A Schwartz type algebra for the Tangent Groupoid

A Sc h w artz t yp e algebra fo r the T angen t group oi d P aulo Carrillo Rouse Abstract. W e construct an algebra of smooth functions ov er the tangent group oid associated to an y Lie group oid. This algebra is a field of algebras o ver the closed interv al [0 , 1] whic h fib er at zero is the algebra of Schw artz functions ov er the Lie algebroid, whereas any fib er out of zero is the conv olution algebra of the initial groupoid. Our motiv ation comes from index theory for Lie group oi ds. In fact, our construction giv es an intermedia te algebra betw een the env eloping C ∗ -algebra and t he conv olution al gebra of compactly supp orted functions of the tangen t group oid; and it will allo ws u s, in a further w ork, to defi ne other analytic ind ex morphisms as a sort o f ”deformations”. Mathematics Sub ject Classification (2000). Primary 58-06, 19-06; Secondary 58H15, 19K56. Keywords. Lie group oids, T angent group oid, K-theory , Index theory . 1. In tro duction The concept of group oid is cen tral in non comm utative geometry . Group oids gener - alize the co ncepts of spaces, groups a nd equiv alence relations. It is clear now adays that group o ids ar e natural substitutes of singular spaces. Many p eople ha ve co n- tributed to r ealizing this idea. W e ca n find fo r instance a g roupoid-like treatment in Dixmier ’s works o n transforma tion groups, [1 1 ], or in Brown-Green-Rieffel’s work on or bit classifica tion of relations , [3]. In fo lia tion theory , several mo dels for the lea f space of a fo liation w ere rea lized using group oids, mainly by p eople like Haefliger ([12 ]) and Wilk elnkemper ([24]), for mention some o f them. There is also the case of Orbifolds, these can b e s een indeed as ´ etale group oids, (see for example Mo erdijk’s pa per [1 7]). There are als o some particular g roupoid mo dels for manifolds with c o rners and conic manifoldss w orked b y people like Month ub ert [18], Deb ord-Lescure- Nistor ([10]) and Aastr up-Melo-Mon thubert-Schrohe ([1 ]) fo r example. The wa y we trea t ”s ingular spaces” in non commutativ e geometr y is by asso- ciating to them algebr as. In the case when the ”sing ular space” is represe n ted by a Lie group oid, we can, for ins ta nce, c o nsider the conv olution alg ebra of differ e n- tiable functions with compact s upp ort ov er the group oid (see Connes o r Paterso n’s bo oks [8] a nd [22]). This las t algebr a plays the role of the algebr a of smo oth func- 2 P aulo Carrillo Rouse tions o ver the ”singular spa ce” represented by the group oid. F rom the conv olution algebra it is a lso p ossible to co nstruct a C ∗ -algebra , C ∗ ( G ), that plays, in some sense, the role of the a lgebra of contin uous functions ov er the ”singular space ” . The idea of ass ociating algebra s in this sense ca n b e traced back in w orks of Dixmier ([11]) for trans formation groups, Connes ([6]) for foliations and Renault ([23]) for lo cally compact gro up oids, fo r mention some o f them. Using metho ds of Noncommutativ e Geometry , we would like to get inv ari- ants of this algebra s, and hence, of the spaces they represent. F or that, Connes show ed that man y gr oupoids a nd algebras asso ciated to them app eared as ‘no n commutativ e ana logues‘ of s paces to whic h man y tools of g e ometry (and topol- ogy) such a s K -theory and Cha racteristic classes co uld be applied ([7], [8]). One classical way to obtain inv ariants in classica l geometry (top ology), is throug h the index theory in the sense of Atiy ah-Singer. In the Lie group oid case, there is a Pseudo differen tial calculus, developed by Connes ([6]), Month ub ert-Pierrot ([19]) and Nistor - W einstein-Xu ([21]) in general. Some interesting pa rticular cas es were treated in the group oid-spirit by Melr o se ([16]), Mo roian u ([20]) a nd others (see [1]). Let G ⇒ G (0) be a Lie g roupoid, there is an a nalytic index morphism, (see [19]), ind a : K 0 ( A ∗ G ) → K 0 ( C ∗ ( G )) , where A G is the L ie alg ebroid of G . The ” C ∗ -index” ind a is a homotopy inv aria n t of the G -pseudo differen tial elliptic op erators and has proved to b e very useful in very different situations, (see [2 ], [9 ], [10] for example). O ne wa y to define the ab ov e index map is using the Connes’ tangent group oid asso ciated to G as ex pla ined b y Hilsum and Sk andalis in [13] or b y Month uber t and Pier rot in [19]. The ta ng en t group oid is a Lie group oid G T ⇒ G (0) × [0 , 1] with G T := A G × { 0 } F G × (0 , 1 ] and the group oid structure is g iv en by the group oid structure of A G at t = 0 and by the gr oupoid structure o f G for t 6 = 0. One of the main features a bout the tangent gro upoid is that its C ∗ − algebra C ∗ ( G T ) is a contin uous field of C ∗ − algebra s over the closed interv al [0 , 1], with asso ciated fiber algebra s C 0 ( A ∗ G ) at t = 0, a nd C ∗ ( G ) for t 6 = 0. In fact, it gives a C ∗ − algebra ic quan tization of the Poisson manifold A ∗ G (in the sense o f [14]), and this is the main p oint why it allows to define the index morphism as a sort of ”defor mation”. Thu s, the tangent g roupoid co nstruction has been very useful in index theory ([2], [1 0 ], [13]) but a ls o for other purp oses ([14], [2 1]) . Now, to understand the purp ose of the pr esen t work, let us fir st say that the indices (in the sense of At iyah-Singer-Co nnes ) hav e not necessarily to b e consider e d as elements in K 0 ( C ∗ ( G )). Indeed, it is po ssible to consider indices in K 0 ( C ∞ c ( G )). The C ∞ c -indices are more refined but they ha ve several inco n v enients (see Alain Connes’ bo ok sectio n 9 .β for a dis c ussion on this matter), nevertheless this kind A Sch w artz typ e algebra for the T angent groupoid 3 of indices hav e the g reat adv antage that one can apply to them the e xisten t to ols (such as pairings with cyclic co cycles o r Chern-Connes character) in orde r to obtain nu merica l in v ariants. In this work we be g in a study of more refined indices . In particular we ar e lo oking for indices b et ween the C ∞ c and the C ∗ -levels; trying to keep the adv a n tages of both approa c hes (see [5 ] for a mor e complete dis cussion). In the cas e of Lie group oids this refinement could mean forget for a momen t the powerful to ols of the theory of C ∗ -algebra s and instead, working in a pure ly a lgebraic and geo metric level. In the pr e sen t article, w e co nstruct an algebra of C ∞ functions over G T , denoted by S r,c ( G T ). This alg ebra is also a field of alg ebras ov er the closed interv al [0 , 1], with asso ciated fib er algebras, S ( A G ), at t = 0, a nd C ∞ c ( G ) for t 6 = 0, where S ( A G ) is the Sch w ar tz algebr a of the Lie algebro id. F urthermore, we will hav e C ∞ c ( G T ) ⊂ S r,c ( G T ) ⊂ C ∗ ( G T ) , (1) as inclusio ns of algebr as. Let us expla in in so me words why we define a n alg ebra ov er the tangent gro upoid such that in zero it is Sch w ar tz: The ”Schw a rtz a lg ebras” hav e in g eneral the go od K − theory gro ups. F or exa mple, we are in terested in the symbols of G -PDO and mo re pr ecisely in their homo top y c la sses in K -theory , that is, we are interested in the gr oup K 0 ( A ∗ G ) = K 0 ( C 0 ( A ∗ G )). Here it would not b e enough to ta k e the K − theor y of C ∞ c ( A G ) (see the exa mple in [8] p.1 42), how ever it is enough to consider the Sch wartz algebr a S ( A ∗ G ). Indeed, the F ourier trans f or m shows that this last algebra is s table under holomorphic calculus on C 0 ( A ∗ G ) and so it has the ”go od” K -theory , meaning that K 0 ( A ∗ G ) = K 0 ( S ( A ∗ G )). None of the inclus ions in (1) is stable under holomor phic calculus, but that is prec is ely what we w anted beca us e our algebra S r,c ( G T ) hav e the rema r k able prop ert y that its ev aluation a t zer o is stable under holo morphic calculus while its ev aluation at one (for example) is not. The algebr a S r,c ( G T ) is, as vector space, a