Pseudodifferential extensions and adiabatic deformation of smooth groupoid actions
aa r X i v : . [ m a t h . OA ] D ec Pseudodifferential extensions and adiabaticdeformation of smooth groupoid actions by Claire Debord and Georges Skandalis Laboratoire de Math´ematiques, UMR 6620 - CNRSUniversit´e Blaise Pascal, Campus des C´ezeaux, BP
F-63171 Aubi`ere cedex, [email protected]´e Paris Diderot, Sorbonne Paris Cit´eSorbonne Universit´es, UPMC Paris 06, CNRS, IMJ-PRGUFR de Math´ematiques, CP - Bˆatiment Sophie Germain5 rue Thomas Mann, 75205 Paris CEDEX 13, [email protected] Abstract
The adiabatic groupoid G ad of a smooth groupoid G is a deformation relating G with itsalgebroid. In a previous work, we constructed a natural action of R on the C*-algebra ofzero order pseudodifferential operators on G and identified the crossed product with a naturalideal J ( G ) of C ∗ ( G ad ). In the present paper we show that C ∗ ( G ad ) itself is a pseudodifferentialextension of this crossed product in a sense introduced by Saad Baaj. Let us point out thatwe prove our results in a slightly more general situation: the smooth groupoid G is assumedto act on a C*-algebra A . We construct in this generalized setting the extension of order0 pseudodifferential operators Ψ( A, G ) of the associated crossed product A ⋊ G . We showthat R acts naturally on Ψ( A, G ) and identify the crossed product of A by the action of theadiabatic groupoid G ad with an extension of the crossed product Ψ( A, G ) ⋊ R . Note thatour construction of Ψ( A, G ) unifies the ones of Connes (case A = C ) and of Baaj ( G is a Liegroup). Keywords:
Noncommutative geometry; groupoids; pseudodifferential calculus.
Alain Connes in [7, Chap. VIII] pointed out that smooth groupoids offer a perfect settingfor index theory. Since then, this fact has been explored and exploited by Connes as well asmany other authors, in many geometric situations (see [11] for a review).In [10, section II.5], A. Connes constructed a beautiful groupoid, that he called the “tangentgroupoid”, which interpolates between the pair groupoid M × M of a (smooth, compact)manifold M and the tangent bundle T M of M . He showed that this groupoid describes theanalytic index on M in a way not involving (pseudo)differential operators at all, and gave aproof of the Atiyah-Singer Index Theorem based on this groupoid.This idea of a deformation groupoid was then used in [15, section III], and extended in [22, 23]to the general case of a smooth groupoid, where the authors associated to every smoothgroupoid G an adiabatic groupoid G ad , which is obtained by applying the “deformation to thenormal cone” construction to the inclusion G (0) → G of the unit space of G into G . Moreover,it was shown in [22, Th´eor`eme 2.1] that this adiabatic groupoid still describes the analyticindex of the groupoid G in this generalized situation. AMS subject classification: Primary 58H05. Secondary 46L89, 58J22.
1n [12], we further explored the relationship between pseudodifferential calculus on G andits adiabatic deformation G ad . An ideal J ( G ) ⊂ C ∗ ( G ad ) which sits in an exact sequence0 → J ( G ) → C ∗ ( G ad ) → C ( G (0) ) → ∗ ( G ) of order 0 pseudodifferential op-erators on G and the crossed product J ( G ) ⋊ R ∗ + of J ( G ) by the natural action of R ∗ + .It appeared that J ( G ) is canonically isomorphic to the crossed product Ψ ∗ ( G ) ⋊ R associ-ated with a natural action of R on the algebra Ψ ∗ ( G ). A natural question is then: can onerecognize the C ∗ -algebra C ∗ ( G ad ) in these terms ?In the present paper, we answer this question, thanks to [3, 4], where Baaj constructed anextension of pseudodifferential operators of order 0 of the crossed product of a C ∗ -algebra A by the action of a Lie group H - with Lie algebra H . Denote by S ∗ H the sphere in H ∗ . Baaj’sexact sequence reads 0 → A ⋊ H −→ Ψ ∗ ( A, H ) σ −→ C ( S ∗ H ) ⊗ A → . Let µ : C ( G (0) ) → Ψ ∗ ( G ) be the inclusion by multiplication operators. In the present paper,we construct a commutative diagram, whose first line is Baaj’s exact sequence:0 / / Ψ ∗ ( G ) ⋊ R / / Ψ ∗ (Ψ ∗ ( G ) , R ) σ / / Ψ ∗ ( G ) ⊕ Ψ ∗ ( G ) / / / / J ( G ) / / ≃ O O C ∗ ( G ad ) / / O O C ( G (0) ) µ O O / / µ ( f ) = ( µ ( f ) , R ∗ + : • We consider R ∗ + as the dual group of R and thus it acts on the crossed product Ψ ∗ ( G ) ⋊R via the dual action. This dual action extends (uniquely) to Baaj’s pseudodifferentialextension Ψ ∗ (Ψ ∗ ( G ) , R ) and is trivial at the quotient level. • The action of R ∗ + on the second line is the canonical action on the adiabatic groupoidby the natural rescaling, and the crossed product C ∗ ( G ad ) ⋊ R ∗ + is the C ∗ -algebra of the“gauge adiabatic groupoid” G ga considered in [12].In particular, this allows us to give also a description of the algebra C ∗ ( G ga ) as a pseudodif-ferential extension.As a side construction, we define the pseudodifferential extension of an action α of a smoothgroupoid G - in the setting introduced by Le Gall in [20, 21]. This is a short exact sequence0 → A ⋊ α G −→ Ψ ∗ ( A, α, G ) σ α −→ A ⊗ C ( M ) C ( S ∗ A G ) → . (2)This construction generalizes both the pseudodifferential calculus on a smooth groupoid of[7, 10, 22, 23] and the pseudodifferential calculus of a crossed product by a Lie group of [3, 4].Our main result, Theorem 5.6, is stated (and proved) in this general frame: in diagram (1) weallow the groupoid G to act on a C ∗ -algebra A and replace groupoid C ∗ -algebras by crossedproducts. We should note that the connecting map of extension (2) is the analytic index inthis context. In the same way as in [22, 23], the crossed product by the adiabatic groupoidallows to define the analytic index too.Here are some examples of natural actions of smooth groupoids which are relevant to ourconstructions. 2. Already an interesting case appears when A = C ( X ) where X is a smooth manifold,endowed with a smooth submersion p : X → M = G (0) and G acts on the fibers. Theaction of G is given by a diffeomorphism α : G s × p X → X p × r G of the form ( γ, x ) ( α γ ( x ) , γ ), which satisfies α γ γ = α γ α γ . Here, G s × p X is a smooth groupoid G X withobjects X , source and range maps given by s ( γ, x ) = x , r ( γ, x ) = α γ ( x ) composition( γ ′ , α γ ( x ))( γ, x ) = ( γ ′ γ, x ) and inverse ( γ, x ) − = ( γ − , α γ ( x )). In that case, the crossedproduct A ⋊ α G , the extension Ψ ∗ ( A, α, G ), the crossed product ( A ⊗ R + ) ⋊ G ad identifyrespectively with the groupoid C ∗ -algebra C ∗ ( G X ), the pseudodifferential extensionΨ ∗ ( G X ) and the C ∗ -algebra C ∗ (( G X ) ad ) of the adiabatic deformation of the groupoid G X .2. Let G be a Lie group acting on a C ∗ -algebra A . The corresponding adiabatic and gaugeadiabatic deformations of G are groupoids with objects R + . They naturally act on the C ( R + ) algebra A ⊗ C ( R + ) - and the associated action is an important piece in ourconstructions - see section 4.3.2.3. An interesting family of examples of groupoid actions comes from 1-cocycles (general-ized morphisms in the sense of [15, section I], [20, Section 2.2]) of a groupoid G to aLie group. For instance, an equivariant vector bundle is equivalent to a cocycle from G to GL n ( R ). Then every algebra A endowed with an action of G gives rise to a G -algebra. This construction is studied in [20] where several examples connected with K -theory and index theory are studied. The corresponding pseudodifferential extensionand associated actions of the adiabatic groupoid appear very naturally in this context.The paper is organized as follows:In the second section, we briefly review the action of a locally compact groupoid and thecorresponding full and reduced crossed products ( cf. [20, 21, 29, 28, 24]).In the third section, we review Baaj’s construction and discuss the dual action.In the fourth section we generalize Baaj’s construction to the case of actions of smoothgroupoids.The fifth section establishes the above mentioned equivariant commutative diagram.Finally, we gathered a few rather well known facts on unbounded multipliers in an appendix. Notation 1.1. If A is a C ∗ -algebra, we denote by M ( A ) its multiplier algebra.Recall that, if A and B are C ∗ -algebras, a morphism f : A → M ( B ) is said to be nondegenerate if f ( A ) .B = B ; a non degenerate morphism extends uniquely to a morphism˜ f : M ( A ) → M ( B ) - this extension is strictly continuous ( i.e. continuous with respect tothe natural topologies of the multipliers).Recall that an ideal J of a C ∗ -algebra A is said to be essential if the morphism A → M ( J )is injective, i.e. if a ∈ A is such that aJ = { } then a = 0. Remark 1.2.
