Construction of a Natural Transformation from a Classical to a Quantum 0-Species
aa r X i v : . [ m a t h . F A ] D ec Construction of a Natural Transformation from aClassical to a Quantum -Species Benedetto Silvestri
Mathematics Subject Classification.
Primary 46M99, 46H35, 47D06, 46Kxx;Secondary 46F05
Key words and phrases. dynamical patterns, natural transformations, topological ∗ -algebras of linear operators, C -semigroups on locally convex spaces, topological ∗ -algebras of test functions. A bstract . A natural transformation J between functors valued in the category Chdv is assembled. Chdv is obtained by replacing both the categories ptls and ptsa withthe category of topological linear spaces in the defining properties of the category Chdv introduced in one of our previous papers. By letting a dp -valued functor be (classical)quantum whenever every its value is a dynamical pattern whose set map takes valuesin the set of (commutative) noncommutative topological unital ∗− algebras, and letting a(classical) quantum 0-species be a Chdv -valued functor factorizing through the canonicalfunctor from dp to Chdv into a (classical) quantum dp -valued functor, we have that thedomain and codomain of J are a classical and a quantum 0-species respectively. ontents Notation 4Introduction 61. Construction of the topological ∗ -algebra B ( M ) and the C -group Γ U M x and z J from x to z Notation
In this section we fix the notation and collect general facts we shall use often withoutany further mention in the paper including the Introduction. If X , Y are topologicalspaces, then C ( X , Y ) is the set of continuous maps from X to Y . If the set underlying X isa subset of the set underlying Y , then X ֒ → Y means ı YX ∈ C ( X , Y ) with ı YX the inclusionmap of X into Y . vct is the category of complex vector spaces and linear maps, top is thecategory of topological spaces and continuous maps.All topological vector spaces are considered over C , tls is the category of topolog-ical vector spaces and continuous linear maps. If E , F ∈ tls, then we let L ( E , F ) ≔ Mor tls ( E , F ) = C ( E , F ) ∩ Mor vct ( E , F ), L ( E ) ≔ L ( E , E ) and E ′ ≔ L ( E , C ). If E and F arelocally convex spaces and G is a family of bounded subsets of E , then L G ( E , F ) is thelocally convex space L ( E , F ) endowed with the topology of uniform convergence overthe sets in G . L s ( E , F ), L b ( E , F ) and L pc ( E , F ) stand for L G ( E , F ) with G respectively thefamily of finite, bounded and precompact subsets of E . If p is a continuous seminormon F and B is a bounded set of E , then p B : T sup x ∈ B p ( Tx ) is a continuous seminorm of L b ( E , F ). ptls is the subcategory of tls of preordered topological vector spaces and linearcontinuous positive maps, r ∈ Fct(ptls , tls) is the forgetful functor.For any unital ∗ -algebra A , F max (relative to a fixed wedge in the hermitian elementsof A ) is defined in [ , p.24], while A -invariant non-empty subsets of F max are defined in[ , p.25]. If H and G are Hilbert spaces, then for every densely defined linear operator in H with values in G , let S ⊺ denote the adjoint of S . The concept of O ∗ -algebra A on a denselinear subspace D of a Hilbert space is provided in [ , Def. 2.1.6], the locally convexspace D A is defined in [ , Def. 2.1.1] and its topology is called the graph topology of A on D . The linear space L ( D A , D + A ) is defined in [ , Def. 3.2.1] and the bounded topology τ b on it in [ , p.76].tsa is the category of topological ∗ -algebras and continuous ∗ -morphisms, tsa is thecategory of topological unital ∗ -algebras and continuous unit preserving ∗ -morphisms.Let ˜ q ∈ Fct(tsa , tls), q ∈ Fct(tsa , tls) and q ∈ Fct(tsa , top) be the forgetful functors. If A is a subcategory of B , then we let I BA denote the inclusion functor. Given an object A witha structure we often use, as we did above, the common abuse of language of denotingby A each of its underlying structure. So for instance if A and B are topological unital ∗ -algebras, L ( A , B ) stands for L ( q ( A ) , q ( B )) while C ( A , B ) stands for C ( q ( A ) , q ( B )). Atopological unital sub ∗ -algebra A of B ∈ tsa here always means that A ∈ tsa so that A is a topological subspace of B , A is a sub ∗ -algebra of B and A holds the same unit of B . Given A ∈ tsa we let A ∈ tsa be the unitization of A [ , p.38] whose topology bydefinition is the product topology. In case A is a locally convex ∗ -algebra whose topologyis generated by the set S of seminorms, then ˜ S = { ˜ r | r ∈ S } generates the locally convextopology of A , where ˜ p ( a , λ ) ≔ p ( a ) + | λ | for every seminorm p on A . In particular if p isa continuous seminorm on A , then ˜ p is continuous on A . Thus if A ∈ tsa , and B ∈ tsaare both locally convex and T ∈ Mor tls (˜ q ( A ) , q ( B )), then(1) (cid:16) ( a , λ ) Ta + λ B (cid:17) ∈ Mor tls ( q ( A ) , q ( B )); OTATION 5 since for every continuous seminorm q of B there exists a continuous seminorm p of A such that for all ( a , λ ) ∈ A we have q ( Ta + λ B ) ≤ q ( Ta ) + k | λ | ≤ p ( a ) + k | λ | with k = q ( B ),thus if k ,
0, then q ( Ta + λ B ) ≤ k e p ′ ( a , λ ) with p ′ = k − p , otherwise q ( Ta + λ B ) ≤ ˜ p ( a , λ ).Given two top-quasi enriched categories A and B , let Fct top ( A , B ) denote the set offunctors of top-quasi enriched categories from A to B [ , Section 1.2].If X is a locally compact space, then K ( X ) denotes the locally convex space of complexvalued continuous maps on X with compact support endowed with the usual inductivelimit topology [ , Ch. 3, §
1, n. 1]. In this paper a measure on X always means an elementof K ( X ) ′ [ , Ch. 3, §
1, n. 3, Def. 2].All manifolds are smooth and finite dimensional, hence locally compact. All vectorfields are smooth. Let M be a manifold. C ∞ ( M ) denotes the locally convex spaceof complex valued smooth maps on M endowed with the Frechet topology [ , p.12]here denoted by τ ∞ ( M ) or simply τ ∞ . D ( M ) denotes the locally convex space of complexvalued smooth maps on M with compact support endowed with the usual inductive limittopology [ , p.13] here denoted by τ ∞ c ( M ) or simply τ ∞ c . We have D ( M ) ֒ → C ∞ ( M ) [ , Rmk.1.1.13]. D ( M ) is a Montel space [ , example 6, p.241] so barrelled, sequentially complete[ , Thm. 1.1.11(i)] topological ∗− algebra [ , 28.7, 28.12] such that D ( M ) ֒ → K ( M ) [ ,p.241]. Let h D ( M ) , · i denote the natural left C ∞ ( M )-module.If N and M are manifolds and φ : N → M is a smooth proper map [ , Def. 1.1.16],then we shall consider φ ∗ defined on D ( M ), thus φ ∗ ∈ L ( D ( M ) , D ( N )) [ , Prp. 1.1.17].For any k ∈ Z + ∗ let Di ff Op k ( M , N ) be the set of the restrictions at D ( M ) of the elementsin Di ff Op k ( M × C , N × C ) where for every vector bundle A on M and B on N , we letDi ff Op k ( A , B ) be the set of di ff erential operators of order k from A to B [ , Def. 1.2.1].Set Di ff Op k ( M ) = Di ff Op k ( M , M ). h Di ff Op k ( M , N ) , · i is naturally a left C ∞ ( N )-module,since Di ff Op k ( A , B ) it is so, where for every F ∈ C ∞ ( N ) and T ∈ Di ff Op k ( M , N ) we set( F · T ) : D ( M ) → D ( N ), h F · T ( h ). If T ∈ Di ff Op k ( M , N ), then supp( D f ) ⊆ supp( f ) forevery f ∈ D ( M ) [ , Rmk.1.2.2(iv)]. We have Di ff Op k ( M , N ) ⊂ L ( D ( M ) , D ( N )) [ , Thm.1.2.10].Let X ( M ) be the set of vector fields of M , if U ∈ X ( M ), then let £ U be the restriction at D ( M ) of the Lie derivative on C ∞ ( M ) associated with U here denoted by £ > U ; so £ U f = U f for every f ∈ D ( M ) and in particular £ U ∈ Di ff Op ( M ). If V ∈ X ( N ), U ∈ X ( M ), φ : N → M is smooth and V and U are φ -related, then £ V ◦ φ ∗ = φ ∗ ◦ £ U . Whenever U is complete,we let θ U : R → Di ff ( M ) be the flow on M generated by U and for every t ∈ R let η UM ( t ) ≔ ( θ U ( − t )) ∗ ∈ L ( D ( M )) namely the map f f ◦ θ U − t .Let M = ( M , g ) be a semi-Riemannian manifold, thus for every f ∈ C ∞ ( M ) let grad M ( f )be the gradient of f w.r.t. g , thus grad M ( f ) ∈ X ( M ) such that h grad M ( f ) , Y i M = £ Y ( f )for every Y ∈ X ( M ) where h · , · i M : X ( M ) × X ( M ) → C ∞ ( M ) is the C ∞ ( M )-bilinear mapcorresponding to the metric g . Let µ g denote the measure on M associated via [ , Thm.4.7] with the density relative to g [ , Prp. 2.1.15(ii)], set H g ≔ L ( M , d µ g ). We have K ( M ) ֒ → H g since for instance [ , Thm. 1.1.11(iv)], since µ g is continuous on K ( M )and since k f ∗ f k = k f k for every compact K and every f ∈ C ( X , K ), being a C ∗ − algebra CONTENTS the normed space C ( X , K ) of complex valued continuous maps on X with support in K endowed with the sup − norm. The inclusion K ( M ) ֒ → H g is dense [ , Ch. 4, §
