓ-invariantVE-spacelinearisation ofk.Then n j,i=1 [k(xi, xj)hj, hi]H= n j,i=1 [V (xi)∗V (xj)hj, hi]H = [ n j=1 V (xj)hj, n j=1 V (xj)hj]H≥ 0,
for all n∈ N, x1,. . . ,xn ∈ X, and h1,. . . ,hn ∈ H, hencek is positive semidefinite.It wasshownin(2.17) thatk isinvariantundertheactionofΓ onX.
(2)⇒(3). This followsfrom Proposition 2.4with thefollowing observation: with no-tation asintheproof ofthatproposition,forallx,y∈ X andh∈ H wehave
kξ·x(y)h = k(y, ξ· x)h = V (y)∗V (ξ· x)h = V (y)∗π(ξ)V (x)h,
hence,lettingρ(ξ)= U π(ξ)U−1,whereU :K → R istheunitaryoperatordefinedasin
(2.11),weobtaina∗-representationofΓ ontheVH-spaceR suchthatkξ·x = ρ(ξ)kxfor allξ∈ Γ andx∈ X.
(3)⇒(2).Letρ : Γ→ L∗(R) isa∗-representationsuchthatkξ·x= ρ(ξ)kxforallξ∈ Γ and x ∈ X. Again, we use Proposition 2.4. Letting π = ρ, it is then easy to see that (R;π;V ) isaΓ-invariant VE-spacelinearisationofthekernel k. 2
Remarks 2.9. (1)Given k : X× X → L∗(H) apositive semidefinitekernel, as a conse-quenceofTheorem 2.8wecandenote,withoutanyambiguity,byRktheuniqueminimal
H-reproducingkernelVE-spaceonX associatedto k.
(2)Theconstructionintheproofof(1)⇒(2)inTheorem 2.8isessentiallyaminimal
H-reproducing kernel VE-space one. More precisely, we first note that, for arbitrary
f ∈ F(X;H),f = Kg withg∈ F0(X;H),wehave f = x∈X k(y, x)g(x) = x∈X kx(y)g(x), (2.30)
henceG(X;H)= Lin{kxh| x∈ X, h∈ H}.Then,forarbitraryf ∈ G we have [f, kxh]K= [f, kxh]G = [f, Khx]G =
y∈X
[f (y), (hx)(y)]H = [f (x), h]H= [f, kxh]R(K), x∈ X, h ∈ H,
hence [·,·]K = [·,·]R(K) on G(X;H)= Lin{kxh| x ∈ X, h∈ H}, thatcoincides with bothK andR(K).Therefore,wecantakeK = R(K)=G(X;H) tobeaVE-space,with theadvantagethatitconsistsentirely ofH-valuedfunctionson X.
Thisideawasusedin[11] aswelland thesourceof inspirationis[32].
3. ResultsthatTheorem 2.8unifies
In this section we obtain, as consequences of the main result, different versions of knowndilationtheorems innon-topological versions.
3.1. Positivesemidefinite mapson∗-semigroups
GivenaVE-spaceH overanordered∗-spaceZ anda∗-semigroupΓ,amap ϕ : Γ→ L∗(H) is called positive semidefinite or of positive type if,for alln ∈ N,ξ
1,. . . ,ξn ∈ Γ andh1,. . . ,hn ∈ H,wehave n i,j=1 [ϕ(ξi∗ξj)hj, hi]H≥ 0. (3.1)
Givenamapϕ : Γ→ L∗(H) weconsider thekernelk : Γ× Γ→ L∗(H) defined by
andobservethatϕ ispositivesemidefinite,inthesenseof(3.1),ifandonlyifk ispositive semidefinite,inthesenseof(2.4).
On the other hand, considering the action of Γ on itself by left multiplication, the kernel k,asdefinedat (3.2),is Γ-invariant,inthesense of(2.16).Indeed,
k(ξ, α· ζ) = ϕ(ξ∗αζ) = ϕ((α∗ξ)∗ζ) = k(α∗· ξ, ζ), α, ξ, ζ ∈ Γ.
Therefore, thefollowingcorollaryisadirectconsequenceofTheorem 2.8.
Corollary3.1.Letϕ : Γ→ L∗(H) beamap,forsome∗-semigroupΓ andsomeVE-space H over anordered ∗-space Z.The followingassertionsare equivalent:
(1) The mapϕ ispositivesemidefinite.
(2) ThereexistsaVE-spaceK overZ,amapV : Γ→ L∗(H,K),anda∗-representation π : Γ→ L∗(K),suchthat:
(i) ϕ(ξ∗ζ)= V (ξ)∗V (ζ) for allξ,ζ∈ Γ.
(ii) V (ξζ)= π(ξ)V (ζ) for allξ,ζ∈ Γ.
In addition, if this happens, then thetriple (K;π;V ) can alwaysbe chosenminimal, in the sense that K is the linear span of theset V (Γ)H, and any two minimal triples as before areunique,modulo unitary equivalence.