Note that if π : A → B is a surjective morphism of C ∗ -algebras and J anessential ideal in B then π − ( J ) is essential in A .3 Actions of locally compact groupoids and crossedproducts
In this section we briefly recall a few facts about actions of locally compact groupoids andthe corresponding crossed products as defined by Le Gall in [20, 21]. See also [29, 28, 24]. ( X ) -algebrasC ( X ) -algebras. Recall ([13], [16, Def. 1.5]) that if X is a locally compact space, a C ( X )-algebra is a pair ( A, θ ), where A is a C ∗ -algebra and θ is a non degenerate ∗ -homomorphism θ : C ( X ) → Z M ( A ) from C ( X ) to the center of the multiplieralgebra of A . Fibers. If A is a C ( X )-algebra, we define its fiber A x for every point x ∈ X by setting A x = A/C x A where C x = { h ∈ C ( X ); h ( x ) = 0 } . Let a ∈ A and denote by a x ∈ A x its class; we have k a k = sup x ∈ X k a x k . In particular a is completely determined by thefamily ( a x ) x ∈ X and the bundle A is semi-continuous in the sense that for all a ∈ A themap x
7→ k a x k is upper semi-continuous. C ( X ) -morphisms. A C ( X )-linear homomorphism α : A → B of C ( X )-algebras deter-mines for each x ∈ X a ∗ -homomorphism α x : A x → B x . Since α ( a ) is determined bythe family ( α ( a )) x = α x ( a x ), the morphism α is determined by the family ( α x ) x ∈ X . Restriction to locally closed sets; pull back.
More generally, if U ⊂ X is an open sub-set, we define the C ( U )-algebra A U by putting A U = C ( U ) A ; if F ⊂ X is a closedsubset, we define the C ( F )-algebra A F = A/A X \ F ; if Y = U ∩ F is a locally closedsubset of X we put A Y = ( A U ) Y (which is canonically isomorphic to ( A F ) Y ).Recall that if f : Y → X is a continuous map between locally compact spaces and A isa C ( X )-algebra, we may define f ∗ ( A ) in the following way: we restrict the C ( X × Y )-algebra A ⊗ C ( Y ) to the graph { ( x, y ) ∈ X × Y ; f ( y ) = x } of f which is a closedsubset of X × Y canonically homeomorphic with Y . Notation 2.1. As f ∗ ( A ) is a quotient of A ⊗ C ( Y ), we have a non degenerate morphism a a ◦ f from A to the multiplier algebra of f ∗ ( A ), where a ◦ f is the image of a ⊗ f ∗ ( A ) of A ⊗ C ( Y ). ([21, Definition 2.2]). Let G be a locally compact groupoid with basis X . A continuous action of G on a C ( X )-algebra A is an isomorphism of C ( G )-algebras α : s ∗ A → r ∗ A such that, for all ( γ , γ ) ∈ G (2) we have α γ γ = α γ ◦ α γ . Remark 2.3.
An action of a non Hausdorff groupoid G on a C ( X )-algebra A (with X = G (0) )is given by isomorphisms α U : s ∗ U ( A ) → r ∗ U ( A ) for every Hausdorff open subset U of X -where s U , r U are the restrictions of r and s to U . These isomorphisms must agree on theintersection U ∩ V of two such sets. It follows that the family ( r U ) gives rise to isomor-phisms α γ : A s ( γ ) → A r ( γ ) for γ ∈ G . We further impose that these isomorphisms satisfy4 γ γ = α γ ◦ α γ for all ( γ , γ ) ∈ G (2) .In the sequel of the paper, we will consider Hausdorff groupoids for simplicity of the expo-sition. Nevertheless, all our constructions and results extend in the usual way to the nonHausdorff case [9, section 6], see also [17, section I.B]. Note that the non trivial part of anykind of pseudodifferential calculus concentrates in a Hausdorff neighborhood of the space ofunits. The (full and reduced) crossed product A ⋊ α G of an action α of a groupoid G with (right)Haar system ( ν x ) x ∈ X on a C ∗ -algebra A is defined in [21, 25]. Let us briefly recall theseconstructions. The vector space C c ( r ∗ A ) = C c ( G ) .r ∗ ( A ) of elements of r ∗ A with compact support is naturallya convolution ∗ -algebra. For f, g ∈ C c ( r ∗ A ) and γ ∈ G , we have( f ∗ g ) γ = Z G r ( γ ) f γ α γ ( g γ − γ ) dν r ( γ ) ( γ ) and ( f ∗ ) γ = α − γ ( f γ − )There is a k k norm given by k f k = sup x ∈ X max (cid:18)Z G x k f γ k dν x ( γ ) , Z G x k f γ − k dν x ( γ ) (cid:19) on this algebra and the corresponding completion is a Banach ∗ -algebra L ( r ∗ A, ν ) (recallthat X is the basis G (0) of G ).The full crossed product A ⋊ α G is the enveloping C ∗ -algebra of L ( r ∗ A, ν ). The algebras A and C ∗ ( G ) sit in the multipliers of A ⋊ α G in a non degenerate way, and A ⋊ α G is the closedvector span of products a.f with a ∈ A and f ∈ C ∗ ( G ). Note that C ( X ) sits both in themultipliers of C ∗ ( G ) and of A ; its images in M ( A ⋊ α G ) agree. The representations of A ⋊ α G can easily be described as in [26, Theorem 1.21, p. 65]. Sucha representation gives rise to representations of A and C ∗ ( G ). We thus obtain: • The representation of C ( X ) corresponds to a measure µ on X and a measurable fieldof Hilbert spaces ( H x ) x ∈ X . • The representation of the C ( X )-algebra A is given by a measurable family π = ( π x ) x ∈ X where π x : A x → L ( H x ) is a ∗ -representation. • The representation of C ∗ ( G ) gives rise to a representation of G in the sense of [26,def. 1.6, p. 52]. In other words, the measure µ is quasi-invariant ( i.e. µ ◦ ν is quasi-invariant by the map γ γ − ) and we have a measurable family U = ( U γ ) γ ∈G where U γ : H s ( γ ) → H r ( γ ) is (almost everywhere) unitary and satisfies (almost everywhere) U γ γ = U γ U γ . • The covariance property then reads: π r ( γ ) ◦ α γ = Ad U γ ◦ π s ( γ ) (almost everywhere).Conversely, such data ( µ, H, π, U ) can be integrated to a representation of A ⋊ α G .5 .2.3 The reduced crossed product (see [26, 17]) The reduced crossed product A ⋊ α,red G is the quotient of A ⋊ α G corresponding to the familyof regular representations on the Hilbert modules A x ⊗ L ( G x ; ν x ) for x ∈ X .If G is amenable (see [1] for a discussion on amenability of groupoids) then the morphism A ⋊ α G → A ⋊ α,red G is an isomorphism.The reduced crossed product has a faithful representation on the Hilbert A -module E = L ( G ; ν ) ⊗ C ( X ) A where L ( G ; ν ) is the Hilbert C ( X ) module described in [17, Theorem 2.3](if G is Hausdorff). The module E is the completion of C c ( G ; s ∗ A ) with respect to the A -valuedinner product satisfying ( h ξ | η i ) x = Z G x ξ ∗ γ η γ dν x ( γ ) (cid:0) where ( ν x ) x ∈ X is the corresponding leftHaar system given by Z f ( γ ) dν x ( γ ) = Z f ( γ − ) dν x ( γ )) and, right action given by ( ξa ) γ = ξ γ a s ( γ ) (cid:1) .Denote by λ the action of C ∗ red ( G ) by (left) convolution on the Hilbert C ( X )-module L ( G ; ν );the left action of C ∗ ( G ) is given by f λ ( f ) ⊗ C ( X )
1. The action of A is given by a.ξ = (cid:16) α − ( a ◦ r ) (cid:17) ξ : in other terms ( a.ξ ) γ = α − γ ( a r ( γ ) ) ξ γ .It follows, that if π = Z ⊕ X π x dµ ( x ) is a faithful representation of A , the correspondingrepresentation of A ⋊ α,red G on Z ⊕ X L ( G x , ν x ) ⊗ H x dµ ( x ) is faithful. Let J ⊂ A be an ideal in A . Note that both J and A/J are then C ( X ) algebras - recall that X = G (0) . Assume that J is invariant under the action of G which means that α ( s ∗ ( J )) = r ∗ ( J ). Then α yields actions of G on J and A/J . Lemma 2.4. [25, Theorem 3] We have an exact sequence of full crossed products: → J ⋊ α G → A ⋊ α G → ( A/J ) ⋊ α G → . Proof.
The only thing which is not completely obvious in this sequence is that the morphism( A ⋊ α G ) / ( J ⋊ α G ) → A/J ⋊ α G is injective. To see that, take a faithful representation of( A ⋊ α G ) / ( J ⋊ α G ); it is a covariant representation of A and G which vanishes on J , andtherefore a covariant representation of A/J and G .If J is a G -invariant essential ideal in A , then at the level of reduced crossed products, theideal J ⋊ α,red G of A ⋊ α,red G is essential. Let U be an open subset of G , which is saturated for G ( i.e. for all γ ∈ G , we have s ( γ ) ∈ U ⇐⇒ r ( γ ) ∈ U ). Put F = X \ U . Define the subgroupoids G U = s − ( U ) = r − ( U ) and G F = s − ( F ) = r − ( F ). The action α of G on A gives actions α U of G U on A U and α F of G F on A F . We may note that A U ⋊ α U G U = A U ⋊ α G and A F ⋊ α v G F = A F ⋊ α G . Let us quotesome results that we will use: 6) We have an exact sequence of full crossed products:0 → A U ⋊ α U G U → A ⋊ α G → A F ⋊ α F G F → . b) If G F is amenable, the same is true for the reduced crossed products - exactness at themiddle terms follows from the diagram0 / / A U ⋊ α U G U / / (cid:15) (cid:15) A ⋊ α G / / (cid:15) (cid:15) A F ⋊ α F G F → / / ≃ (cid:15) (cid:15) / / A U ⋊ α U ,red G U / / A ⋊ α,red G / / A F ⋊ α F ,red G F / / A U is an essential ideal in A , then A U ⋊ α U ,red G U is an essential ideal in A ⋊ α,red G .d) It follows from Rem. 1.2 that, if G F is amenable and A U is an essential ideal in A , then A U ⋊ α U G U is an essential ideal in A ⋊ α G . In this section, we briefly review Baaj’s construction of the pseudodifferential extension ofa crossed product by a Lie group G . We note that the dual action extends to the pseu-dodifferential extension (and is trivial at the symbol level) and discuss the correspondingcrossed product. Although this is not necessary in our framework, we will not assume G tobe abelian, so that this dual action is a coaction of G , since this doesn’t really add any diffi-culty. We then establish an isomorphism between the crossed product of the algebra of thepseudodifferential operators by the dual action and a natural pseudodifferential extension.Finally, we examine the case where the Lie group is R - which is the relevant case for ourresults of section 5. Let us begin by recalling the extension of pseudodifferential operators associated with acontinuous action α by automorphisms of a Lie group G on a C ∗ -algebra A ([3, 4], the resultsof Baaj concern the case G = R n - but immediately generalize to the general case of a Liegroup).Recall first that the order 0 pseudodifferential operators on a Lie group G give rise to anexact sequence 0 → C ∗ ( G ) −→ Ψ ∗ ( G ) σ −→ C ( S ∗ g ) → C ∗ ( G ) is the (full) group C ∗ -algebra of G and S ∗ g denotes the (compact) space of halflines in the dual space g ∗ of the Lie algebra g .Now, the algebras A and C ∗ ( G ) sit in the multiplier algebra of A ⋊ α G in a non degenerateway, and the elements ax with a ∈ A and x ∈ C ∗ ( G ) span a dense subspace of A ⋊ α G . Thisholds for the full group algebra and crossed product, as well as for the reduced group algebraand crossed product. Note however that, at the level of full C ∗ -algebras, the morphism C ∗ ( G ) → M ( A ⋊ α G ) needs not be injective in general - it is easily seen to be injective at the7evel of reduced C ∗ -algebras. We will somewhat abusively identify C ∗ ( G ) and A with theirimages in the multiplier algebra M ( A ⋊ α G ).In what follows, since we will consider the crossed product by the dual action, we will mainlyuse the reduced crossed product. Note also that we will mainly use Baaj’s construction inthe case where G is R which is amenable and there is no distinction between the full and thereduced case. In particular the morphism C ∗ ( G ) → M ( A ⋊ α G ) is injective in that case (if A = { } ).The nondegenerate morphism C ∗ ( G ) → M ( A ⋊ α G ) extends to the multiplier algebra of C ∗ ( G ) and in particular to the subalgebra Ψ ∗ ( G ) of order 0 pseudodifferential operators of G . We still identify (abusively) the elements of Ψ ∗ ( G ) with their images in M ( A ⋊ α G ).Recall that we have: Proposition 3.1. [3, section 4] a) For every P ∈ Ψ ∗ ( G ) and a ∈ A , the commutator [ P, a ] belongs to A ⋊ α G . b) The closure of the linear span of products of the form
P a with P ∈ Ψ ∗ ( G ) and a ∈ A is a C ∗ -subalgebra Ψ ∗ ( A, α, G ) ⊂ M ( A ⋊ α G ) and we have an exact sequence: → A ⋊ α G −→ Ψ ∗ ( A, α, G ) σ α −→ C ( S ∗ g ) ⊗ A → . (1)Let us briefly discuss some naturality properties of this construction: Proposition 3.2.
Let ( A, G, α ) and ( B, G, β ) be C ∗ -dynamical systems and γ : A → M ( B ) a G -equivariant morphism a) We obtain a morphism b γ : Ψ ∗ ( A, α, G ) → M (Ψ ∗ ( B, β, G )) and a commutative diagram Ψ ∗ ( A, α, G ) σ α / / b γ (cid:15) (cid:15) C ( S ∗ g ) ⊗ A id ⊗ γ (cid:15) (cid:15) M (Ψ ∗ ( B, β, G )) f σ β / / M ( C ( S ∗ g ) ⊗ B ) . both for the full and the reduced versions - where we denoted by f σ β the extension of σ β to the multipliers. b) If γ ( A ) ⊂ B then b γ (Ψ ∗ ( A, α, G )) ⊂ Ψ ∗ ( B, β, G ) . Moreover, if γ : A → B is anisomorphism, then b γ : Ψ ∗ ( A, α, G ) → Ψ ∗ ( B, β, G ) is an isomorphism. c) If γ is injective then so is the reduced version of b γ .Proof. a) By construction the inclusion of B in Ψ ∗ ( B, β, G ) is a nondegenerate morphism( i.e. B Ψ ∗ ( B, β, G ) = Ψ ∗ ( B, β, G )). It therefore extends to a morphism M ( B ) →M (Ψ ∗ ( B, β, G )). In this way, we find a representation b γ : A → M (Ψ ∗ ( B, β, G )). Nowthe images of A and G in M ( B ⋊ β G ) ⊃ M (Ψ ∗ ( B, β, G )) form a covariant representationso that we get a morphism A ⋊ α G → M ( B ⋊ β G ) (both for the reduced and full versionsof the crossed products). The image of this morphism is spanned by elements a.h with a ∈ A and h ∈ C ∗ ( G ); it therefore sits in M (Ψ ∗ ( B, β, G )). Finally, upon replacing A bythe algebra obtained by adjoining a unit, we may assume that γ is non degenerate. Itfollows that b γ : A ⋊ α G → M ( B ⋊ β G ) is non degenerate and therefore uniquely extendsto the multiplier algebra. We thus get a morphism b γ : Ψ ∗ ( A, α, G ) → M ( B ⋊ β G ).The image of a.P is b γ ( a ) .P (for a ∈ A and P ∈ Ψ ∗ ( G )) and therefore b γ (Ψ ∗ ( A, α, G )) ⊂M (Ψ ∗ ( B, β, G )). 8) This is obvious.c) If γ is one to one, then the reduced version γ red : A ⋊ α,red G → M ( B ⋊ β,red G ) isinjective. Therefore ker b γ red ∩ A ⋊ α,red G = { } whence ker b γ red = { } since A ⋊ α,red G is an essential ideal in Ψ ∗ red ( A, α, G ) - see prop. 4.3.
We now restrict to the reduced group algebras and crossed products.The coproduct of C ∗ red ( G ) is a non degenerate morphism δ : C ∗ red ( G ) → M ( C ∗ red ( G ) ⊗ C ∗ red ( G )).It therefore extends to a morphism ˜ δ : M ( C ∗ red ( G )) → M ( C ∗ red ( G ) ⊗ C ∗ red ( G )). Proposition 3.3.
The restriction of ˜ δ to Ψ ∗ ( G ) , is a coaction: for P ∈ Ψ ∗ red ( G ) and f ∈ C ∗ red ( G ) , we have ˜ δ ( P )(1 ⊗ f ) ∈ Ψ ∗ red ( G ) ⊗ C ∗ red ( G ) and the span of such products is densein Ψ ∗ red ( G ) ⊗ C ∗ red ( G ) . Moreover, for P ∈ Ψ ∗ red ( G ) and f ∈ C ∗ red ( G ) , we have (˜ δ ( P ) − P ⊗ ⊗ f ) ∈ C ∗ ( G × G ) .Proof. Let ( X i ) ≤ i ≤ d be an (orthonormal) basis of g and let ∆ = − X i X i be the associated(positive) laplacian, seen as an unbounded (elliptic, positive) multiplier of C ∗ red ( G ).The non degenerate morphism δ has an extension ˇ δ to unbounded multipliers: for 1 ≤ i ≤ d ,set p i = X i (1 + ∆) − / ∈ Ψ ∗ red ( G ).We let now C ∗ red ( G × G ) act faithfully on L ( G × G ). The following equalities hold on theinfinite domain of the laplacian of the group G × G , which is a dense subspace of L ( G × G ).We have ˇ δ ( X i ) = X i ⊗ ⊗ X i . It follows that ˇ δ (∆) = ∆ ⊗ ⊗ ∆ − X i X i ⊗ X i . For f ∈ C ∞ c ( G ) (acting as a convolution operator), we may then write:(1 ⊗ f )(˜ δ ( p i ) − p i ⊗
1) = (1 ⊗ f X i ) δ ((1 + ∆) − / ) + ( X i ⊗ f )( δ ((1 + ∆) − / ) − (1 + ∆) − / ⊗ . Now f X i and (1+∆) − / extend to elements of C ∗ red ( G ) therefore C i = (1 ⊗ f X i ) δ ((1+∆) − / )extends as well to an element of C ∗ red ( G × G ). We write (1 + ∆) − / as an integral ( cf. [5]):(1 + ∆) − / = 2 π Z + ∞ (1 + ∆ + λ ) − dλ. Write also(1+∆+ λ ) − ⊗ − δ (1+∆+ λ ) − = ((1+∆+ λ ) − ⊗ ⊗ ∆+2 X j X j ⊗ X j ) δ (1+∆+ λ ) − Putting D i = ( X i ⊗ f ) (cid:16) (1 + ∆) − / ⊗ − δ ((1 + ∆) − / ) (cid:17) , we find D i = 2 π ( X i ⊗ f ) Z + ∞ ((1 + ∆ + λ ) − ⊗ − δ (1 + ∆ + λ ) − dλ = 2 π Z + ∞ ( X i (1 + ∆ + λ ) − ⊗ f ∆) δ (1 + ∆ + λ ) − dλ − π X j Z + ∞ ( X i (1 + ∆ + λ ) − X j ⊗ f X j ) δ (1 + ∆ + λ ) − dλ Now all the terms appearing are bounded operators:9 X i (1 + ∆ + λ ) − is pseudodifferential of order − X i (1 + ∆ + λ ) − ∈ C ∗ red ( G ); • f ∆ and f X j are smoothing therefore in C ∗ red ( G ); • (1 ⊗ f X j ) δ (1 + ∆ + λ ) − ∈ C ∗ red ( G ) ⊗ C ∗ red ( G ).It follows that the integrand extends to an element of C ∗ red ( G ) ⊗ C ∗ red ( G ).Furthermore, X k (1 + ∆ + λ ) − / = X k (1 + ∆) − / h λ (∆) where k h λ k ∞ ≤
1, whence k X i (1 +∆ + λ ) − k and k X i (1 + ∆ + λ ) − X j k are bounded independently of λ . Hence, this integralis norm convergent and D i extends to an element ¯ D i of C ∗ red ( G ) ⊗ C ∗ red ( G ).Thus, we have proved that (1 ⊗ f )(˜ δ ( p i ) − p i ⊗