3, n. 4,Def. 2] as well the inclusion D ( M ) ֒ → H g . If ( N , g ′ ) is a semi-Riemannian manifold and D ∈ Di ff Op k ( M , N ), then D ⊺ is well-set since D ( M ) is dense in H g , moreover by [ , Thm.1.2.15] we deduce that D ( N ) ⊆ Dom( D ⊺ ) and D ⊺ D ( N ) ⊆ D ( M ). If φ : N → M is a smoothdi ff eomorphism such that φ ∗ g = g ′ , thus φ ∗ (on D ( M )) extends to a unitary operator from H g onto H g ′ still denoted φ ∗ such that ( φ ∗ ) ⊺ = ( φ − ) ∗ .Here by a C -semigroup on a topological vector space Y is meant a map U ∈ C ( R + , L s ( Y )) such that U ( s + t ) = U ( s ) U ( t ) for all t , s ∈ R + and U (0) = . In additionby letting τ the topology of Y , U is called τ − equicontinuous or simply equicontinuousif { U ( t ) | t ∈ R + } is a ( τ, τ )-equicontinuous set. Similar definitions for a C -group byreplacing R + with R .If X is a sequentially complete locally convex space with topology τ and T ∈ L ( X )such that { T n | n ∈ Z ∗ + } is ( τ, τ )-equicontinuous, then it is well-known that there exists a C -semigroup exp TX on X such that:(1) T is the infinitesimal τ − generator of exp TX ;(2) exp TX ( t ) x = P ∞ k = tT ) k k ! x convergence in X for every t ∈ R + and x ∈ X ;(3) by letting exp TX : R + ∋ t exp( − t ) exp TX ( t ) we have that exp TX is an equicontinu-ous C -semigroup on X .Since the equicontinuity hypothesis it is clear that the series in (2) extends to t ∈ R , soexp TX extends to a C -group on X still denoted by the same symbol, moreover exp TX is anequicontinuous C -semigroup on X , where exp TX : R + ∋ t exp( − t ) exp TX ( − t ). Introduction
In [ , Cor.1.6.43] and the discussion after we have shown that the existence of a naturaltransformation, from the classical gravity species a to a strict quantum gravity species,satisfying certain constraints would render the dark energy hypothesis unnecessaryin explaining the actual cosmic acceleration. This paper is one step toward a betterunderstanding of the way to construct such a natural transformation.In order to describe our results we need some additional terminology.First of all we recall that an object of the category dp of dynamical patterns ([ ,Cor. 1.4.5 and Def. 1.4.1]) is a functor of top-quasi enriched categories (i.e. a functorwhose morphism map is continuous ) valued in the top-quasi enriched category tsa ofunital topological ∗ -algebras (enriched by endowing the morphism set of every twoobjects of tsa with the topology of simple convergence). A morphism of dynamicalpatterns is a couple ( f , T) formed by a functor f of top-quasi enriched categories fromthe domain of the second dynamical pattern to the domain of the first one, and by anatural transformation T from the composition of the first dynamical pattern with f tothe second dynamical pattern.Next let Chdv be the category introduced in Prp. 1.26 and obtained by replacing inthe defining properties of the category Chdv ([ , Cor. 1.4.18 and Def. 1.4.17]) both the NTRODUCTION 7 categories ptls and ptsa with the category of topological linear spaces tls. Similarly at
ΨΨΨ there exists the (canonical) functor
ΨΨΨ from dp to Chdv . A 0-species is a functor valuedin Chdv which factorizes through dp (Def. 1.27) in particular a 0-species is a 1-cell of the2-category 2 − dp . A dynamical pattern is called quantum (respectively classical) if allits values are noncommutative (respectively commutative) algebras. A functor valuedin dp is called quantum (respectively classical) if all its values are quantum (respectivelyclassical) dynamical patterns. Finally a 0-species is called quantum (respectively classi-cal) if it factorizes through ΨΨΨ into a quantum (respectively classical) functor valued in dp . Thus we have what follows.In Thm. 2.2 and
Thm. 2.4 we construct two functors valued in dp , the first x classicaland the second z quantum.Then in Thm. 3.10 we establish our main result: The existence of the natural trans-formation J from the classical 0-species x to the quantum 0-species z , where x factorizesto the left and to the right through x and z factorizes to the left and to the right through z . Now the following observation is worthwhile.Since in the present paper we are decisively dealing with the categories dp and Chdv , specifically with the construction of the functors x and z and the constructionof the natural transformation J , statements concerning continuity acquire a distinctivevalue. Specifically we refer to:(1) The C -continuity of the semigroup Γ U M (Thm. 1.24(2)) at the core of the objectmap of z .(2) The continuity of the ∗ -morphism T ( φ ) (Thm. 1.15(2)) at the core of the morphismmap of z .(3) The continuity of the map f £ grad M ( f ) (Cor. 3.6) at the core of J .In the remaining of this introduction we shall briefly outline the main steps to arriveat our main result. Thm. 1.15 and
Thm. 1.24 are the main results of section 1. In Thm. 1.15(1) weprove that B ( M ) is a unital topological ∗ -algebra and in Thm. 1.15(2) we prove that φ implements via T a morphism of unital topological ∗ -algebras. In Thm. 1.24(2)we establish the existence of Γ U M a C -group on B ( M ) of ∗− automorphisms and in Thm.1.24(3) we prove that Γ and T are equivariant namely (8) holds true. Here M = ( M , g ) and N = ( N , g ′ ) are semi-Riemannian manifolds and φ : N → M is a smooth di ff eomorphismsuch that φ ∗ g = g ′ . Our construction of B is calibrated to ensure that T and Γ possessthe above properties.We start by defining exp UM as the exponential C -group, on the sequentially completelocally convex space D ( M ) (remember D ( M ) is endowed with the inductive limit topol-ogy τ ∞ c ( M )), generated by £ U provided { £ kU } k ∈ Z + be ( τ ∞ c , τ ∞ c )-equicontinuous, and let Λ UM be the corresponding action on L ( D ( M )) namely Λ UM : t ( T exp UM ( t ) ◦ T ◦ exp UM ( − t )) . CONTENTS
Next we define the set B ( M ) underlying B ( M ) in Def. 1.7 as the subset of those linearand continuous operators on D ( M ) whose Hilbert space adjoint in H g is such that itsdomain contains D ( M ), maps D ( M ) into itself and its restriction to D ( M ) is continuous: B ( M ) ≔ { T ∈ L ( D ( M )) | D ( M ) ⊆ Dom( T ⊺ ) , T ⊺ D ( M ) ⊆ D ( M ) , T † ≔ T ⊺ ↾ D ( M ) ∈ L ( D ( M )) } ;where T ⊺ is the H g − adjoint of the operator T . In Prp. 1.9 we show that B ( M ) is a O ∗ -algebra on D ( M ), D ( M ) seen in this context as a dense linear subspace of H g , inparticular B ( M ) is a unital ∗ -algebra. Then in Def. 1.11 we define a set of functionals F M over B ( M ) and prove in Prp. 1.12 that F M is a B ( M )-invariant non-empty subset of F max (relative to the wedge of finite sums of positive elements of B ( M )). This result alongwith the general result [ , Lemma 1.5.7] applied to our ∗ -algebra B ( M ) and our set F M ,enables us to show in Thm. 1.15(1) that(2) B ( M ) ≔ B ( M )[ τ M ] ∈ tsa;where τ M (Def. 1.13) is the locally convex topology on B ( M ) generated by the followingset of seminorms { q B | B ∈ Bounded( D ( M )) } ; q B : B ( M ) ∋ T sup f ∈ B |h f , T f i H g | . There is another tsa-structure over B ( M ), indeed B ( M ) endowed with the topologyrelative to the bounded topology on L ( D ( M ) B ( M ) , D ( M ) + B ( M ) ) is a unital topological ∗ -algebra and this topology is stronger than τ M (Prp. 1.17). Next by letting(3) T ( φ )( T ) ≔ φ ∗ ◦ T ◦ ( φ − ) ∗ ;we prove in Thm. 1.15(2) that(4) T ( φ ) ∈ Mor tsa ( B ( M ) , B ( N )) . In Cor. 1.6 we prove that Λ UM is a C -group on the locally convex space L b ( D ( M )) ofcontinuous linear maps on D ( M ) endowed with the topology of uniform convergenceover the bounded subsets of D ( M ) namely(5) Λ UM ∈ C (cid:16) R , L s ( L b ( D ( M ))) (cid:17) ; Λ UM is a one-parameter group . This result is a consequence of Lemma 1.5, a more general result important in its own,enlightening the twofold essential role palyed by the Montel space D ( M ) in obtainingCor. 