(3) ThereexistanH-reproducingkernelVE-spaceR onΓ anda∗-representationρ : Γ→ L∗(R) suchthat:
(i) R hasthereproducing kernelΓ× Γ (ξ,ζ)→ ϕ(ξ∗ζ)∈ L∗(H). (ii) ρ(α)ϕ(·ξ)h= ϕ(·αξ)h for allα,ξ∈ Γ andh∈ H.
In addition, the reproducing kernel VE-space R as in (3) can be always constructed minimaland inthis caseitisuniquelydetermined by ϕ.
Ascanbeobservedfromcondition(2).(i)inCorollary 3.1,wedonothavea represen-tation ofϕ onthewhole∗-semigroupΓ butonlyonits∗-subsemigroup{ξ∗ζ| ξ,ζ∈ Γ},
which maybe strictly smallerthan Γ.This situationcanbe remedied, forexample, in casethe∗-semigroupΓ hasaunit,whenthepreviouscorollarytakesaformsimilarwith B. Sz.-Nagy Theorem,cf. [32].
Corollary3.2.Assumethatthe∗-semigroupΓ hasaunit.Letϕ : Γ→ L∗(H) be amap, forsomeVE-spaceH overanordered∗-spaceZ.Thefollowingassertionsareequivalent:
(1) The mapϕ ispositivesemidefinite.
(2) There exist a VE-space K over Z, a linear operator W ∈ L∗(H,K), and a unital ∗-representation π : Γ→ L∗(K), suchthat:
In addition,if this happens,then thetriple (K;π;V ) can alwaysbe chosen minimal, in thesense that K is thelinearspan of theset π(Γ)WH, andany two minimaltriples as beforeare unique,modulo unitary equivalence.
3.2. Positivesemidefinite linearmaps
Given a ∗-algebra A, a linear map ϕ :A → L∗(H), for some VE-space H over an ordered ∗-space Z, is called positive semidefinite if for all a1,. . . ,an ∈ A and all
h1,. . . ,hn ∈ H wehave
n i,j=1
[ϕ(a∗iaj)hj, hi]H≥ 0, (3.4)
where the inequality is understood in Z with respect to the given cone Z+ and the underlying partial order, see Subsection 1.1. Observe that for aHermitian linearmap
ϕ :A→ L∗(H) onecandefineaHermitian kernelkϕ:A× A→ L∗(H) byletting
kϕ(a, b) = ϕ(a∗b), a, b∈ A.
Also,observe thatthe ∗-algebraA can be viewed asamultiplicative ∗-semigroupand, letting A act on itself by multiplication, the kernel kϕ is invariant under this action. Withthisnotation,anotherconsequenceofTheorem 2.8isthefollowing:
Corollary 3.3. Let ϕ : A → L∗(H) be a linear map, for some ∗-algebra A and some VE-space H overanordered ∗-space Z.The followingassertionsare equivalent:
(1) Themapϕ ispositivesemidefinite.
(2) There exist a VE-space K over the ordered ∗-space Z, a linear map V : A → L∗(H,K),and a∗-representationπ :A→ L∗(K),suchthat:
(i) ϕ(a∗b)= V (a)∗V (b) foralla,b∈ A.
(ii) V (ab)= π(a)V (b) foralla,b∈ A.
In addition,if this happens,then thetriple (K;π;V ) can alwaysbe chosen minimal, in the sense that K is thelinear span of theset V (A)H, and any twominimal triples as beforeare unique,modulo unitary equivalence.
(3) There exist an H-reproducing kernel VE-space R on A and a ∗-representation ρ :A→ L∗(R) suchthat:
(i) R hasthereproducing kernel A× A (a,b)→ ϕ(a∗b)∈ L∗(H). (ii) ρ(a)ϕ(·b)h= ϕ(·ab)h for alla,b∈ A andh∈ H.
In addition, the reproducing kernel VE-space R as in (3) can be always constructed minimalandinthis caseitisuniquely determinedby ϕ.
In case the ∗-algebra has a unit, the previous corollary yields a Stinespring type Representation Theorem, cf. [30], or its generalisations [11]. More precisely, letting e
denote theunitofthe∗-algebraA andwiththenotationasinCorollary 3.3.(2),letting
W = V (e), wehave
Corollary 3.4.LetA beaunital∗-algebraA andϕ :A→ L∗(H) alinearmap, forsome VE-space H over anordered ∗-spaceZ.The followingassertionsare equivalent:
(i) ϕ is positivesemidefinite.
(ii) There exist K a VE-space over the same ordered ∗-space Z, a ∗-representation π :A→ L∗(K),and W ∈ L∗(H,K) suchthat
ϕ(a) = W∗π(a)W a∈ A. (3.5)
In addition, ifthis happens,then thetriple (K;π;W ) canalwaysbe chosenminimal,in the sensethat K isthelinear span oftheset π(A)W H, andany twominimaltriples as before areunique,modulo unitary equivalence.