1) = C i + ¯ D i belongs to C ∗ red ( G ) ⊗ C ∗ red ( G ).The set A of P ∈ Ψ ∗ red ( G ) such that (1 ⊗ C ∗ red ( G ))(˜ δ ( P ) − P ⊗ ⊂ C ∗ red ( G ) ⊗ C ∗ red ( G ) and(1 ⊗ C ∗ red ( G ))(˜ δ ( P ∗ ) − P ∗ ⊗ ⊂ C ∗ red ( G ) ⊗ C ∗ red ( G ) is a closed ∗ -subalgebra of Ψ ∗ red ( G ); itcontains C ∗ red ( G ). As p i + p ∗ i ∈ C ∗ red ( G ), it follows by the above calculation that p i ∈ A .Since the symbols of the p i ’s generate a dense subalgebra of the symbol algebra C ( S ∗ g ) weconclude that A = Ψ ∗ red ( G ).Finally, the closed vector span of (1 ⊗ f )˜ δ ( P ) contains the closed vector span of (1 ⊗ f ) δ ( h )(with f, h ∈ C ∗ ( G )) hence, C ∗ red ( G ) ⊗ C ∗ red ( G ). Therefore (1 ⊗ f )˜ δ ( P ) − P ⊗ f is in thisspan: the same holds for P ⊗ f . Let α be an action of a Lie group G on a C ∗ -algebra A . Denote by ˆ α the dual action onthe reduced crossed product A ⋊ α,red G as well as its extension to Ψ ∗ red ( A, α, G ) discussedabove. Recall that in the context on non abelian groups, B ⋊ b G is just a notation for thecrossed product by a dual action, - it is a C ∗ -algebra generated by products bf with b ∈ B and f ∈ C ( G ) subject to the equivariance condition.The Takesaki-Takai duality ([27]) for non abelian groups, (see [19, 18]), is an isomorphism( A ⋊ α,red G ) ⋊ ˆ α b G ≃ A ⊗ K which is based on the following facts:a) There are natural morphisms of the C ∗ -algebras A and C ( G ) to the multiplier algebra M (( A ⋊ α,red G ) ⋊ ˆ α b G ), as well as a (strictly continuous) morphism of the group G to theunitary group of this multiplier algebra, yielding a morphism of C ∗ r ( G ) to M (( A ⋊ α,red G ) ⋊ ˆ α b G ).The double crossed product ( A ⋊ α,red G ) ⋊ ˆ α b G is generated by the products f.a.h with a ∈ A , h ∈ C ∗ r ( G ) and f ∈ C ( G ) (sitting in the multiplier algebra of ( A ⋊ α,red G ) ⋊ ˆ α b G ).Now, since the dual action is trivial on A , the images of A and C ( G ) commute so thatwe find in the multiplier algebra of ( A ⋊ α,red G ) ⋊ ˆ α b G a copy of the C ∗ -tensor product A ⊗ C ( G ). The group G acts on A ⊗ C ( G ) through the action α ⊗ λ (where λ denotesthe action of G on C ( G ) by left translation).The morphisms of the C ∗ -algebra A and the group G ( resp. of C ( G ) and G ) to M (( A ⋊ α,red G ) ⋊ ˆ α b G ) form a covariant representation of the C ∗ -dynamical system( A, G, α ) ( resp. ( C ( G ) , G, λ )). It follows that the morphisms of A ⊗ C ( G ) and G in the multiplier algebra M (( A ⋊ α,red G ) ⋊ ˆ α b G ) form a covariant representation of the C ∗ -dynamical system ( A ⊗ C ( G ) , G, α ⊗ λ ).In this way, we get an isomorphism ( A ⋊ α,red G ) ⋊ ˆ α b G ≃ ( A ⊗ C ( G )) ⋊ α ⊗ λ,red G .10) Now, on A ⊗ C ( G ), the actions α ⊗ λ and id ⊗ λ are conjugate through the automorphism γ of C ( G ; A ) = A ⊗ C ( G ) given by the formula ( γf )( x ) = α x ( f ( x )) for f ∈ C ( G ; A )and x ∈ G . We find an isomorphism ( A ⊗ C ( G )) ⋊ α ⊗ λ,red G ≃ ( A ⊗ C ( G )) ⋊ id ⊗ λ G .c) Finally ( A ⊗ C ( G )) ⋊ id ⊗ λ G ≃ A ⊗ ( C ( G ) ⋊ λ G ) ≃ A ⊗ K . Proposition 3.4.
The isomorphism f : ( A ⋊ α,red G ) ⋊ ˆ α b G ∼ −→ ( A ⊗ C ( G )) ⋊ α ⊗ λ,red G extendsto an isomorphism Ψ ∗ red ( A, α, G ) ⋊ ˆ α b G ≃ Ψ ∗ red ( A ⊗ C ( G ) , α ⊗ λ, G ) .Proof. Since A ⋊ α,red G is an essential ideal in Ψ ∗ red ( A, α, G ) (see 4.3), the algebra Ψ ∗ red ( A, α, G ) ⋊ ˆ α b G sits in the multiplier algebra M (( A ⋊ α,red G ) ⋊ ˆ α b G ) . In the same way, the algebra Ψ ∗ red ( A ⊗ C ( G ) , α ⊗ λ, G ) sits also in M (( A ⊗ C ( G )) ⋊ α ⊗ λ,red G ).Both algebras are generated by products aP h where a ∈ A , P ∈ Ψ ∗ red ( G ) and h ∈ C ( G ).Now the inclusions of A and of C ( G ) in M correspond to each other under the extension ˜ f of f to the multipliers. As the inclusions of C ∗ red ( G ) to M (( A ⋊ α,red G ) ⋊ ˆ α b G ) and M (( A ⊗ C ( G )) ⋊ α ⊗ λ,red G ) correspond to each other under ˜ f , the same holds for the extension to themultipliers, and in particular for the inclusions of Ψ ∗ red ( G ).The actions α ⊗ λ and id ⊗ λ of G on A ⊗ C ( G ) are conjugate. Using prop. 3.2, we deduceisomorphisms Ψ ∗ red ( A, α, G ) ⋊ ˆ α b G ≃ A ⊗ Ψ ∗ red ( C ( G ) , λ, G ) ≃ A ⊗ (Ψ ∗ red ( G ) ⋊ ˆ λ b G ) . Definition 3.5.
Let B be a subalgebra of C ( S ∗ g ) ⊗ A . We denote by Ψ ∗ red ( A, α, G ; B ) the B -valued pseudodifferential extension of α i.e. the subalgebraΨ ∗ red ( A, α, G ; B ) = { P ∈ Ψ ∗ red ( A, α, G ); σ ( P ) ∈ B } of Ψ ∗ red ( A, α, G ).In the case of the trivial action, Ψ ∗ red ( A, id , G ; B ) = { P ∈ A ⊗ Ψ ∗ red ( G ); ( σ ⊗ id)( P ) ∈ B } . R When G = R , then g ∗ = R which has two half lines, i.e. C ( S ∗ g ) = C ⊕ C .Extension (1) reads therefore0 → A ⋊ α R −→ Ψ ∗ ( A, α, R ) σ ± −→ A ⊕ A → , where σ + and σ − are morphisms from Ψ ∗ ( A, α, R ) → A .It is helpful for our discussion to identify the dual group of R with R ∗ + through the pairing h t | u i = u it for u ∈ R ∗ + and t ∈ R . Under this identification, C ∗ ( R ) ≃ C ( R ∗ + ) and Ψ ∗ ( R ) ≃ C ([0 , + ∞ ]). The maps σ − and σ + correspond to evaluation at 0 and + ∞ in the sense that σ − ( P a ) = P (0) a and σ + ( P a ) = P (+ ∞ ) a , where a ∈ A and P ∈ C ([0 , + ∞ ]) ≃ Ψ ∗ ( R ).The algebra A sits in M ( A ⋊ α R ) and we have a strictly continuous family ( u t ) t ∈ R in M ( A ⋊R ). Then we can write u t = Q itα where Q α is a regular unbounded, selfadjoint, positivemultiplier with dense range - i.e. such that Q − α is also densely defined, and therefore aregular unbounded, selfadjoint, positive multiplier. The algebra A ⋊ R is spanned by af ( Q α )with f ∈ C ( R ∗ + ) and Ψ ∗ ( A, α, R ) is spanned by af ( Q α ) with a ∈ A and f ∈ C ([0 , + ∞ ]).11 efinition 3.6. Let A be a C ∗ -algebra and let α = ( α t ) t ∈ R be a continuous action of R on A by ∗ -automorphisms. Let B be a C ∗ -subalgebra of A . We setΨ ∗ ( A, α, R , , B ) = { x ∈ Ψ ∗ ( A, α, R ); σ − ( x ) ∈ B, σ + ( x ) = 0 } . The algebra Ψ ∗ ( A, α, R , , B ) is spanned by elements af ( Q α ) + b (1 + Q α ) − for a ∈ A, b ∈ B, f ∈ C ( R ∗ + ) = C ∗ ( R ) all sitting naturally as multipliers of A ⋊ α R . In this section, we recall a few facts on smooth groupoids: the pseudodiffelential calculus,the adiabatic groupoid G of a smooth groupoid G [22, 23], its ideal J ( G ) ([12, section 4.1]),the action of R ∗ + . We then extend all these to the case of an action of G on a C ∗ -algebra A .Recall that A G denotes the total space of the normal bundle of the inclusion of G (0) ⊂ G , A ∗ G the total space of its dual bundle, and S ∗ A G the associated sphere bundle, i.e. the setof half lines in A ∗ G . On every Lie groupoid G , there is a (longitudinal) pseudodifferential calculus. For every m ∈ R (and even for m ∈ C - [30, section 3]) we have a space P m ( G ) of classical pseudod-ifferential operators of order m (with polyhomogeneous symbol σ ∼ + ∞ X k =0 a m − k where a m − k ishomogeneous of order m − k ) and a symbol map which is a linear map σ m from P m ( G ) tohomogeneous functions of order m defined on A ∗ G (outside the zero section) - with kernel P m − ( G ).The smooth functions of M = G (0) define elements of P ( G ); the sections of the algebroiddefine elements of P ( G ). The algebra generated by these is the algebra of differential opera-tors. Given a positive definite quadratic form q on the bundle A ∗ G , we may find a (positive)laplacian ∆ G ∈ P ( G ) which is a positive and whose principal symbol is q .At the level of C ∗ -algebras we obtain an extension Ψ ∗ ( G ) of C ∗ ( G ) and an exact sequence oforder 0 pseudodifferential operators0 → C ∗ ( G ) −→ Ψ ∗ ( G ) σ −→ C ( S ∗ A G ) → . Recall ( cf. [7, 22, 23]) that Ψ ∗ ( G ) is the closure of the algebra P ( G ) of order zero pseu-dodifferential operators on G in the multiplier algebra of C ∗ ( G ) and σ is the (extension bycontinuity of the) principal symbol map. J ( G ) Let G be a Lie groupoid. We denote by M = G (0) its set of objects. The associated adiabaticgroupoid G ad is obtained by applying the “deformation to the normal cone” construction tothe inclusion M → G of the unit space of G into G . This construction was introduced by12onnes in the case of a pair groupoid G = M × M ([10, section II.5]), and generalized in[22, 23].As a set, and as a groupoid, G ad = A G × { } ∪ G × R ∗ + where A G is (the total space of) theLie algebroid of G , i.e. the normal bundle of the inclusion in G of the space of objects M of G ; its groupoid structure is given by addition of vectors - source and range coincide and arejust the bundle map A G → M . These sets are glued using an exponential map A G → G (see[22, 6, 12] for further details).The C ∗ -algebra of the adiabatic groupoid of G sits in an exact sequence0 → C ∗ ( G ) ⊗ C ( R ∗ + ) −→ C ∗ ( G ad ) ev −→ C ( A ∗ G ) → , where A ∗ G denotes the total space of the dual bundle to the Lie algebroid A G of G . Considerthe morphism ǫ : C ( A ∗ G ) → C ( M ) which associates to a function on A ∗ G its value onthe 0-section M of the bundle A ∗ G - i.e. the trivial representation of the group A x G . Wedenote by J ( G ) the kernel of ǫ ◦ ev , which is an ideal of C ∗ ( G ad ). We therefore have an exactsequence: 0 → J ( G ) → C ∗ ( G ad ) → C ( M ) → . Remark 4.1.