1.6: the first directly by its definition, the second permitting to use of the Banach-Steinhaus Thm. since any Montel space is barrelled.In Def. 1.10 we define the category vf of the couples ( M , U ) with the followingproperties: M = ( M , g ) is a semi-Riemannian manifold, U is a vector field of M such that NTRODUCTION 9 { £ nU | n ∈ Z ∗ + } is ( τ ∞ c , τ ∞ c )-equicontinuous, and the following property of invariance holdstrue(6) µ g ◦ £ U = . While φ is a morphism from ( M , U ) to ( N , V ) i ff φ : N → M is smooth, φ ∗ g = g ′ , and U and V are φ -related with N = ( N , g ′ ). vf is the subcategory of vf with the same objectset and di ff eomorphisms as morphisms. Now the reason of introducing the abovecategories stands on Thm. 1.24(2,3) establishing that whenever ( M , U ) , ( N , V ) ∈ vf and φ ∈ Mor vf (( M , U ) , ( N , V )), Λ UM restricts to a C -group Γ U M on B ( M ) of ∗− automorphismssuch that T and Γ are equivariant, namely(7) Γ U M ∈ C (cid:16) R , L s ( B ( M )) (cid:17) , Γ U M is a one-parameter group of ∗− automorphisms;and for every t ∈ R (8) T ( φ ) ◦ Γ U M ( t ) = Γ V N ( t ) ◦ T ( φ ) . Let us outline the essential steps yielding to (7). Firstly in Cor. 1.19 we show thatwhenever ( M , U ) ∈ vf , the group exp UM extends to a unique C -group exp U M on H g ofunitary operators whose infinitesimal generator extends £ U .It is worthwhile remarking that the unitary extension is essentially due to (6) (proofof Lemma 1.18). Thus Cor. 1.19 ensures that Λ UM restricts to a group Γ U M on B ( M ) of ∗− automorphisms (Cor. 1.20).Now in the fundamental Lemma 1.22 we prove that the topology τ M is generated bya collection of seminorms extending to L b ( D ( M ))-continuous seminorms. While Γ U M ( t ) isa continuous linear map on B ( M ) since exp UM ( t ) maps bounded sets into bounded sets.Therefore (5) implies that Γ U M is a C -group on B ( M ). We remark that in showing Lemma1.22 the fact that D ( M ) is barrelled is essential.In addition to the above results, by an application of the Banach-Steinhaus Thm.and of the fact that D ( M ) is specifically a Montel space, in Thm. 1.24(1) we prove thatexp UM : R → U ( M ) is a continuous morphism of groups, where U ( M ) is the group ofunitary elements of B ( M ) endowed with the relative topology.In conclusion of section 1 we determine in Prp. 1.26 the category Chdv and thefunctor ΨΨΨ , while 0-species are introduced in Def. 1.27. Thm. 2.2 and
Thm. 2.4 are the main results of section 2, where by using Thm. 1.15and Thm. 1.24, we construct two functors x and z from the category vf to the category ofdynamical patterns dp , classical x and quantum z .Let us delineate what above said for the more interesting quantum functor z , but firstof all we outline the main structures involved.For every ( M , U ) ∈ vf let h M , U i be the top-quasi enriched category of subsets of M such that for all X , Y ∈ h M , U i we have Mor h M , U i ( X , Y ) = { ( X , Y ) } × mor h M , U i ( X , Y )endowed with the topology inherited by R wheremor h M , U i ( X , Y ) = { t ∈ R | exp UM ( t ) D ( M , X ) = D ( M , Y ) } ; and D ( M , X ) is the topological sub ∗ -algebra of D ( M ) of those maps whose support iscontained in X . Next let B ( M , X ) be the topological unital sub ∗ -algebra of B ( M ) of those T such that T D ( M , X ) ⊆ D ( M , X ) and T † D ( M , X ) ⊆ D ( M , X ). Thus we can define themaps ( F h M , U i ) o and ( F h M , U i ) m on the object and morphism set of h M , U i respectively as ( F h M , U i ) o : X B ( M , X ) , ( F h M , U i ) m : (( X , Y ) , t ) (cid:16) B ( M , X ) → B ( M , Y ) T Γ U M ( t ) T (cid:17) . While for every ( M , U ) , ( N , V ) ∈ vf and φ ∈ Mor vf (( M , U ) , ( N , V )) we can set the maps f φ o and f φ m over the object and the morphism set of h N , V i respectively such that f φ o : Y φ ( Y ); f φ m : (( Y , Z ) , s ) (( φ ( Y ) , φ ( Z )) , s ) . and define the map T over the morphism set of vf such that T φ : Y (cid:16) B ( M , φ ( Y )) → B ( N , Y ) T T ( φ ) T (cid:17) . Thus we are able to define z on the category vf such that ( z o : ( M , U )
7→ h h M , U i , F h M , U i i ; z m : φ ( f φ , T φ ) . Now we have to see that e ff ectively(9) z ∈ Fct( vf , dp ) . What happens is that (7) is the core of the proof that the object map z o is well-set namely F h M , U i ∈ Fct top ( h M , U i , tsa);that Lemma 1.4 implies that the first component of the morphism map z m is well-set,namely f φ ∈ Fct top ( h N , V i , h M , U i );that (4) and the equivariance (8) are the core of the proof that also the second componentof z m is well-set, namely(10) T φ ∈ Mor
Fct( h N , V i , tsa) ( F h M , U i ◦ f φ , F h N , V i );finally that T ( φ ◦ ψ ) = T ( ψ ) ◦ T ( φ ) implies that z m preserves the morphism composition,and (9) follows.About the classical functor x the main novelties and advantages with respect to thefunctor a constructed in [ , Thm. 1.6.24] are represented by two facts: Firstly in orderto construct a group associated with a vector field U , here U needs not to be complete,rather we require { £ kU | k ∈ Z + } to be ( τ ∞ c , τ ∞ c )-equicontinuous, by obtaining in this way theadditional C -property of exp UM . Secondly here we select a specific topology on D ( M , X ),then by force on its unitization D ( M , X ), by de facto avoiding the problem of introducingwhat in [ , Def. 1.6.18] we called a vf-topology. This because for what just above said NTRODUCTION 11 and since φ ∗ is ( τ ∞ c , τ ∞ c )- continuous, the τ ∞ c -topology satisfies the requirements of a vf-topology provided η UM , the adjoint on D ( M ) of the flow on M generated by a completevector field U of M , be replaced by the group exp UM . Said that the construction of x mimics the one of a , by replacing η UM with exp UM . At the end of this Introduction we shallsee that in special cases exp UM equals η UM . Thm. 3.10 establishes the main result of section 3 and of the entire work, namely theexistence of the natural transformation(11) J ∈ Mor
Fct( Vf , Chdv ) ( x , z );between the classical 0-species x ≔ ΨΨΨ ◦ x ◦ I vfVf and the quantum 0-species z ≔ ΨΨΨ ◦ z ◦ I vfVf ,uniquely determined by J : Obj( Vf ) ∋ ( M , U ) J ( M , U ) = ( h M , U i , J † ( M , U ) , J ( M , U ) ); J ( M , U ) : Obj( h M , U i ) ∋ X Z X ( M , U ) ; J † ( M , U ) : Obj( h M , U i ) ∋ X ( Z X ( M , U ) ) † ;where Z X ( M , U ) : D ( M , X ) → B ( M , X )( f , λ ) £ [grad M ( f ) , U ] + λ ;and where Vf is the full subcategory of vf of those ( M , U ) for which there exists a frame { E i } of orthonormal fields of M such that(12) ( ∀ i ∈ [1 , dim M ] ∩ Z )([ U , E i ] = ) . Three are the fundamental steps to establish (11).First of all Cor. 3.6 by stating that (cid:16) f £ grad M ( f ) (cid:17) ∈ L ( D ( M , X ) , B ( M , X ));ensures that J ( M , U ) ( X ) is a continuous linear map.Then what right now stated and Thm. 3.1 by establishing that(13) exp UM ( t ) ◦ £ [grad M ( f ) , U ] = £ [(grad M ◦ exp UM ( t ))( f ) , U ] ◦ exp UM ( t );ensure that J ( M , U ) ∈ Mor
Fct( h M , U i , tls) ( q ◦ F h M , U i , q ◦ F h M , U i );which together its adjoint imply(14) J ( M , U ) ∈ Mor
Chdv ( x ( M , U ) , z ( M , U )) . It is in order to determine (13) that we require the use of the category Vf rather than vf .Specifically hypothesis (12) ensures that the following term n X i = ε i · £ E i ( f )£ [ E i , U ] in the right side of (24) vanishes. Finally Lemma 3.3 states that φ ∗ ◦ £ grad M ( f ) = £ (grad N ◦ φ ∗ )( f ) ◦ φ ∗ . which together (14) represent the core of the proof of the commutativity of the followingdiagram in the category Chdv x ( N , V ) J ( N , V ) / / z ( N , V ) x ( M , U ) J ( M , U ) / / x ( φ ) O O z ( M , U ) z ( φ ) O O and (11) follows. Cor. 4.6 is the main result of the closing section 4, where under the hypothesis that U is complete and an additional equicontinuity condition on £ U , we answer in Cor. 4.6(2)the natural question in the a ffi rmative on whether exp UM equals the adjoint action on D ( M ) of the flow on M generated by U . In the same section we also prove in Lemma4.2 that under the obvious additional equicontinuity request over £ > U the Lie derivativeof U on C ∞ ( M ), the exponential one-parameter group Exp UM generated by £ > U extendsexp UM . As a result in Prp. 4.4(2) we obtain that Λ UM ( t ) restricts to a morphism A → A t of left C ∞ ( M )-modules where A ⊂ L ( D ( M )) is naturally a left C ∞ ( M )-module such that Λ UM ( t ) A ⊆ A while A t is the left C ∞ ( M )-module whose underlying group is A and externallaw is given by F · Q Exp UM ( t )( F ) · Q .