Remarks3.5.(1)Indilationtheory,oneencountersalsothenotionofcompletelypositive,
e.g. see[25]. Inoursetting, we canconsider alinear map ϕ :V → L∗(E), where V is a
∗-spaceandE issomeVE-spaceoveranordered∗-spaceZ.Foreachn onecanconsider the ∗-spaceMn(V) ofalln× n matriceswith entriesinV.Thenthe n-thamplification
map ϕn: Mn(V)→ Mn(L∗(E))=L∗(En) isdefinedby
ϕn([ai,j]ni,j=1) = [ϕ(ai,j)]i,j=1n , [ai,j]ni,j=1∈ Mn(V). (3.6) Basically,ϕ wouldbecalledcompletelypositive ifϕnis“positive”forall n,where “posi-tive”shouldmeanthat,whenever[ai,j]ni,j=1is“positive”inMn(V) thenϕn([ai,j]ni,j=1) is positiveinMn(L∗(E)).SincepositivityinMn(L∗(E)) isperfectlydefined,seeRemark 1.6, the onlyproblem is to definepositivity inMn(V). Oneof thepossible approaches, e.g. see[6],istoassumeV beamatrixquasiordered∗-space,thatis,thereexists{Cn}n≥1 a
matrix quasiordering ofV,inthefollowing sense (mo1) Foreachn∈ N,Cn isaconeonMn(V).
(mo2) Foreachm,n∈ N andeachm×n matrixwithcomplexentries,wehaveT∗CmT ⊆
Cn,where multiplicationistheusualmatrixmultiplication.
Inthespecialcasewhen(mo1)ischangedsuchthatforeachn∈ N,theconeCnisstrict, onehastheconceptofmatrixordering and,respectively,of matrixordered∗-space,e.g. see [26]. For example, L∗(E) has a natural structure of matrix ordered ∗-algebra, see
Remark 1.6.Observethat,inthelattercase,eachMn(V) isanordered∗-spacehence,in this case,theconcept ofcompletely positivemapϕ :V → L∗(E) makesperfectly sense.
In theformer case, that of matrix quasiordered ∗-spaceV, the concept of completely positivemap ϕ makessenseaswell.
(2) Assuming that instead of V we have a ∗-algebra A and that the concept of a completely positive map ϕ :A → L∗(E) is defined, a natural question is what is the relationof this conceptwith thatofpositivesemidefinitemap ϕ. Byinspection,itcan beobservedthat,inorderto relatethetwo concepts,thematrix(quasi)orderingonA shouldbe relatedwith thatof∗-positivity,seeRemark 1.1.Moreprecisely,observe first that∗-positivityprovides inanaturalwayamatrixquasiordering ofA. Then,onecan provethatifϕ iscompletelypositive,withdefinitionasinitem(1)andwithrespectto the∗-positivity,thenϕ ispositivesemidefinite,withdefinitionasin(3.4).Theconverse isevenmoreproblematic,dependingonwhetherany∗-positivematrix[ai,j]ni,j=1canbe representedasasumofmatricesa∗a,wherea isaspecialmatrixwithonlyonenon-null row.ThisspecialsituationhappensforC∗-algebras [30],orevenforlocally C∗-algebras
[13],butitmayfail evenforpreC∗-algebras,ingeneral.
3.3. Linearmapswith valuesadjointableoperatorson VE-modules
Given an ordered ∗-algebra A and aVE-module E over A, anE-reproducing kernel VE-module overA is justan E-reproducingkernel VE-spaceoverA, withdefinitionas inSubsection2.1,whichisalsoaVE-module overA.
Proposition3.6.LetΓ bea∗-semigroupthatactsonthenonemptysetX andletk : X×
X → L∗(H) be a kernel, for some VE-module H over an ordered ∗-algebra A. The followingassertionsare equivalent:
(1) k ispositivesemidefinite,in thesense of (2.4),andinvariantunder theaction ofΓ
on X,that is, (2.16) holds.
(2) k hasaΓ-invariant VE-module(over A)linearisation (K;π;V ).
(3) k admitsanH-reproducing kernelVE-moduleR and thereexistsa∗-representation ρ : Γ→ L∗(R) suchthat ρ(ξ)kxh= kξ·xh forallξ∈ Γ,x∈ X, h∈ H.
In addition, in case any of the assertions (1), (2), or (3) holds, then a minimal
Γ-invariant VE-module linearisation can be constructed, any minimal Γ-invariant VE-module linearisation is unique up to unitary equivalence, a pair (R;ρ) as in assertion
(3) with R minimal can be alwaysobtained and, in this case, itis uniquely determined byk as well.
Proof. We use the notation as in the proof of Theorem 2.8. We actually prove only theimplication(1)⇒(2)since,as observedinRemark 2.9, thatconstruction providesa Γ-invariant reproducingkernel VE-spacelinearisation, while theother implications are notmuchdifferent.