It follows from [17, Corollary 2.4], since M × R ∗ + is dense in M × R + that theideal C ( R ∗ + ) ⊗ C ∗ red ( G ) is essential in C ∗ red ( G ad ).Thanks to remark 1.2 we deduce that C ( R ∗ + ) ⊗ C ∗ ( G ) is also an essential ideal in C ∗ ( G ad ).As it contains C ( R ∗ + ) ⊗ C ∗ ( G ), the ideal J ( G ) is essential in C ∗ ( G ad ) both for the reducedand the full C ∗ -norm.Note also that the subset A ∗ G \ M is dense in A ∗ G (unless the groupoid G is r -discrete in thesense of [26, def. 2.6, p. 18] - i.e. the dimension of the algebroid is 0), and therefore ker ǫ isessential in C ( A ∗ G ). In this way we have another proof that J ( G ) is essential in C ∗ ( G ad ).We denote by τ the action of the group R ∗ + by groupoid automorphisms on G ad . Thisaction is given by τ t ( γ, u ) = ( γ, tu ) for γ ∈ G and t, u ∈ R ∗ + τ t ( x, U,
0) = ( x, t − U,
0) for( x, U ) ∈ A G ( x ∈ M ).We therefore get an action still denoted by τ of R ∗ + on C ∗ ( G ad ). Note that J ( G ) is invariantunder this action and that the quotient action of R ∗ + on C ∗ ( G ad ) /J ( G ) = C ( M ) is trivial.We will also use from [12, section 3.1] the dense subspaces S ( G ad ) of C ∗ ( G ad ) and J ( G ) of J ( G )consisting of smooth functions with Schwartz decay properties. Recall ([12, Theorem 3.7])that for f ∈ J ( G ) and m ∈ R , the operator Z + ∞ f t dtt m +1 is an order m pseudodifferentialoperator of the groupoid G i.e. an element of P m ( G ); its principal symbol σ is given by σ ( x, ξ ) = Z + ∞ ˆ f ( x, tξ, dtt m +1 · We now extend Baaj’s construction of the pseudodifferential extension to the case of an action α of a smooth groupoid G on a C ∗ -algebra A - in the sense of [20, 21] - see section 2.1.13 .3.1 Smooth elements Let G be a smooth groupoid with base M acting on a C ( M ) algebra A . We denote by α : s ∗ A → r ∗ A this action.We may define elements of A which are smooth along the action in the following way: • Let W be an open subset in G diffeomorphic to U × V where U ⊂ M is open and V is an open ball in R k , and such that r ( u, v ) = u . Then the C ( W ) algebra ( r ∗ A ) W isisomorphic to C ( V ; A U ); an element a ∈ r ∗ A is said to be of class C ∞ , if for everysuch W and f ∈ C ∞ c ( W ), we have f a ∈ C ∞ c ( V ; A U ) ⊂ C ( V ; A U ) ≃ A W . • An element a ∈ A is said to be smooth for the action of G if for all f ∈ C ∞ c ( G ), theelement α ( f. ( a ◦ s )) of r ∗ A is of class C ∞ , . Here f. ( a ◦ s ) is the class of a ⊗ f in s ∗ A - i.e. the restriction of a ⊗ f to the graph of s . In other words, we have (cid:16) α ( f. ( a ◦ s )) (cid:17) γ = f ( γ ) α γ ( a s ( γ ) ) . The smooth elements form a dense sub-algebra A ∞ of A . Indeed, if a ∈ A and f ∈ C ∞ c ( G ),the element f ∗ a given by ( f ∗ a ) x = Z G x f ( γ ) α γ a s ( γ ) dν x ( γ ) is easily seen to be smooth. Takethen a sequence f n with f n ∈ C ∞ c ( G ) positive with support tending to M and such that ν x ( f n ) = 1: we have f n ∗ a → a . Let G be a smooth groupoid with base M acting on a C ( M ) algebra A . Consider themorphism G ad → G × R + which is the identity on G × R ∗ + and satisfies ( x, ξ, ( x,
0) for x ∈ M = G (0) ⊂ G and ξ ∈ g x . Using this morphism, the adiabatic groupoid G ad acts onthe C ( R + × M )-algebra C ( R + ) ⊗ A : we have A x,t = A x (for t ∈ R + and x ∈ M ) and, for t ∈ R ∗ + , γ ∈ G and b ∈ A s ( γ ) , we have α γ,t ( b ) = α γ ( b ); for x ∈ M , ξ ∈ g x and b ∈ A x , we have α x,ξ, ( b ) = b .We have an exact sequence0 → ( A ⋊ α G ) ⊗ C ( R ∗ + ) → ( A ⊗ C ( R + )) ⋊ α G ad → A ⊗ C ( M ) C ( A ∗ G ) → . As the groupoid A G is amenable, the same exact sequence holds with reduced crossed prod-ucts.Note also that the action τ of R ∗ + extends on ( A ⊗ C ( R + )) ⋊ α G ad : it acts naturally on( A ⊗ C ( R + )) = C ( R + ; A ) by ( τ t ( a ))( u ) = a ( t − u ).We will also use the ideal J ( G , A ) ⊂ ( A ⊗ C ( R + )) ⋊ α G ad which is the kernel of the morphism( A ⊗ C ( R + )) ⋊ α G ad → A obtained as the composition( A ⊗ C ( R + )) ⋊ α G ad → A ⊗ C ( M ) C ( A ∗ G ) → A ⊗ C ( M ) C ( M ) = A. It is the closed vector span of elements f.a with f ∈ J ( G ) and a ∈ A . It is an essential idealin A ⊗ C ( R + )) ⋊ α G ad (see remark 4.1). Lemma 4.2. If a ∈ A is smooth for the G action and f ∈ S c ( G ad ) (cf. [12, section 3.1]),then k [ f t , a ] k A ⋊ α G = O ( t ) . roof. Note that f.a, a.f are in ( A ⊗ C ( R + )) ⋊ α G ad and since they are equal in A ⊗ C ( M ) C ( A ∗ G ), we find that k [ f t , a ] k A ⋊ α G → θ : V ′ → V be an “exponential map” which is a diffeomorphism of a (relatively compact)neighborhood V ′ of the 0 section M in A G onto a tubular neighborhood V of M in G .We assume that r ( θ ( x, U )) = x for x ∈ M and U ∈ A x G . Let W ′ = { ( x, U, t ) ∈ A G × R + ; ( x, tU ) ∈ V ′ } and W be the open subset W = A G × { } ∪ V × R ∗ + of G ad ; finally letΘ : W ′ × R + → W be the diffeomorphism defined by Θ( x, U,
0) = ( x, U,
0) and Θ( x, U, t ) =( θ ( x, tU ) , t ).If f ∈ S c ( R ∗ + × G ), then we have k [ f t , a ] k A ⋊ α G = O ( t n ) for all n .We may therefore assume that f is of the form g ◦ Θ where g ∈ S c ( W ′ ); then [ f t , a ] is theimage in A ⋊ α G of the function b t ∈ r ∗ A , where ( b t ) γ = f t ( γ ) (cid:16) a r ( γ ) − α γ (cid:0) a s ( γ ) (cid:1)(cid:17) .Note that there is a well defined element c ∈ ( r ◦ Θ) ∗ ( A ⊗ C ( R + )) given by c ( x,U,t ) = g ( x, U, t ) 1 t (cid:16) a x − α θ ( x,tU ) (cid:0) a s ( θ ( x,tU )) (cid:1)(cid:17) for t = 0 and − c ( x,U, is the derivative at 0 of t α θ ( x,tU ) (cid:0) a s ( θ ( x,tU )) (cid:1) , and f. ( c ◦ Θ − ) gives an element d ∈ ( A ⊗ C ( R + )) ⋊ α G ad ; we have td t = [ f t , a ]. a) For P ∈ Ψ ∗ ( G ) and a ∈ A sitting in M ( A ⋊ α G ) , we have [ P, a ] ∈ A ⋊ α G . b) The closed vector span of products aP where a ∈ A and P ∈ Ψ ∗ ( G ) is a C ∗ -subalgebra Ψ ∗ ( A, α, G ) ⊂ M ( A ⋊ α G ) . c) We have an exact sequence → A ⋊ α G −→ Ψ ∗ ( A, α, G ) σ α −→ A ⊗ C ( M ) C ( S ∗ A G ) → . Proof. a) We can assume P is in a dense subalgebra of Ψ ∗ ( G ) and a smooth. Whence, by[12, Theorem 3.7], we may choose P = Z + ∞ f t dtt where f = ( f t ) ∈ J ( G ). Then, byLemma 4.2, [ P, a ] is a norm converging integral of elements in A ⋊ α G .b) This closed subspace contains A ⋊ α G and its image in M ( A ⋊ α G ) / ( A ⋊ α G ) is a C ∗ -algebra since Ψ ∗ ( G ) and A commute in this quotient.c) Using (a) and the compatibility of the inclusions of C ( M ) in Ψ ∗ ( G ) and in M ( A ), wefind a morphism ̟ : C ( S ∗ A G ) ⊗ C ( M ) A → M ( A ⋊ α G ) / ( A ⋊ α G ) such that ̟ ( σ ( P ) ⊗ a )is the class of P a . We just have to show that ̟ is injective.Equivalently, we wish to show that A ⋊ α G is an essential ideal in the fibered product e Ψ ∗ ( G ; A ) = Ψ ∗ ( G ; A ) × ̟ ( C ( S ∗ A G )) C ( S ∗ A G ).We have a representation of e Ψ( G , A ) as multipliers of J ( G , A ) given, for ( T, σ ) ∈ e Ψ ∗ ( G ; A ), by (( T, σ ) f ) t = T f t for t = 0 and \ (( T, σ ) f ) )( x, ξ ) = σ ( x, ξ ) b f ( x, ξ ), where T ∈ Ψ ∗ ( G , A ) and σ ∈ C ( S ∗ A G ). This representation is faithful: indeed, if ( T, σ ) is inits kernel, taking its value at 0 it follows that σ = 0; therefore T ∈ A ⋊ α G ; but therepresentation of A ⋊ α G in J ( G ; A ) is faithful since A ⋊ α G ⊗ C ( R ∗ + ) ⊂ J ( G , A ).15ow as C ( R ∗ + ) ⊗ A ⋊ α G is an essential ideal in J ( G ; A ), it follows that the representation P ⊗ P of e Ψ ∗ ( G ; A ) on C ( R ∗ + ) ⊗ A ⋊ α G is faithful, whence A ⋊ α G is essential in e Ψ ∗ ( G ; A ). Let G be a smooth groupoid acting on the C ∗ -algebra A . In this section we prove the mainresults of this paper: • We construct an action of R on the associated C ∗ -algebra Ψ ∗ ( G , A ) of pseudodifferentialoperators - extending a construction sketched in [12, Remark 4.10]. • We establish the isomorphism J ( G , A ) ≃ Ψ ∗ ( G , A ) ⋊ β R - which was sketched in [12,Remark 4.10] in the case where A = C ( M ) and the action is trivial. • Finally we identify ( A ⊗ C ( R + )) ⋊ e α G ad as a pseudodifferential extension of the abovecrossed product. D of C ∗ ( G ad ) We first recall the construction of an unbounded multiplier D of C ∗ ( G ad ) which was given in[12, section 4.4].Let G be a longitudinally smooth groupoid with compact space of objects M = G (0) .Fix a metric on A G (and therefore on A ∗ G ) and choose a positive invertible pseudodifferentialoperator D on G with principal symbol σ D ( x, ξ ) = k ξ k . It is shown in [30, Prop. 21] that D is a regular multiplier of C ∗ ( G ). Proposition 5.1. (cf. [12, Prop. 4.8]) Let G be a Lie groupoid with compact set of objects G (0) = M and G ad its adiabatic groupoid. Fix a metric on A G (and therefore on A ∗ G )and choose a positive invertible pseudodifferential operator D on G with principal symbol σ D ( x, ξ ) = k ξ k . There is a unique regular unbounded multiplier D of C ∗ ( G ad ) satisfying: (i) the evaluation at of D is D ; (ii) we have β u ( D ) = uD for u ∈ R ∗ + .Moreover, a) The evaluation at of D , D , is the unbounded multiplier q of C ( A ∗ G ) = C ∗ ( A G ) where q ( x, ξ ) = k ξ k . b) The multiplier (1 + D ) − is in fact a strictly positive element of C ∗ ( G ad ) . c) For all f ∈ C ( R ∗ + ) we have f ( D ) ∈ J ( G ) . Moreover, the representation f f ( D ) isnon degenerate: if h ∈ C ( R ∗ + ) is strictly positive in R ∗ + , then f ( D ) is a strictly positiveelement of J ( G ) .. roof. If D satisfies (i) and (ii), then D u = uD for all u >
0, and this establishes uniquenessof D .Choose a finite family ( X , . . . , X m ) of sections of A G in such a way that the embedding ξ
7→ h X i | ξ i is an isometry from A ∗ G to the trivial bundle. In [12, prop. 4.8], we constructedan unbounded multiplier, call it e D such that e D = (cid:16) X X ∗ i X i + 1 (cid:17) / , e D = q and e D u = u e D for u ∈ R ∗ + . Now, D − e D is a 0-order operator, whence bounded. We may then define anunbounded multiplier D by putting D u = e D u + u ( D − e D ) and D = e D .Let us prove property (b).Let c ∈ R ∗ + . Since M × [0 , c ] is compact and D is elliptic of order 1 ([30, Th. 18 and Prop.21]), the restriction of (1+ D ) − to ( G ad ) | [0 ,c ] is in C ∗ ( G ad ) | [0 ,c ] . Let m ∈ R ∗ + such that D ≥ m ,we have 1 + D u ≥ um and therefore k (1 + D u ) − k ≤ (1 + um ) − . It follows that (1 + D ) − belongs to C ∗ ( G ad ).Now, (1 + D ) − C ∗ ( G ad ) is the domain of the multiplier D , whence it is dense, and (1 + D ) − is strictly positive.Property (c) follows from [12, Prop. 4.8.b)]. Note that our D here is slightly more generalthan the one used there, but the same proof applies. R on Ψ ∗ ( G , A ) Let S ∈ P / ( G ) be a positive elliptic pseudodifferential operator of order 1 / S such that σ / ( S ) = ( σ (∆ G )) / where ∆ G is a laplacian as defined in the section 4.1. Denoteby ∂ S the associated derivation on M ( A ⋊ α G ) (see appendix - facts 6.3). Lemma 5.2.
Every smooth element a ∈ A and every classical pseudodifferential P on G oforder are in the domain of the derivation ∂ S .Proof. We may write S = R + S where S = Z + ∞ f t t − / dt , ( f t ) is a positive element in J ( G ) and R ∈ P − / ( G ). This integral means that dom S is the set of x ∈ A ⋊ α G suchthat the integral Z + ∞ f t x t − / dt converges in norm to some y ∈ A ⋊ α G and then S x = y .(Indeed, by [30, Prop. 21], S is selfadjoint regular and it is clear that ( x, y ) is then in thegraph of S ∗ ).Since a is assume to be smooth, the integral Z + ∞ [ f t , a ] t − / dt converges in norm (by Lemma4.2) to some element b ∈ A ⋊ α G . Then, for x ∈ dom S = dom S , the sequence Z + ∞ /n f t ax t − / dt = Z + ∞ /n ( af t + [ f t , a ]) x t − / dt converges in norm to aS x + bx . Thus ax ∈ dom S = dom S . It follows that a ∈ dom ∂ S and ∂ S ( a ) = b + [ R, a ].If P ∈ P ( G ), the operator ( S + 1) / P ( S + 1) − / ∈ Ψ ∗ ( G ); it follows that P dom S ⊂ dom S . Moreover, [ S, P ] ∈ P / ( G ) and since σ / [ S, P ] = [ σ / ( S ) , σ / ( P )] = 0, we find[ S, P ] ∈ P − / ( G ) ⊂ C ∗ ( G ). 17 roposition 5.3. Let D ∈ P ( G ) be any positive invertible pseudodifferential operator el-liptic of order . Then we have an action β of R on Ψ ∗ ( G ; A ) given by β t ( P ) = D it P D − it .This action is trivial at the symbol level.Proof. By [30, Theorem 41] there exists S ∈ P / positive elliptic of order 1 / T ∈ C ∗ ( G )such that p D = S + T . It follows by Lemma 5.2, that with a, P as above P a ∈ dom ∂ √ D .Since D − / ∈ A ⋊ α G , it follows from Lemma 6.6, that P a ∈ dom ∂ ln D and [ln D , aP ] ∈ A ⋊ α G . The conclusion follows from Lemma 6.4. Ψ ∗ ( G , A ) ⋊ R ≃ J ( G , A ) In [12, prop. 4.2.b)], we constructed a morphism φ : Ψ ∗ ( G ) → M ( J ( G )) such that, for P ∈ Ψ ∗ ( G ) and f = ( f u ) in J ( G ) we have ( φ ( P )( f )) u = P ∗ f u , for u = 0 and ( φ ( P )( f )) = σ ( P ) f thanks to [12, prop. 4.2.b)]. Now J ( G ) sits in a non degenerate way in M ( J ( G , A )). Also,by definition A embeds in a compatible way in M ( J ( G , A )).In this way, we find a morphism φ : Ψ ∗ ( G , A ) → M ( J ( G , A )) such that, for P ∈ Ψ ∗ ( G , A )and f = ( f t ) in J ( G , A ) we have ( φ ( P )( f )) u = P ∗ f u , for u = 0 and ( φ ( P )( f )) = σ ( P ) f .Furthermore, the operator D recalled in section 5.1 yields a one parameter group ( D it ) t ∈ R in M ( J ( G )); we will still denote by ( D it ) t ∈ R its image in M ( J ( G , A )).As D u and D are scalar multiples of each other, we find in this way a covariant representationof the pair (Ψ ∗ ( G ) , β, R ) (prop. 5.3).Associated to this covariant representation is a morphism from Ψ ∗ ( G , A ) ⋊ β R into the mul-tiplier algebra of J ( G ), but since the image of C ∗ ( R ) ⊂ Ψ ∗ ( G ) ⋊ β R is contained in J ( G ), weget a homomorphism ϕ : Ψ ∗ ( G , A ) ⋊ β R → J ( G , A ). For P ∈ Ψ ∗ ( G , A ) ⊂ M (Ψ ∗ ( G , A ) ⋊ β R )and f ∈ C ∗ ( R ) = C ( R ∗ + ) ⊂ M (Ψ ∗ ( G , A ) ⋊ β R ), we have ( ϕ ( P f )) = φ ( P ) f ( D ). Proposition 5.4.