1. Construction of the topological ∗ -algebra B ( M ) and the C -group Γ U M D efinition Define vf ⋆ to be the category such that its object set is the set of the couples ( M , U ) where M is a manifold and U is a vector field on M such that { £ nU | n ∈ Z ∗ + } is ( τ ∞ c , τ ∞ c ) -equicontinuous. For every ( M , U ) , ( N , V ) ∈ vf ⋆ , Mor vf ⋆ (( M , U ) , ( N , V )) is the set of propersmooth maps φ : N → M so that U and V are φ -related, while for every ( Q , K ) ∈ vf ⋆ and ψ ∈ Mor vf ⋆ (( N , V ) , ( Q , K )) we set ψ ◦ vf ⋆ φ ≔ φ ◦ ψ . Since D ( M ) is sequentially complete we can set the followingD efinition Let ( M , U ) ∈ vf ⋆ define exp UM ≔ exp £ U D ( M ) , set exp UM : R + ∋ t exp( − t ) exp UM ( t ) and exp UM : R + ∋ t exp( − t ) exp UM ( − t ) . Moreoverdefine Λ UM : R → End vct ( L ( D ( M ))) such that Λ UM : t ( T exp UM ( t ) ◦ T ◦ exp UM ( − t )) . . CONSTRUCTION OF THE TOPOLOGICAL ∗ -ALGEBRA B ( M ) AND THE C -GROUP Γ U M R emark M , U ) ∈ vf ⋆ . Thus for every constant map c on M and f ∈ D ( M )we have exp UM ( t )( c · f ) = c · exp UM ( t )( f ), since £ U ( c · f ) = c · £ U ( f ) being £ > U ( c ) = , by D ( M ) ֒ → C ∞ ( M ) and since C ∞ ( M ) is a topological algebra. Moreover exp UM is a groupof ∗ -automorphisms of D ( M ) indeed its infinitesimal τ ∞ c -generator £ U is a ∗ -preservingderivation on D ( M ) then the statement follows.L emma Let ( M , U ) , ( N , V ) ∈ vf ⋆ and φ ∈ Mor vf ⋆ (( M , U ) , ( N , V )) thus φ ∗ ◦ exp UM ( t ) = exp VN ( t ) ◦ φ ∗ and £ U ◦ exp UM ( t ) = exp UM ( t ) ◦ £ U for every t ∈ R . P roof . Since U and V are φ -related we have that φ ∗ ◦ £ U = £ V ◦ φ ∗ , thus the firstequality follows since φ ∗ is τ ∞ c -continuous, the second equality follows since £ U is τ ∞ c -continuous. (cid:3) L emma Let X be a Montel space, Y a topological space, U , V : Y → L s ( X ) continuousat t ∈ Y and such that { U ( t ) | t ∈ Y } is equicontinuous. If the neighbourhood filter of t in Yadmits a countable basis, then Z U , V : Y → L s ( L b ( X )) is continuous at t , where Z U , V : t ( T U ( t ) ◦ T ◦ V ( t )) . P roof . In this proof we let Z denote Z U , V which is well-defined namely Z ( t ) ∈ L ( L b ( X ))for every t ∈ Y . Indeed let p be a continuous seminorm on X , B a bounded subset of X and T ∈ L ( X ), thus p B ( Z ( t ) T ) = ( q t ) C t ( T ) with q t = p ◦ U ( t ) and C t = V ( t ) B . But q t is acontinuous seminorm of X , while C t is bounded in X since V ( t ) is linear and continuous,thus q tC t is a continuous seminorm of L b ( X ) and then Z ( t ) ∈ L ( L b ( X )). Next assume thatthe neighbourhood filter of t in Y admits a countable basis. Now X is a Montel spacethus it is su ffi cient to prove that for every sequence { t n } in Y converging at t and every T ∈ L ( X ), we have that Z t n T converges at T in L pc ( X ). Now since the equicontinuityhypothesis, there exists a continuous seminorm q of X such that for all x ∈ X and n ∈ Z ∗ + we have p ( Z t n ( T ) x − Z t ( T ) x ) ≤ p ( U t n ( TV t n x − TV t x )) + p (( U t n − U t ) TV t x ) ≤ q ( T ( V t n x − V t x )) + p (( U t n − U t ) TV t x );so p ( Z t n ( T ) x − Z t ( T ) x ) converges at 0, but p is an arbitrary continuous seminorm on X ,thus Z t n T converges at Z t T in L s ( X ). Finally X is barrelled being Montel, thus by theBanach-Steinhaus Thm. we deduce that Z t n T converges at Z t T in L pc ( X ) which is whatwe claimed to prove. (cid:3) C orollary Let ( M , U ) ∈ vf ⋆ , thus Λ UM is a C -group on L b ( D ( M )) . P roof . By letting U + = exp UM , V + : R + ∋ t exp( t ) exp UM ( − t ), U − = exp UM and V − : R + ∋ t exp( t ) exp UM ( t ) the statement follows by Lemma 1.5 and since Λ UM ↾ R + = Z U + , V + and Λ UM ↾ R − = Z U − , V − ◦ i where i : R − ∋ λ → − λ ∈ R + . (cid:3) Since D ( M ) is dense in H g we can set the followingD efinition B ( M )). Let M = ( M , g ) be a semi-Riemannian manifold, define B ( M ) ≔ { T ∈ L ( D ( M )) | D ( M ) ⊆ Dom( T ⊺ ) , T ⊺ D ( M ) ⊆ D ( M ) , T † ≔ T ⊺ ↾ D ( M ) ∈ L ( D ( M )) } ; where T ⊺ is the H g − adjoint of the operator T. R emark M = ( M , g ) be a semi-Riemannian manifold, thus since the discussionin Notation we deduce that Di ff Op k ( M ) ⊂ B ( M ) for every k ∈ Z ∗ + .P roposition B ( M ) is a O ∗ -algebra on D ( M )). Let M = ( M , g ) be a semi-Riemannianmanifold, thus B ( M ) is a O ∗ -algebra on D ( M ) in particular it is a unital ∗ -algebra with involution ( · ) † . P roof . Let S , T ∈ B ( M ), thus T is closable since T ⊺ is densely defined. Since ( S + T ) ⊺ ⊇ S ⊺ + T ⊺ we obtain D ( M ) ⊆ Dom(( S + T ) ⊺ ) and ( S + T ) † = S † + T † ∈ L ( D ( M )). Nextsince ( ST ) ⊺ ⊇ T ⊺ S ⊺ we obtain D ( M ) ⊆ Dom(( ST ) ⊺ ) and ( ST ) † = T † S † ∈ L ( D ( M )). Finally T = ( T ⊺ ) ⊺ ⊆ ( T † ) ⊺ since T † ⊆ T ⊺ , so D ( M ) ⊆ Dom(( T † ) ⊺ ) and ( T † ) † = T ∈ L ( D ( M )). (cid:3) D efinition vf and vf ). Define vf to be the unique category whoseobject set consists of the couples ( M , U ) where M = ( M , g ) is a semi-Riemannian manifold, ( M , U ) ∈ vf ⋆ and µ g ◦ £ U = . The morphism set of vf is such that Mor vf (( M , U ) , ( N , V )) consists of those φ ∈ Mor vf ⋆ (( M , U ) , ( N , V )) such that φ ∗ g = g ′ where N = ( N , g ′ ) , and whose composition is givenby φ ◦ vf ψ = ψ ◦ φ with ◦ the map composition. Let vf be the subcategory of vf with the sameobject set and Mor vf (( M , U ) , ( N , V )) consisting of those φ ∈ Mor vf (( M , U ) , ( N , V )) such that φ is a di ff eomorphism. D efinition Let M = ( M , g ) be a semi-Riemannian manifold, define ω f : B ( M ) → C , T
7→ h f , T f i H g , f ∈ D ( M ); ω Qf : B ( M ) → C , T ω f ( Q † TQ ) , f ∈ D ( M ) , Q ∈ B ( M ); F M ≔ {{ ω f | f ∈ B } | B ∈ Bounded( D ( M )) } . P roposition Let M = ( M , g ) be a semi-Riemannian manifold, B ∈ Bounded( D (M)) ,T , Q ∈ B ( M ) . Thus {| ω f ( T ) | | f ∈ B } is bounded and { ω Qf | f ∈ B } ∈ F M . P roof . T ( B ) is τ ∞ c -bounded since T is ( τ ∞ c , τ ∞ c )-continuous, thus B and T ( B ) are k · k H g -bounded since D ( M ) ֒ → H g , so the first part of the statement follows sincesup f ∈ B |h f , T f i H g | ≤ sup f ∈ B k f k H g sup f ∈ B k T f k H g . The second part follows since ω Qf = ω Q f and Q ( B ) is τ ∞ c -bounded. (cid:3) The first part of Prp. 1.12 allows us to give the followingD efinition τ M on B ( M )). Let M = ( M , g ) be a semi-Riemannianmanifold, define τ M to be the locally convex topology on B ( M ) generated by the set of seminorms { q B | B ∈ Bounded( D ( M )) } where q B : B ( M ) → R + T sup f ∈ B | ω f ( T ) | . R emark M = ( M , g ) be a semi-Riemannian manifold, thus by the polar-ization formula and since for any locally convex space X , λ, µ ∈ C and B , C boundedsubsets of X , the set λ B + µ C is bounded in X we deduce that the topology τ M is . CONSTRUCTION OF THE TOPOLOGICAL ∗ -ALGEBRA B ( M ) AND THE C -GROUP Γ U M generated by the following set of seminorms { q B , C | B , C ∈ Bounded( D ( M )) } , where q B , C ( T ) ≔ sup ( f , g ) ∈ B × C |h f , Tg i H g | .T heorem ∗ -algebra B ( M ) and the morphism T ( φ )). Let M = ( M , g ) be a semi-Riemannian manifold, thus (1) B ( M )[ τ M ] ∈ tsa(2) if N = ( N , g ′ ) is a semi-Riemannian manifold and φ : N → M is a smooth di ff eomor-phism such that φ ∗ g = g ′ , then by letting ( T ( φ ) : B ( M ) → B ( N ) , T φ ∗ ◦ T ◦ ( φ − ) ∗ ; we have that T ( φ ) ∈ Mor tsa ( B ( M )[ τ M ] , B ( N )[ τ N ]) . P roof . St. (1) follows by Prp. 1.12 and [ , Lemma 1.5.7] applied to the unital ∗ -algebra B ( M ) and the set F M . Next let T ∈ B ( M ) thus we have what follows. T ( φ )( T ) ∈ L ( D ( N ))since φ ∗ , T and ( φ − ) ∗ are all linear and τ ∞ c -contiuous operators. Next by recalling theproperty ( φ ∗ ) ⊺ = ( φ − ) ∗ discussed in Notation, we have T ( φ )( T ) ⊺ = (( φ − ) ∗ ) ⊺ ◦ T ⊺ ◦ ( φ ∗ ) ⊺ = φ ∗ ◦ T ⊺ ◦ ( φ − ) ∗ . Thus D ( N ) ⊆ Dom( T ( φ )( T ) ⊺ ), T ( φ )( T ) ⊺ D ( N ) ⊆ D ( N ) and T ( φ )( T ) † = T ( φ )( T † ) ∈ L ( D ( N )).Therefore T ( φ ) is well-set namely T ( φ )( T ) ∈ B ( N ), moreover T ( φ ) is a ∗ -morphism.Finally let us prove the continuity of T ( φ ). For every f ∈ D ( N ) we have that h f , T ( φ )( T ) f i H g ′ = h ( φ ∗ ) ⊺ f , T (( φ − ) ∗ f ) i H g = h ( φ − ) ∗ f , T (( φ − ) ∗ f ) i H g ;but ( φ − ) ∗ is ( τ ∞ c ( N ) , τ ∞ c ( M ))-continuous, therefore ( φ − ) ∗ ( B ) is τ ∞ c ( M )-bounded for every τ ∞ c ( N )-bounded set B hence T ( φ ) is ( τ M , τ N )-continuous and st. (2) follows. (cid:3) The above result justifies the followingD efinition
Let M = ( M , g ) be a semi-Riemannian manifold, define B ( M ) to be theunital topological ∗ -algebra B ( M )[ τ M ] . P roposition Let M = ( M , g ) be a semi-Riemannian manifold, thus B ( M ) ⊂ L ( D ( M ) B ( M ) , D ( M ) + B ( M ) ) and B ( M )[ τ b ] is a unital topological ∗ -algebra such that B ( M )[ τ b ] ֒ → B ( M ) . P roof . Let T ∈ B ( M ), thus ( f , h )
7→ h
T f , h i H g is clearly jointly continuous w.r.t. thegraph topology of B ( M ) on D ( M ) so the inclusion in the statement follows, in particular B ( M )[ τ b ] is well-set and it is a topological ∗ -algebra since [ , Prp. 3.3.10]. Next it iswell-known that the graph topology of a O ∗ -algebra A on a dense subspace D of a Hilbert space H is the weakest among all the locally convex topologies τ on D such that A ⊂ L ( D [ τ ] , H ). But B ( M ) ⊂ L ( D ( M ) , H g ), therefore D ( M ) ֒ → D ( M ) B ( M ) . In particularBounded( D ( M )) ⊆ Bounded( D ( M ) B ( M ) ) which implies B ( M )[ τ b ] ֒ → B ( M ). (cid:3) L emma Let ( M , U ) ∈ vf with M = ( M , g ) and t ∈ R , then exp UM ( t ) extends uniquelyto a unitary operator exp U M ( t ) on H g such that exp U M ( t ) ⊺ = exp U M ( − t ) . P roof . D ( M ) ֒ → K ( M ) and µ g ◦ £ U = imply that µ g ◦ exp UM ( t ) = µ g for all t ∈ R ,therefore for every f , h ∈ D ( M ) we have(15) h exp UM ( t ) f , exp UM ( t ) h i H g = µ g (exp UM ( t ) f exp UM ( t ) h ) = ( µ g ◦ exp UM ( t ))( f h ) = µ g ( f h ) = h f , h i H g ;where the second equality follows since Rmk. 1.3. Thus exp UM ( t ) is a unitary operatoron the dense subspace D ( M ) of H g , which then extends uniquely to a unitary operator exp U M ( t ) on H g . Next since exp U M ( t ) is unitarity and since exp UM ( t ) − = exp UM ( − t ), we havethat exp U M ( t ) ⊺ ↾ D ( M ) = exp U M ( − t ) ↾ D ( M ) and the equality in the statement follows. (cid:3) C orollary Let ( M , U ) ∈ vf with M = ( M , g ) , then there exists a unique C -group exp U M on H g of unitary operators extending exp UM and whose infinitesimal generator l U extends £ U . P roof . Since Lemma 1.18 it remains only to prove the C -property and l U ⊇ £ U .To this end let f ∈ D ( M ), thus t exp U M ( t ) f is k · k H g -continuous since t exp UM ( t ) f is τ ∞ c -continuous by construction and since D ( M ) ֒ → H g . Next exp U M ( R ) is ( k · k H g , k ·k H g )-equicontinuous since isometric, while D ( M ) is dense in H g . Therefore since overequicontinuous sets the uniform structure of simple convergence coincides with theuniform structure of simple convergence over a total set, we conclude that for every h ∈ H g the map t exp U M ( t ) h is k · k H g -continuous namely exp U M is a C -group on H g .Finally l U extends the infinitesimal τ ∞ c -generator £ U of exp UM since exp U M extends exp UM and since D ( M ) ֒ → H g . (cid:3) C orollary Let ( M , U ) ∈ vf with M = ( M , g ) and t ∈ R , then exp UM ( t ) ∈ B ( M ) suchthat exp UM ( t ) † = exp UM ( − t ) , in particular Λ UM ( t ) B ( M ) ⊆ B ( M ) . P roof . Since Cor. 1.19. (cid:3) Cor. 1.20 allows to give the followingD efinition Γ U M ). Let ( M , U ) ∈ vf , define Γ U M : R → End vct ( B ( M )) suchthat Γ U M : t ( T Λ UM ( t )( T )) , where M is the manifold underlying M . . CONSTRUCTION OF THE TOPOLOGICAL ∗ -ALGEBRA B ( M ) AND THE C -GROUP Γ U M L emma Let M be a semi-Riemannian manifold, thus the topology τ M is generated by acollection of seminorms extending to L b ( D ( M )) -continuous seminorms. P roof . Let B and C be bounded subsets of D ( M ) and let ζ B : D ( M ) → R + h sup f ∈ B |h f , h i H g | finite since D ( M ) ֒ → H g . Now since D ( M ) ֒ → K ( M ) and h · µ g ∈ K ( M ) ′ for every h ∈ K ( M ) we have h f , · i H g ◦ ı H g D ( M ) = ( f · µ g ) ◦ ı K ( M ) D ( M ) ∈ D ( M ) ′ . Therefore ζ B is lower τ ∞ c -semicontinuous since superior envelop of τ ∞ c -continuous maps, hence ζ B is τ ∞ c -continuous since D ( M ) is barrelled. Thus ( ζ B ) C is a continuous seminorm of L b ( D ( M ))so the statement follows by the fact that ( ζ B ) C = q B , C and by Rmk. 1.14. (cid:3) D efinition Let M be a semi-Riemannian manifold, define U ( M ) ≔ { T ∈ B ( M ) | T † = T − } endowed with the relative topology of B ( M ) and with the group structure inherited by themultiplication on B ( M ) . Notice that in general U ( M ) needs not to be a topological group.T heorem Γ U M is a C -group on B ( M ) of ∗− automorphisms). Let ( M , U ) , ( N , V ) ∈ vf and φ ∈ Mor vf (( M , U ) , ( N , V )) , thus by letting M be the manifold underlying M , we have (1) exp UM ∈ C ( R , U ( M )) morphism of groups; (2) Γ U M is a C -group on B ( M ) of ∗− automorphisms; (3) T ( φ ) ◦ Γ U M ( t ) = Γ V N ( t ) ◦ T ( φ ) , for every t ∈ R . P roof . exp UM is a morphism of the groups involved in the statement since Cor. 1.20.Let us prove the continuity. Since exp UM is a C -group on D ( M ) by construction and since D ( M ) is barrelled it follows by the Banach-Steinhaus Thm. that exp UM ∈ C ( R + , L pc ( D ( M ))),then exp UM ∈ C ( R + , L b ( D ( M ))) since D ( M ) is a Montel space, therefore st. (1) follows byLemma 1.22. Next let B be a τ ∞ c -bounded set and t ∈ R , thus since exp UM ( t ) † = exp UM ( − t )by Cor. 1.20, we have q B ◦ Γ U M ( t ) = q exp UM ( − t ) B , but exp UM ( − t ) B is τ ∞ c -bounded since exp UM ( − t )is τ ∞ c -continuous, therefore Γ U M ( t ) ∈ L ( B ( M )). Thus Γ U M is a C -group on B ( M ) since Cor.1.6 and Lemma 1.22. Finally Γ U M ( t ) is a ∗ -automorphism of B ( M ) since Cor. 1.20, so st.(2) is proven. St. 3 follows since Lemma 1.4. (cid:3) Next we set the following definition of † here used instead of the analog one givenin [ , Def. 1.4.14]. Clearly the corresponding of [ , Cor. 1.4.16] holds.D efinition Let D be a category, a , b ∈ Fct( D , tls) and T ∈ Q d ∈ D Mor tls (a( d ) , b( d )) ,then define T † ∈ Q d ∈ D Mor tls (b( d ) ′ , a( d ) ′ ) such that T † ( e ) ≔ ( T ( e )) † for all e ∈ D, whereS † ( ω ) ≔ ω ◦ S. We conclude this section with the existence of the category
Chdv obtained by for-getting in the category Chdv uniquely determined in [ , Cor. 1.4.18] the category ptlsinto the category tls and the category ptsa into the category tls. Then we obtain thecorresponding functor ΨΨΨ in analogy with the functor ΨΨΨ in [ , Thm. 1.4.19] Before thenext result let us recall that for any A ∈ dp , A † is defined in [ , Def. 1.4.13] and that since[ , Thm. 1.4.15] r ◦ σ † A is well-set. P roposition Chdv ). There exists a unique category
Chdv whose objectset equals the object set of dp and whose morphism set is such that for every A , B , C ∈ Chdv wehave Mor
Chdv ( A , B ) = a f ∈ Fct top (G B , G A ) Mor
Fct(G op B , tls) ( r ◦ σ † B , r ◦ σ † A ◦ f ) × Mor
Fct(G B , tls) ( q ◦ σ A ◦ f , q ◦ σ B )(16) and (17) ( ◦ ) : Mor Chdv ( B , C ) × Mor
Chdv ( A , B ) → Mor
Chdv ( A , C ) , ( g , L , S ) ◦ ( f , H , T ) ≔ ( f ◦ g , ( H ∗ g ) ◦ L , S ◦ ( T ∗ g )) . Moreover the maps A A and ( f , T ) ( f , ( q ∗ T ) † , q ∗ T ) determine uniquely an element ΨΨΨ ∈ Fct( dp , Chdv ) . Next in analogy with [ , Def. 1.5.8] we give the followingD efinition − species). Define Sp the fibered categoryover − dp such that for all D ∈ − dpSp ( D ) = − dp ( D , Chdv ) , moreover set Sp ∗ ≔ { (a , b) ∈ Sp × Sp | d (a) = d (b) } .
2. Construction of the classical and quantum -species x and z Since exp UM is a C -group on D ( M ) we immediatedly obtain the followingP roposition Let ( M , U ) ∈ vf . Thus there exists a unique h h M , U i , F h M , U i i ∈ dp with the following properties. h M , U i is the unique top -quasi enriched category with thefollowing properties. The object set of h M , U i is the set of all subsets of M, the morphismset of h M , U i is such that for every X , Y ∈ h M , U i we have Mor h M , U i ( X , Y ) = { ( X , Y ) } × mor h M , U i ( X , Y ) , with mor h M , U i ( X , Y ) = { t ∈ R | exp UM ( t ) D ( M , X ) = D ( M , Y ) } , where we let M be the manifold underlying M and D ( M , X ) be the topological sub ∗ -algebra of D ( M ) of those maps whose support is contained in X. The topology on Mor h M , U i ( X , Y ) is thatinduced by the topology on R , while the composition is that inherited by the addition in R . While F h M , U i ∈ Fct top ( h M , U i , tsa) such that for every t ∈ mor h M , U i ( X , Y ) we have F h M , U i ( X ) = D ( M , X ) , F h M , U i (( X , Y ) , t ) : D ( M , X ) → D ( M , Y ) , ( f , λ ) (exp UM ( t ) f , λ ); with D ( M , X ) ∈ tsa the unitization of D ( M , X ) . . CONSTRUCTION OF THE CLASSICAL AND QUANTUM 0-SPECIES x AND z T heorem There exists a unique x ∈ Fct( vf , dp ) such that for all ( M , U ) , ( N , V ) ∈ vf and φ ∈ Mor vf (( M , U ) , ( N , V ))(1) x (( M , U )) = h h M , U i , F h M , U i i , (2) x ( φ ) = ( f φ , T φ ) ;where f φ ∈ Fct top ( h N , V i , h M , U i ) and (18) T φ ∈ Mor
Fct( h N , V i , tsa) (F h M , U i ◦ f φ , F h N , V i ); such that for all Y , Z ∈ h N , V i and t ∈ mor h N , V i ( Y , Z )(1) f φ ( Y ) = φ ( Y ) ; (2) f φ (( Y , Z ) , t ) = (( φ ( Y ) , φ ( Z )) , t ) ; (3) T φ ( Y ) : D ( M , φ ( Y )) → D ( N , Y ) h φ ∗ h; (4) T φ ( Y ) : D ( M , φ ( Y )) → D ( N , Y ) ( h , λ ) ( φ ∗ h , λ ) ;where M and N are the manifolds underlying M and N respectively. In particular ΨΨΨ ◦ x ∈ Sp ( vf ) and ΨΨΨ ◦ x ∈ Sp ( vf ) . P roof . Let us take the properties of the statement as definition of x . Let t ∈ mor h N , V i ( Y , Z ) and f ∈ D ( M , φ ( Y )), so since Lemma 1.4 we have φ ∗ (exp UM ( t ) f ) = exp VN ( t )( φ ∗ f ) ∈ exp VN ( t )( D ( N , Y )) ⊆ D ( N , Z );namely exp UM ( t ) f ∈ ( φ − ) ∗ D ( N , Z ) = ( φ − ) ∗ D ( N , φ − ( φ ( Z ))) ⊆ D ( M , φ ( Z )) . Therefore f φ (( Y , Z ) , t ) ∈ Mor h M , U i ( φ ( Y ) , φ ( Z )), next f φ is clearly continuous and compo-sition preserving so f φ ∈ Fct top ( h N , V i , h M , U i ). Next T φ ( Y ) is continuous since φ ∗ is( τ ∞ c ( M ) , τ ∞ c ( N ))-continuous, moreover for every f ∈ D ( M , φ ( Y )) and λ ∈ C , since Lemma1.4 we have( T φ ( Z ) ◦ F h M , U i ( φ ( Y ) , φ ( Z ) , t ))( f , λ ) = (( φ ∗ ◦ exp UM ( t )) f , λ ) = ((exp VN ( t ) ◦ φ ∗ ) f , λ ) = (F h N , V i ( Y , Z , t ) ◦ T φ ( Y ))( f , λ );which proves (18). Finally x ( ψ ◦ vf φ ) = x ( ψ ) ◦ dp x ( φ ) follows by the same line of reasoningwe use in [ , Thm. 1.6.24] to prove that a ( ψ ◦ vf φ ) = a ( ψ ) ◦ dp a ( φ ). (cid:3) T heorem Let ( M , U ) ∈ vf , thus there exists a unique h h M , U i , F h M , U i i ∈ dp withthe following properties. F h M , U i ∈ Fct top ( h M , U i , tsa) such that for every subset X and Y of Mand every t ∈ mor h M , U i ( X , Y ) we have ( F h M , U i ( X ) = B ( M , X ) , F h M , U i (( X , Y ) , t ) : B ( M , X ) → B ( M , Y ) T Γ U M ( t ) T ; where we let M be the manifold underlying M and B ( M , X ) be the topological unital sub ∗ -algebraof B ( M ) of those T such that T D ( M , X ) ⊆ D ( M , X ) and T † D ( M , X ) ⊆ D ( M , X ) . P roof . Since Thm. 1.24(2). (cid:3) T heorem There exists a unique z ∈ Fct( vf , dp ) such that for all ( M , U ) , ( N , V ) ∈ vf and φ ∈ Mor vf (( M , U ) , ( N , V ))(1) z (( M , U )) = h h M , U i , F h M , U i i , (2) z ( φ ) = ( f φ , T φ ) ;where (19) T φ ∈ Mor
Fct( h N , V i , tsa) ( F h M , U i ◦ f φ , F h N , V i ); such that for all Y , Z ∈ h N , V i we have T φ ( Y ) : B ( M , φ ( Y )) → B ( N , Y ) T T ( φ ) T . In particular
ΨΨΨ ◦ z ∈ Sp ( vf ) and ΨΨΨ ◦ z ∈ Sp ( vf ) . P roof . Let us take the properties of the statement as definition of z . Let Q ∈ B ( M , φ ( Y )) thus since Thm. 2.2(3) we obtain T φ ( Y )( Q ) ↾ D ( N , Y ) = T φ ( Y ) ◦ Q ◦ T φ − ( φ ( Y )) , moreover T ( φ ) B ( M ) ⊆ B ( N ) since Thm. 1.15(2), thus we obtain that T φ ( Y ) B ( M , φ ( Y )) ⊆ B ( N , Y ) and then T φ is well-set. Next T φ ( Y ) is continuous since it is so T ( φ ) according toThm. 1.15(2). Next for every t ∈ mor h N , V i ( Y , Z ) we have by Thm. 1.24(3)( T φ ( Z ) ◦ F h M , U i ( φ ( Y ) , φ ( Z ) , t )) Q = ( T ( φ ) ◦ Γ U M ( t )) Q = ( Γ V N ( t ) ◦ T ( φ )) Q = ( F h N , V i ( Y , Z , t ) ◦ T φ ( Y )) Q ;which proves (19). Finally for every ψ ∈ Mor vf which is vf − composable to the left with φ we have z ( ψ ◦ vf φ ) = z ( ψ ) ◦ dp z ( φ ) since T ( φ ◦ ψ ) = T ( ψ ) ◦ T ( φ ). (cid:3)
3. Construction of the natural transformation J from x to z T heorem Let M be a semi-Riemannian manifold with underlying manifold M, thus forevery f , h ∈ D ( M ) we have (20) £ grad M ( f ) ( h ) = £ grad M ( h ) ( f ) . Next let U be such that ( M , U ) ∈ vf ⋆ . If there exists a frame { E i } n = dim Mi = of orthonormal fields of M such that [ U , E i ] = for every i ∈ [1 , n ] ∩ Z , then for every f ∈ D ( M ) and t ∈ R we have (21) exp UM ( t ) ◦ £ [grad M ( f ) , U ] = £ [(grad M ◦ exp UM ( t ))( f ) , U ] ◦ exp UM ( t ) . . CONSTRUCTION OF THE NATURAL TRANSFORMATION J FROM x TO z P roof . By the same symbol [ , ] we shall denote the Lie braket of vector fields on M and the Lie braket induced by the associative product on L ( D ( M )), namely [ T , Q ] = T ◦ Q − Q ◦ T for T , Q ∈ L ( D ( M )). Next let { E i } n = dim Mi = be a frame of orthonormal fieldsof M , let ε i = h E i , E i i M for every i ∈ [1 , n ] ∩ Z . Thus for every vector field W we have£ W = P ni = ε i h W , E i i M · £ E i . Next let f , h ∈ D ( M ) thus(22) £ grad M ( f ) ( h ) = n X i = ε i h grad M ( f ) , E i i M · £ E i ( h ) = n X i = ε i · £ E i ( f )£ E i ( h ) = n X i = ε i h grad M ( h ) , E i i M · £ E i ( f ) = £ grad M ( h ) ( f );and (20) follows. Next let W ∈ X ( M ), so £ W is ( τ ∞ c , τ ∞ c )-continuous thusexp UM ( t ) ◦ £ W = ∞ X k = t k k ! £ kU £ W , £ W ◦ exp UM ( t ) = ∞ X k = t k k ! £ W £ kU ;both converging in L s ( D ( M )). Since £ : X ( M ) → Der( D ( M )) is a Lie algebra isomorphismonto the Lie algebra of derivations of D ( M ), we have [ W , U ] = ⇔ £ [ W , U ] = ⇔ [£ W , £ U ] = ⇔ ( ∀ n ∈ Z + )([£ W , £ nU ] = ) therefore(23) ( ∀ W ∈ X ( M ))([ W , U ] = ⇒ [£ W , exp UM ( t )] = ) . Next by the defining property of