The homomorphism ϕ is an equivariant isomorphism from (Ψ ∗ ( G , A ) ⋊ β R , ˆ β ) to ( J ( G , A ) , τ ) .Proof. The images of the elements of Ψ ∗ ( G , A ) are translation invariant, i.e. invariant by theextension τ u of τ u to the multiplier algebra, and τ u ( D it ) = u it D it . This shows that ϕ is anequivariant morphism from (Ψ ∗ ( G , A ) ⋊ β R , ˆ β ) to ( J ( G , A ) , τ ).Now β t restricts to an action of R on C ∗ ( G ), and according to [12, prop. 4.2.a)] it follows that ϕ extended to the multipliers defines a morphism from C ∗ ( G ) ⋊ β R into the ideal C ( R ∗ + ) ⊗ C ∗ ( G ) of J ( G ). It follows that ϕ ( A ⋊ α G ) is contained in the ideal A ⋊ α G ⊗ C ( R ∗ + ) of J ( G , A ).We thus have the diagram:0 / / ( A ⋊ α G ) ⋊ β R / / ϕ ′ (cid:15) (cid:15) Ψ ∗ ( G , A ) ⋊ β R / / ϕ (cid:15) (cid:15) ( A ⊗ C ( M ) C ( S ∗ A G )) ⋊ β R / / ϕ ′′ (cid:15) (cid:15) / / ( A ⋊ α G ) ⊗ C ( R ∗ + ) / / J ( G , A ) / / A ⊗ C ( M ) C ( A ∗ G \ M ) / / D is an unbounded invertible multiplier of C ∗ ( G ) and therefore of A ⋊ α G , the action β of R on A ⋊ α G is inner. It follows that the crossed product ( A ⋊ α G ) ⋊ β R identifies with( A ⋊ α G ) ⊗ R ∗ + . This isomorphism is defined in the following way: the canonical multipliersof the crossed product, i.e. the generators a ∈ A ⋊ α G and λ t for t ∈ R map to the functions18 a and u u it D it from R ∗ + to M ( A ⋊ α G ). It follows, the image of af with a ∈ C ∗ ( G )and f ∈ C ∗ ( R ) = C ( R ∗ + ) is af ( D ). This isomorphism identifies thus with ϕ ′ .The action β is trivial on symbols; thus ( A ⊗ C ( M ) C ( S ∗ A G )) ⋊ β R is equal to ( A ⊗ C ( M ) C ( S ∗ A G )) ⊗ C ( R ∗ + ), and ϕ ′′ ( σ ⊗ f ) = σf ( q ) is the isomorphism corresponding to the home-omorphism A ∗ G \ M ≃ S ∗ A G × R ∗ + given by ξ ( ξ/q ( ξ ) , q ( ξ )). The result follows. The algebra A sits in Ψ ∗ ( G , A ) as (the closure of) order 0 differential operators. Denote by ϑ : A → Ψ ∗ ( G ; A ) the corresponding morphism. The element ϑ ( a ) as a multiplier of A ⋊ α G ,is just the multiplication by a . Remark 5.5.
Using at the non degenerate morphism Ψ ∗ ( G , A ) → M ( J ( G ; A )) we thenobtain a morphism ˆ ϑ : A → M (Ψ ∗ ( G , A ) ⋊ R ).Also the algebra A is in the multiplier algebra of A ⊗ C ( R + ) end thus we have an embedding˜ ϑ : A → M (( A ⊗ C ( R + )) ⋊ ˜ α G ad ) - which is a subalgebra of M ( J ( G ; A )) since J ( G ; A ) is anessential ideal in ( A ⊗ C ( R + )) ⋊ ˜ α G ad .We now use the notation of paragraph 3.4. The main result of this paper is : Theorem 5.6.
The isomorphism ϕ : Ψ ∗ ( G , A ) ⋊ β R → J ( G , A ) extends uniquely to an iso-morphism of Ψ ∗ (Ψ ∗ ( G , A ) , β, R , , A ) with ( A ⊗ C ( R + )) ⋊ e α G ad . This isomorphism intertwinesthe actions β and τ of R .Proof. The isomorphism ϕ : Ψ ∗ ( G , A ) ⋊ β R → J ( G , A ) extends to an isomorphism Φ of themultiplier algebras. Since the ideals Ψ ∗ ( G , A ) ⋊ β R ⊂ Ψ ∗ (Ψ ∗ ( G , A ) , β, R ) and J ( G , A ) ⊂ ( A ⊗ C ( R + )) ⋊ α G ad are essential, we just need to show that Φ(Ψ ∗ (Ψ ∗ ( G , A ) , β, R , , A )) =( A ⊗ C ( R + )) ⋊ e α G ad .It follows from proposition 3.2.a) that the morphism Φ coincides on Ψ ∗ ( G , A ) with the mor-phism φ : Ψ ∗ ( G , A ) → M ( J ( G , A )) of section 5.3 and that the image of the unboundedmultiplier Q β (see section 3.4) is D .With the notation introduced in remark 5.5, one easily checks that Φ ◦ ˆ ϑ = ˜ ϑ .We deduce that Φ (cid:16) Ψ ∗ (Ψ ∗ ( G , A ) , β, R , , A ) (cid:17) is spanned by ϕ (Ψ ∗ ( G , A ) ⋊ β R ) = J ( G , A ) and(1 + D ) − ˜ ϑ ( a ) where and a over A .Since (1 + D ) − ∈ C ∗ ( G ad ) (prop. 5.1.b)), and for a ∈ A we have (1 + D ) − ˜ ϑ ( a )) ∈ ( A ⊗ C ( R + )) ⋊ ˜ α G ad ).Finally Φ induces a homomorphism e ϕ : Ψ ∗ (Ψ ∗ ( G , A ) , β, R , , A )) → ( A ⊗ C ( R + )) ⋊ e α G ad .Moreover, since ev ( D ) = q which vanishes at the 0 section of A ∗ G , we find that ǫ ◦ ev ((1 + D ) − ) = 1, whence ǫ ◦ ev (Φ((1 + D ) − θ ( a )) = a . We thus have a commutative diagramwhere the sequences are exact:0 / / Ψ ∗ ( G , A ) ⋊ β R / / ϕ (cid:15) (cid:15) Ψ ∗ (Ψ ∗ ( G , A ) , β, R , , A ) / / Φ (cid:15) (cid:15) A / / id A (cid:15) (cid:15) / / J ( G , A ) / / ( A ⊗ C ( R + )) ⋊ ˜ α G ad ) / / A / / e ϕ is an isomorphism.By uniqueness of the extension to multipliers, we deduce that ˆ β t ◦ e ϕ = e ϕ ◦ τ t for all t ∈ R ∗ + .19ecall that the gauge adiabatic groupoid G ga is the semi-direct product G ga = G ad ⋊ τ R ∗ + . If G acts on A , then G ga acts on A ⊗ C ( R + ). Corollary 5.7.
We have isomorphisms ( A ⊗ C ( R + )) ⋊ α G ga ≃ Ψ ∗ (Ψ ∗ ( G , A ) , β, R , , C ( M )) ⋊ ˆ β R ∗ + ≃ Ψ ∗ (Ψ ∗ ( G , A ) ⊗ C ( R ) , β ⊗ λ, R , , C ( M ) ⊗ C ( R )) . Proof.
We have ( A ⊗ C ( R + )) ⋊ α G ga = (( A ⊗ C ( R + )) ⋊ α G ad ) ⋊ τ R ∗ + . The first isomorphismis a direct consequence of theorem 5.6; the second one comes from prop. 3.4. Remark 5.8.