derivations, the second equality of (22) and Rmk. 1.3we deduce that(24) [£ grad M ( f ) , £ U ] = n X i = ε i · £ E i ( f )£ [ E i , U ] + ( − n X i = ε i · (£ U ◦ £ E i )( f )£ E i . Let A and B denote the left and right sides of the equality (21) respectively, thus A = exp UM ( t ) ◦ [£ grad M ( f ) , £ U ]; B = [£ (grad M ◦ exp UM ( t ))( f ) , £ U ] ◦ exp UM ( t ) . Now if [ U , E i ] = for every i ∈ [1 , n ] ∩ Z , then by (24) and Rmk. 1.3 we obtain A = ( − n X i = ε i · (exp UM ( t ) ◦ £ U ◦ £ E i )( f )(exp UM ( t ) ◦ £ E i ); B = ( − n X i = ε i · (£ U ◦ £ E i ◦ exp UM ( t ))( f )(£ E i ◦ exp UM ( t ));then st. (21) follows since (23). (cid:3) R emark U , W ] = ⇒ ( ∀ t ∈ R )( Λ UM ( t )(£ W ) = £ W ).L emma Let M = ( M , g ) , N = ( N , g ′ ) be semi-Riemannian manifolds and φ : N → Mbe a smooth di ff eomorphism such that φ ∗ g = g ′ . Thus for every f ∈ D ( M ) we have φ ∗ ◦ £ grad M ( f ) = £ (grad N ◦ φ ∗ )( f ) ◦ φ ∗ . P roof . Let { E i } n = dim Mi = be a frame of orthonormal fields of M , and set G i ≔ d ( φ − ) ◦ E i ◦ φ ,thus E i and G i are φ -related, hence φ ∗ ◦ £ E i = £ G i ◦ φ ∗ and { G i } ni = is a frame of orthonormalfields of N since [ , Lemma 1.6.6]. Therefore since again [ , Lemma 1.6.6] we have that φ ∗ ◦ £ grad M ( f ) = n X i = φ ∗ ( h E i , E i i M ) φ ∗ (£ E i ( f ))( φ ∗ ◦ £ E i ) = n X i = h G i , G i i N £ G i ( φ ∗ f )(£ G i ◦ φ ∗ ) = £ (grad N ◦ φ ∗ )( f ) ◦ φ ∗ . (cid:3) D efinition Z ). Let M be a semi-Riemannian manifold, M be the manifoldunderlying M , U ∈ X ( M ) and X ⊆ M. Define Z X ( M , U ) : D ( M , X ) → B ( M , X )( f , λ ) £ [grad M ( f ) , U ] + λ ; where is the unit element of the unital algebra B ( M ) . P roposition Def. 3.4 is well-set namely Z X ( M , U ) ( f ) ∈ B ( M , X ) for every f ∈ D ( M , X ) . P roof . Z X ( M , U ) ( f ) ∈ B ( M ) since Rmk. 1.8 and (1). Next by Notation we know that theelements in Di ff Op ( M ) are local, and that D ∈ Di ff Op ( M ) ⇒ D † ∈ Di ff Op ( M ), thus l ∈ D ( M , X ) implies supp( Z X ( M , U ) ( f )( l )) ⊆ supp( l ) ⊆ X and supp( Z X ( M , U ) ( f ) † ( l )) ⊆ X . (cid:3) C orollary Let M be a semi-Riemannian manifold, M be the manifold underlying M and X ⊆ M, thus (cid:16) f £ grad M ( f ) (cid:17) ∈ L ( D ( M , X ) , B ( M , X )) . . CONSTRUCTION OF THE NATURAL TRANSFORMATION J FROM x TO z P roof . Let ξ be the map in the statement, ˜ ξ : D ( M ) → B ( M ) be the map f £ grad M ( f ) ,and let ξ ≔ ı L ( D ( M )) B ( M ) ◦ ˜ ξ . The above maps are well-set namely ξ ( D ( M , X )) ⊆ B ( M , X ) sincethe proof of Prp. 3.5 while ˜ ξ ( D ( M )) ⊆ B ( M ) since Rmk. 1.8. We claim to show that(25) ˜ ξ ∈ L ( D ( M ) , B ( M ));which would prove our statement since ı B ( M ) B ( M , X ) ◦ ξ = ˜ ξ ◦ ı D ( M ) D ( M , X ) . Now since Lemma 1.22we have that (25) would follow if we prove that ξ ∈ L ( D ( M ) , L b ( D ( M ))). But D ( M ) isbarrelled therefore by applying the Banach-Steinhaus Thm. the above statement wouldfollow if ξ ∈ L ( D ( M ) , L s ( D ( M ))) which at once follows since (20). (cid:3) For any semi-Riemannian manifold M let Onf( M ) denote the set of frames of or-thonormal fields of M .D efinition Let Vf be the unique full subcategory of vf such that Obj( Vf ) = n ( M , U ) ∈ Obj( vf ) (cid:12)(cid:12)(cid:12) ( ∃{ E i } n = dim Mi = ∈ Onf( M ))( ∀ i ∈ [1 , n ] ∩ Z )([ U , E i ] = ) o . D efinition Define x ≔ ΨΨΨ ◦ x ◦ I vfVf and z ≔ ΨΨΨ ◦ z ◦ I vfVf . C orollary − species x and z ). x ∈ Sp ( Vf ) and z ∈ Sp ( Vf ) . P roof . Since Thm. 2.2 and Thm. 2.4. (cid:3) T heorem x to z ). There exists a naturaltransformation J ∈ Mor
Fct( Vf , Chdv ) ( x , z ) uniquely determined by the following properties: J : Obj( Vf ) ∋ ( M , U ) J ( M , U ) = ( h M , U i , J † ( M , U ) , J ( M , U ) ); J ( M , U ) : Obj( h M , U i ) ∋ X Z X ( M , U ) ; J † ( M , U ) : Obj( h M , U i ) ∋ X ( Z X ( M , U ) ) † . P roof . Let us take the properties of J given in the statement as its definition, thenshow that the definition is well-set and that J is a natural transformation from x to z . Let( M , U ) ∈ Obj( Vf ) and X , Y ∈ h M , U i . By £ [grad M ( f ) , U ] = [£ grad M ( f ) , £ U ] and since B ( M , X ) isa topological algebra we obtain by Cor. 3.6 that(26) (cid:16) f £ [grad M ( f ) , U ] (cid:17) ∈ L ( D ( M , X ) , B ( M , X )) . Since (26) and (1) we have(27) J ( M , U ) ( X ) ∈ Mor tls ( q ( D ( M , X )) , q ( B ( M , X ))) . Next (21) is equivalent to say that for all f ∈ D ( M ) we have Γ U M ( t )(£ [grad M ( f ) , U ] ) = £ [(grad M ◦ exp UM ( t ))( f ) , U ] ; therefore by considering (27) we have that the following is a commutative diagram inthe category tls q ( D ( M , Y )) J ( M , U ) ( Y ) / / q ( B ( M , Y )) q ( D ( M , X )) J ( M , U ) ( X ) / / q (exp UM ( t ) ⊕ Id C ) O O q ( B ( M , X )) q ( Γ U M ( t )) O O namely(28) J ( M , U ) ∈ Mor
Fct( h M , U i , tls) ( q ◦ F h M , U i , q ◦ F h M , U i ) . Thus by Def. 1.25 and the corresponding of [ , Cor. 1.4.16](29) J † ( M , U ) ∈ Mor
Fct( h M , U i op , tls) ( r ◦ F †h M , U i , r ◦ F †h M , U i ) . (28) and (29) imply(30) J ( M , U ) ∈ Mor
Chdv ( x ( M , U ) , z ( M , U )) . Therefore the statement will follow if we prove that for every ( N , V ) ∈ Obj( Vf ) and every φ ∈ Mor Vf (( M , U ) , ( N , V )) the following is a commutative diagram in the category Chdv (31) x ( N , V ) J ( N , V ) / / z ( N , V ) x ( M , U ) J ( M , U ) / / ΨΨΨ ( f φ , T φ ) O O z ( M , U ) ΨΨΨ ( f φ , T φ ) O O namely (cid:16) f φ , ( q ∗ T φ ) † , q ∗ T φ (cid:17) ◦ (cid:16) h M , U i , J † ( M , U ) , J ( M , U ) (cid:17) = (cid:16) h N , V i , J † ( N , V ) , J ( N , V ) (cid:17) ◦ (cid:16) f φ , ( q ∗ T φ ) † , q ∗ T φ (cid:17) ;that according to (17) is equivalent to (cid:16) h M , U i ◦ f φ , ( J † ( M , U ) ∗ f φ ) ◦ ( q ∗ T φ ) † , ( q ∗ T φ ) ◦ ( J ( M , U ) ∗ f φ ) (cid:17) = (cid:16) f φ ◦ h N , V i , (( q ∗ T φ ) † ∗ h N , V i ) ◦ J † ( N , V ) , J ( N , V ) ◦ (( q ∗ T φ ) ∗ h N , V i ) (cid:17) . which reduces to the following equality of morphisms of the category Fct( h N , V i , tls)( q ∗ T φ ) ◦ ( J ( M , U ) ∗ f φ ) = J ( N , V ) ◦ (( q ∗ T φ ) ∗ h N , V i ) . Which is equivalent to say that for every Z ∈ h N , V i we have the following equality ofmorphisms of the category tls T φ ( Z ) ◦ J ( M , U ) ( φ ( Z )) = J ( N , V ) ( Z ) ◦ T φ ( Z ); . ADDITIONAL EQUICONTINUITY REQUESTS 25 namely T ( φ ) ◦ Z φ ( Z )( M , U ) = Z Z ( N , V ) ◦ ( φ ∗ ⊕ Id C );equivalent to say that for every ( f , λ ) ∈ D ( M , φ ( Z )) we have φ ∗ (£ [grad M ( f ) , U ] + λ )( φ − ) ∗ = £ [(grad N ◦ φ ∗ )( f ) , V ] + λ ;namely φ ∗ £ [grad M ( f ) , U ] ( φ − ) ∗ = £ [(grad N ◦ φ ∗ )( f ) , V ] ;but £ is a Lie algebra morphism, so the above is equivalent to[ φ ∗ £ grad M ( f ) ( φ − ) ∗ , φ ∗ £ U ( φ − ) ∗ ] = [£ (grad N ◦ φ ∗ )( f ) , £ V ];which follows since Lemma 3.3 and the fact that U and V are φ -related. Thus the diagram(31) is commutative and the statement follows. (cid:3)
4. Additional equicontinuity requests D efinition Let M be a manifold and U ∈ X ( M ) such that { (£ > U ) k | k ∈ Z + } is ( τ ∞ , τ ∞ ) -equicontinuous, define Exp UM ≔ exp £ > U C ∞ ( M ) . Under the same reasoning used in Rmk. 1.3 we have that Exp UM is a group of unitpreserving ∗ -automorphisms of C ∞ ( M ).L emma Let M be a manifold and U ∈ X ( M ) such that { (£ > U ) k | k ∈ Z + } is ( τ ∞ , τ ∞ ) -equicontinuous and { £ kU | k ∈ Z + } is ( τ ∞ ↾ D ( M ) , τ ∞ c ) -equicontinuous. Thus Exp UM ( t ) ◦ ı C ∞ ( M ) D ( M ) = ı C ∞ ( M ) D ( M ) ◦ exp UM ( t ); in particular exp UM ( t ) ∈ L ( D ( M )[ τ ∞ ]) . P roof . By the hypothesis and since D ( M ) ֒ → C ∞ ( M ) we have that { £ kU | k ∈ Z + } is( τ ∞ c , τ ∞ c )-equicontinuous. Thus exp UM ( t ) is well-set and for every f ∈ D ( M ) we haveexp UM ( t ) f = ∞ X k = t k k ! £ kU ( f ) w.r.t. the τ ∞ c -topology = ∞ X k = t k k ! (£ > U ) k ( f ) w.r.t. the τ ∞ -topology = Exp UM ( t ) f . (cid:3) P roposition Under the hypothesis of Lemma 4.2, for every t ∈ R , F ∈ C ∞ ( M ) andf ∈ D ( M ) we have exp UM ( t )( F · f ) = Exp UM ( t )( F ) · exp UM ( t )( f ) . P roof . Since Lemma 4.2 and since Exp UM ( t ) is a ∗ -automorphism of C ∞ ( M ). (cid:3) P roposition Under the hypothesis of Lemma 4.2, let A ⊂ L ( D ( M )) such that A isnaturally a left C ∞ ( M ) -module namely w.r.t. the external product defined by ( F · Q ) : D ( M ) → D ( M ) , h F · Q ( h ) for every Q ∈ A and F ∈ C ∞ ( M ) . Thus for every t ∈ R we have (1) if h ∈ D ( M ) , then Λ UM ( t )( F · Q ) h = Exp UM ( t )( F ) · Λ UM ( t )( Q ) h; (2) if Λ UM ( t ) A ⊆ A , then Λ UM ( t )( F · Q ) = Exp UM ( t )( F ) · Λ UM ( t )( Q ) with the ( · ) operation in A . P roof . St. (2) follows since st. (1). Next we have Λ UM ( t )( F · Q ) h = exp UM ( t ) (cid:16) F · ( Q exp UM ( − t ) h ) (cid:17) = Exp UM ( t )( F ) · Λ UM ( t )( Q ) h ;where the second equality follows by Prp. 4.3. (cid:3) C orollary Let M be a semi-Riemannian manifold, M its underlying manifold and { E i } n = dim Mi be a frame of orthonormal fields of M . Thus under the hypothesis of Lemma 4.2 wehave for every V ∈ X ( M ) , t ∈ R and h ∈ D ( M ) that (32) Γ U M ( t )(£ V ) h = n X i = ε i Exp UM ( t )( h V , E i i M ) · Γ UM ( t )(£ E i ) h . If in addition [ U , E i ] = for every i ∈ [1 , n ] ∩ Z , then (33) Γ U M ( t )(£ V ) = £ V t where V t ≔ P ni = ε i Exp UM ( t )( h V , E i i M ) E i . Moreover the left C ∞ ( M ) -submodule A M of Di ff Op ( M ) generated by the set { £ W | W ∈ X ( M ) } is such that Γ U M ( t ) A M ⊆ A M , and (34) Γ U M ( t )(£ V ) = n X i = ε i Exp UM ( t )( h V , E i i M ) · £ E i ; with the ( · ) operation in A M . P roof . (32) follows since Prp. 4.4(1) applied to the natural left C ( M ) ∞ -moduleDi ff Op ( M ). Next if [ U , E i ] = for every i ∈ [1 , n ] ∩ Z , then by (32) and Rmk. 3.2we obtain(35) Γ U M ( t )(£ V ) h = n X i = ε i Exp UM ( t )( h V , E i i M ) · £ E i h = £ V t h . Hence (33) follows, which together Prp. 4.4(1) imply Γ U M ( t ) A M ⊆ A M . (34) follows sincethe first equality in (35) and the fact that A M is a left C ( M ) ∞ -module; alternatively by Γ U M ( t ) A M ⊆ A M and Prp. 4.4(2). (cid:3) We conclude this section with a result providing su ffi cient conditions on a completevector field U on M ensuring that exp UM = η UM . . ADDITIONAL EQUICONTINUITY REQUESTS 27 C orollary Let ( M , U ) ∈ vf ⋆ with M = ( M , g ) such that U is complete and (36) ( ∀ t ∈ R )( µ g ◦ η UM ( t ) = µ g ◦ ı K ( M ) D ( M ) ) . Thus (1) ( M , U ) ∈ vf and η UM extends to a C -group η U M on H g of unitary operators whoseinfinitesimal generator extends £ U . (2) If { £ kU | k ∈ Z + } is ( k · k H g , k · k H g ) -equicontinuous, then D ( M ) is a core for both theinfinitesimal generators of η U M and exp U M . Thus exp U M = η U M , in particular exp UM = η UM . P roof . Let t ∈ R . η UM ( t ) is an isometry of the H g -dense subspace D ( M ) since η U M is a morphism of ∗ -algebras and since (36), thus η UM ( t ) extends to a unitary operator η U M ( t ) on H g . Next let { t n } n ∈ N be a sequence in R converging at 0 and f ∈ D ( M ),thus lim n ∈ N η U M ( t n )( f ) = f pointwise since the flow θ U is pointwise continuous. But η U M ( t n )( f ) ∈ D ( M ) as well f ∈ D ( M ) and D ( M ) ⊂ H g therefore by applying the LebesgueThm. lim n ∈ N η U M ( t n )( f ) = f w.r.t. the norm topology of H g . But η U M is a semigroup ofnorm continuous operators, therefore we conclude that η U M is a C -group on H g andwe let m U denote its infinitesimal generator. Next by definition of θ U we deduce thatfor every f ∈ D ( M ), £ U f is the pointwise derivative at t = t η U M ( t ) f .Thus by D ( M ) ⊂ H g and by applying the Lebesgue Thm. we conclude that £ U f is thederivative at t = t η U M ( t ) f w.r.t. the norm topology of H g , namely m U is anextension of £ U . Now η U M ( t ) f ∈ L ( M , d µ g ) since µ g ( | η U M ( t ) f | ) = µ g ( η U M ( t ) | f | ) = µ g ( | f | ) < ∞ where the first equality follows at once by the definition of η UM , the second equalityfollows by (36), the inequality follows since D ( M ) ⊂ K ( M ) ⊂ L ( M , d µ g ). Moreover£ U f ∈ D ( M ) ⊂ L ( M , d µ g ) thus by applying the Lebesgue Thm. similarly as above, weobtain that £ U f is the derivative at t = t η U M ( t ) f . w.r.t. the norm topologyof L ( M , d µ g ). Next µ g extends to an element of L ( M , µ g ) ′ , therefore what right nowproven and (36) imply that µ g ◦ £ U = and st. (1) follows. Now st. (1) and Cor. 1.19imply that there exists a unique C -group exp U M on H g of unitary operators extendingexp UM and whose infinitesimal generator l U extends £ U , in particular(37) m U ↾ D ( M ) = l U ↾ D ( M ) = £ U . Therefore the additional equicontinuity hypothesis in st. (2) implies that D ( M ) is a set ofanalytic elements for both the generators m U and l U , moreover D ( M ) is dense in H g w.r.t.the norm topology thus also w.r.t. the σ ( H g , H ′ g )-topology, and £ U D ( M ) ⊆ D ( M ), thus weconclude that D ( M ) is a core of m U and l U by applying well-known general results aboutthe core of generators of C -semigroups, and then the first sentence of st. (2) follows.The first sentence of st. (2) and (37) imply m U = l U and then the second sentence ofst. (2) follows by the well-known uniqueness of the generator of a equicontinuous C -semigroup. (cid:3) ibliography
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