Let us drop the algebra A . The exact sequence0 → C ∗ ( G ) ⊗ K → C ∗ ( G ga ) ⋊ R ∗ + → C ( A ∗ G ) ⋊ R ∗ + → KK ( C ( A ∗ G ) ⋊ R ∗ + , C ∗ ( G ) ⊗ K ). Using Connes’ Thom isomor-phism ( cf. [8, 14]), this group is isomorphic to KK ( C ( A ∗ G ) , C ∗ ( G )). In fact, using again theThom isomorphism, this element corresponds to the ext element in KK ( C ( A ∗ G ) , C ∗ ( G ) ⊗ C ( R ∗ + )) of the exact sequence0 → C ∗ ( G ) ⊗ C ( R ∗ + ) → C ∗ ( G ad ) → C ( A ∗ G ) → . One easily sees (using e.g. [22, Theorem 2.1]) that this element is the analytic index.Let µ : C ( M ) → Ψ ∗ ( G ) be the inclusion, and let C µ be the corresponding mapping cone. Wehave an exact sequence 0 → Ψ ∗ ( G ) ⊗ C ( R ∗ + ) → C µ → C ( M ) → . The quotient of C µ by the ideal C ∗ ( G ) ⊗ C ( R ∗ + ) is the cone of the inclusion C ( M ) → C ( S ∗ g ),which is naturally isomorphic to C ( A ∗ G ). We thus find an exact sequence0 → C ∗ ( G ) ⊗ C ( R ∗ + ) → C µ → C ( A ∗ G ) → . The corresponding KK -element can be seen again to be the analytic index element in KK ( C ( A ∗ G ) , C ∗ ( G )). Taking crossed product by the natural action of R ∗ + on C µ (justby rescaling), we find an exact sequence0 → K → C µ ⋊ R ∗ + → C ( T ∗ M ) ⋊ R ∗ + → . In the case of the pair groupoid, we deduce an isomorphism C µ ⋊ R ≃ C ∗ ( G ga ) thanks toVoiculescu’s theorem ([31, Theorem 1.5]).It is a natural question to decide whether this isomorphism extends to the general case. Onthe other hand, this isomorphism is not “natural”. Indeed, C µ and C ∗ ( G ad ) are not isomorphicin general, whence there is no isomorphism C µ ⋊ R ≃ C ∗ ( G ga )(= C ∗ ( G ad ) ⋊ R ∗ + ) equivariantwith respect to the dual actions. 20 Appendix: some facts on unbounded operators
In this appendix, we recall a few rather classical abstract facts about unbounded operatorsthat we used in the text. These facts are presented here in a form suitable for our expositionand certainly not in their most general forms. They can be found in (or deduced directlyfrom) [2, 32] - see also [30].Let E be a C ∗ -module (over a C ∗ -algebra) and L a regular (densely defined, unbounded)self-adjoint operator on E . Facts 6.1.
Let us recall a few facts about unbounded functional calculus, f f ( L ) ( cf. [2, 32]).a) Put h ( t ) = ( i + t ) − ; there exists a unique morphism π L : f f ( L ) from C ( R ) to L ( E ) such that π L ( h ) = ( L + i id E ) − .b) Since h ( L ) has a dense range (dom L ), this morphism is non degenerate, it extends toa morphism f f ( L ) from C b ( R ) = M ( C ( R )) to L ( E ).c) If f ∈ C ( R ), define the operator f ( L ) whose domain is the range of g ( L ) where g ( t ) =( | f ( t ) | + 1) − and such that f ( L ) g ( L ) = ( f g )( L ).d) If f, g ∈ C ( R ) are such that f | g | + 1 is bounded, then dom g ( L ) ⊂ dom f ( L ).e) If ( f n ) is an increasing sequence of positive elements of C b ( R ) converging simply (andtherefore uniformly on compact subsets of R ) to a continuous function f , then thedomain of f ( L ) is the set of x ∈ E such that ( f n ( L ) x ) converges (in norm) and then f ( L ) x is the limit of this sequence.Indeed, as f n + 1 f + 1 = h n converges to 1 for the topology of C b ( R ) : • if x is in the domain of f ( L ), it is of the form x = ( f ( L ) + 1) − z , and x + f n ( L ) x = h n ( L ) z converges to z , therefore f n ( L ) x converges to z − x ; • ( f ( L )+id E ) − ( f n ( L ) x + x ) = h n ( L ) x converges to x ; assume that f n ( L ) x convergesto y ∈ E , then ( x, x + y ) is the limit of the sequence (cid:16) h n ( L ) x, ( f n ( L ) x + x ) (cid:17) ofelements of the graph of f ( L ) + id E ; therefore y = f ( L ) x since the graph of f ( L )is closed. Lemma 6.2.
We have an equality L = Z + ∞ (cid:16) s − ( e L + s ) − (cid:17) ds − Z ( e L + s ) − ds which means that dom L is the set of x ∈ E such that the integrals Z + ∞ (cid:16) s − ( e L + s ) − (cid:17) x ds and Z ( e L + s ) − x ds are norm convergent and Lx is then the difference of these two integrals. roof. Put f n ( t ) = Z n (cid:16) s − ( e t + s ) − (cid:17) ds and f ( t ) = lim f n ( t ) = ln( e t + 1); put also g n ( t ) = Z n ( e t + s ) − ds and g ( t ) = lim g n ( t ) = ln( e t + 1) − t .Then as ln( e t + 1) | t | + 1 is bounded, dom L = dom f ( L ) ∩ dom g ( L ) (by fact 6.1.d). The conclusionfollows from fact 6.1.e). Fact 6.1.f ).
Assume L is positive with resolvent in K ( E ). Then f f ( L ) is a morphism π L : C ( R ∗ + ) → K ( E ). Note that, for t ∈ R ∗ + , we have π tL = π L ◦ λ t where λ t is theautomorphism of C ( R ∗ + ) induced by the regular representation. Since t tt + 1 is astrictly positive element of C ( R ∗ + ), it follows that π L ( C ( R ∗ + )) E is the closure of theimage of L ( L + 1) − . Facts 6.3 (about derivations) . We will consider the (unbounded, skew adjoint) derivation ∂ L associated with L : its domain is the ∗ -subalgebra of the elements a ∈ L ( E ), such thatthere exists ∂ L ( a ) ∈ L ( E ) with aL ⊂ La + ∂ L ( a ) (in other words a dom L ⊂ dom L and [ a, L ]defined on dom L extends to an operator ∂ L ( a ) ∈ L ( E )).Put u t = exp( itL ) and define for a ∈ L ( E ), β t ( a ) = u t au ∗ t .a) For a ∈ L ( E ), the map t β t is of class C (for the norm topology) if and only if a ∈ dom ∂ L and, in that case d/dt ( β t ( a )) = i∂ L ( β t ( a )) = iβ t ( ∂ L ( a )).b) The closure dom ∂ L of dom ∂ L is a C ∗ -subalgebra of L ( E ) and t β t ( a ) is a continuousaction of R on it. Lemma 6.4.
Let Q be the norm closure of { a ∈ dom ∂ L ; ∂ L a ∈ K ( E ) } . It is a C ∗ -subalgebraof dom ∂ L invariant under the action β of R . The quotient action of R on Q/ K ( E ) is trivial.In particular, every C ∗ -subalgebra of Q containing K ( E ) is invariant by β .Proof. Denote by q : L ( E ) → L ( E ) / K ( E ) the quotient map. If a ∈ dom ∂ L satisfies ∂ L a ∈K ( E ), then t β t ( a ) is C , and the derivative of t q ( β t ( a )) is zero. All other statementsare clear. Lemma 6.5.
Let a ∈ dom ∂ e L ∩ ∂ e − L . Then a ∈ dom ∂ L . If the resolvent of L is in K ( E ) ,then ∂ L ( a ) ∈ K ( E ) .Proof. The integral Z + ∞ h s − ( e L + s ) − , a i ds = Z + ∞ ( e L + s ) − [ e L , a ]( e L + s ) − ds is normconvergent (since k ( e L + s ) − k ≤ s − ), as well as − Z h ( e L + s ) − , a i ds = Z ( e L + s ) − h e L , a i ( e L + s ) − ds = − Z e L ( e L + s ) − h e − L , a i e L ( e L + s ) − ds. (since k e L ( e L + s ) − k ≤ f n − g n )( L ) , a ] converges to an element b = Z + ∞ ( e L + s ) − [ e L , a ]( e L + s ) − ds . If x ∈ dom L , then ( f n − g n )( L ) ax converges to aLx + bx ; therefore ax ∈ dom L and ∂ L ( a ) = b .Assume L has compact resolvent ( i.e. in K ( E )). Put q s = ( e L + s ) − [ e L , a ]( e L + s ) − . Notethat e L q s is bounded and, since q s = − ( e L + s ) − e L h e − L , a i e L ( e L + s ) − , e − L q s is also bounded.If L has compact resolvant, then ( e L + e − L ) − ∈ K ( E ), whence q s ∈ K ( E ). Lemma 6.6.
Assume L is positive. Let a ∈ L ( E ) such that a dom e L ⊂ dom e L and e − L/ [ e L , a ] defined on dom e L extends to an element of L ( E ) . Then a ∈ dom ∂ L . If moreoverthe resolvant of L is in K ( E ) , then ∂ L ( a ) ∈ K ( E ) .Proof. The integral Z + ∞ h s − ( e L + s ) − , a i ds = Z + ∞ ( e L + s ) − [ e L , a ]( e L + s ) − ds is normconvergent, since k ( e L + s ) − [ e L , a ]( e L + s ) − k ≤ k e L/ ( e L + s ) − kk e − L/ [ e L , a ] kk ( e L + s ) − k ≤ s − / Cs − for C = k e − L/ [ e L , a ] k .Of course the integral − Z h ( e L + s ) − , a i ds is also norm convergent.It follows, with the notation of Lemma 6.2 that [( f n − g n )( L ) , a ] converges to an element b = Z + ∞ ( e L + s ) − [ e L , a ]( e L + s ) − ds . If x ∈ dom L , then ( f n − g n )( L ) ax converges to aLx + bx ; therefore ax ∈ dom L and ∂ L ( a ) = b .Assume L has compact resolvant. Then, since L is positive, ( e L + s ) − ∈ K ( E ), whence( e L + s ) − [ e L , a ]( e L + s ) − ∈ K ( E ). References [1] C. Anantharaman-Delaroche and J. Renault,
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