pa r ticular ca s e of a mor e genera l construction that we do for ” Def or mation to the normal c one manifolds” from which the tangent group oid is a sp ecial case (see [4] and [13]). A defor ma tion to the normal cone manifold (DNC fo r simplify) is a manifold as sociated to an injective immersion X ֒ → M that is considered as a so rt o f blo w up in differen tial geometry . The co nstruction of a DNC manifold has v ery nice functoria l pro perties (se c tion 3) which we exploit to achieve our constr uction. W e think that o ur construction could b e used als o for other purp oses, for example, it se ems tha t it could help to give more under standing in quantization theory (see again [4]). The article is org anized as follows. In the second section w e r e c all the ba sic facts ab out Lie g roupoids . W e explain very briefly how to define the co n volution algebra C ∞ c ( G ). In the third section we explain the ” deformation to the normal cone” construction a ssocia ted to an injective immer s ion. E v en if this could be considered as clas sical materia l, we do it in some detail since we will use in the 4 P aulo Carrillo Rouse sequel very explicit descriptions that we co uld not find els ewhere. W e also review some functorial pro perties asso ciated to thes e deformations. A particular case of this cons tr uction is the tangent gr oupoid a s sociated to a Lie group oid. In the four th section we start by constructing a vector spa ce S r,c ( D M X ) for a n y Deforma tion to the normal cone manifold D M X ; this space alrea dy exhibits the c hara cteristic of being a field of vector spaces o ver the closed in terv al [0 , 1], suc h that in zer o we hav e a Sc hw artz space while o ut o f zero we have C ∞ c ( M ). W e then define the algebra S r,c ( G T ), the main result is pre c isely that the product is well defined. The last section is devoted to motiv a te the co nstruction of our alg ebra by expla ining in a few words some fur ther dev elop emen ts that will immediately follow from this work. All the res ults of the pres e n t work ar e part of the author ’s PHD thesis. I wan t to tha nk m y PHD adviso r, Geo r ges Sk a ndalis, for all the ideas that he shared with me. I would also like to thank him for all the co mmen ts and remark s he made to the present work. I would also like to thank the referee for the useful comments he made for improving this pap er. 2. Lie group oids Let us recall wha t a group oid is: Definition 2 . 1. A gr oup oid consists of the following data: t wo sets G and G (0) , and maps · s, r : G → G (0) called the s ource and target ma p resp ectiv ely , · m : G (2) → G called the product map (where G (2) = { ( γ , η ) ∈ G × G : s ( γ ) = r ( η ) } ), · u : G (0) → G the unit ma p and · i : G → G the in verse ma p such that, if we note m ( γ , η ) = γ · η , u ( x ) = x a nd i ( γ ) = γ − 1 , we have 1. γ · ( η · δ ) = ( γ · η ) · δ , ∀ γ , η , δ ∈ G when this is p ossible. 2. γ · x = γ and x · η = η , ∀ γ , η ∈ G with s ( γ ) = x a nd r ( η ) = x . 3. γ · γ − 1 = u ( r ( γ )) and γ − 1 · γ = u ( s ( γ )), ∀ γ ∈ G . 4. r ( γ · η ) = r ( γ ) a nd s ( γ · η ) = s ( η ). Generally , w e denote a gro upoid by G ⇒ G (0) where the pa rallel arrows are the source and target maps a nd the o ther maps are given. Now, a Lie group oid is a g roupoid in which ev ery set and map app earing in the last definition is C ∞ (po ssibly with b o rders), and the source and target maps are submersions. F o r A, B subse ts o f G (0) we use the notation G B A for the subset { γ ∈ G : s ( γ ) ∈ A, r ( γ ) ∈ B } . All along this paper , G ⇒ G (0) is going to b e a Lie g roupoid. W e recall how to define an a lgebra structure in C ∞ c ( G ) using smo oth Haa r systems. A Sch w artz typ e algebra for the T angent groupoid 5 Definition 2.2. A smo oth Haar system over a Lie group oid c o nsists of a family of measures µ x in G x for each x ∈ G (0) such that, • for η ∈ G y x we hav e the following compatibility condition: Z G x f ( γ ) dµ x ( γ ) = Z G y f ( γ ◦ η ) dµ y ( γ ) • for ea ch f ∈ C ∞ c ( G ) the map x 7→ Z G x f ( γ ) dµ x ( γ ) belo ngs to C ∞ c ( G (0) ) A Lie group oid alwa ys p osses a smo oth Haar system. In fact, if we fix a smo oth (po sitiv e) sec tion of the 1-density bundle asso ciated to the Lie algebroid we obtain a smo oth Haar system in a canonica l wa y . The a dv a n tage of using 1-densities is that the meas ures are lo cally equiv alent to the Leb esgue meas ure. W e suppose for the rest of the pap er a given smo oth Haar system given by 1-densities (for complete details see [2 2 ]). W e can now define a conv olution pro duct on C ∞ c ( G ): Let f , g ∈ C ∞ c ( G ), we set ( f ∗ g )( γ ) = Z G s ( γ ) f ( γ · η − 1 ) g ( η ) dµ s ( γ ) ( η ) This gives a well defined as s ociative pro duct. Remark 2 .3. There is a way to av oid the Haa r system when one works with Lie group oids, using half densities (see Connes’ bo ok [8]). 3. Deformation to the normal cone Let M b e a C ∞ manifold and X ⊂ M b e a C ∞ submanifold. W e denote by N M X the normal bundle to X in M , i . e . , N M X := T X M /T X . W e define the follo wing set D M X := N M X × 0 G M × (0 , 1] . (2) The purp ose of this section is to r ecall how to define a C ∞ -structure with b oundary in D M X . This is mor e or less clas sical, for example it was extensively us e d in [13]. Here we a r e only going to do a sketch. Let us first consider the case where M = R n and X = R p × { 0 } (where we ident ify ca nonically X = R p ). W e denote b y q = n − p and by D n p for D R n R p as ab o ve. In this case w e clearly hav e that D n p = R p × R q × [0 , 1 ] (as a set). Consider the bijection Ψ : R p × R q × [0 , 1] → D n p (3) 6 P aulo Carrillo Rouse given by Ψ( x, ξ , t ) =  ( x, ξ , 0) if t = 0 ( x, tξ , t ) if t > 0 which inv erse is given explicitly by Ψ − 1 ( x, ξ , t ) =  ( x, ξ , 0) if t = 0 ( x, 1 t ξ , t ) if t > 0 W e can co nsider the C ∞ -structure with b order on D n p induced b y this bijection. In the gene r al case. Let ( U , φ ) b e a lo cal chart in M and supp ose it is an X -slice, so that it satisfies 1) φ : U ∼ = → U ⊂ R p × R q 2) If U ∩ X = V , V = φ − 1 ( U ∩ R p × { 0 } ) (we note V = U ∩ R p × { 0 } ) With this no ta tion we have that D U V ⊂ D n p is a n op en subset. W e ma y de fine a function ˜ φ : D U V → D U V in the following wa y: F or x ∈ V we ha ve φ ( x ) ∈ R p × { 0 } . If we write φ ( x ) = ( φ 1 ( x ) , 0), then φ 1 : V → V ⊂ R p is a diffeomorphism, where V = U ∩ ( R p ×{ 0 } ). W e set ˜ φ ( v , ξ , 0) = ( φ 1 ( v ) , d N φ v ( ξ ) , 0) and ˜ φ ( u, t ) = ( φ ( u ) , t ) for t 6 = 0. Here d N φ v : N v → R q is the normal comp onent of the deriv ate dφ v for v ∈ V . It is clear that ˜ φ is also a bijection (in pa r ticular it induces a C ∞ structure with b order ov er D U V ). Let us define, with the same notations as ab o ve, the following set Ω U V = { ( x, ξ , t ) ∈ R p × R q × [0 , 1] : ( x, t · ξ ) ∈ U } . which is an open subset of R p × R q × [0 , 1 ] and th us a C ∞ manifold (with b order). It is immediate that D U V is diffeomorphic to Ω U V through the restric tion of Ψ , used in (3). No w we co ns ider an atlas { ( U α , φ α ) } α ∈ ∆ of M consisting of X − slices. It is clear tha t D M X = ∪ α ∈ ∆ D U α V α (4) and if we take D U α V α ϕ α → Ω U α V α defined as the c o mposition D U α V α φ α → D U α V α Ψ − 1 α → Ω U α V α then w e obtain the following result. Prop osition 3. 1. { ( D U α V α , ϕ α ) } α ∈ ∆ is a C ∞ atlas with b or der over D M X . In fact the pro position can b e prov ed dir ectly from the following elementary lemma A Sch w artz typ e algebra for the T angent groupoid 7 Lemma 3.2. L et F : U → U ′ a C ∞ diffe omorphism wher e U ⊂ R p × R q and U ′ ⊂ R p × R q ar e op en subsets . We write F = ( F 1 , F 2 ) and we supp ose that F 2 ( x, 0) = 0 . Then t h e function ˜ F : Ω U V → Ω U ′ V ′ define d by ˜ F ( x, ξ , t ) =  ( F 1 ( x, 0) , ∂ F 2 ∂ ξ ( x, 0) · ξ , 0) if t = 0 ( F 1 ( x, tξ ) , 1 t F 2 ( x, tξ ) , t ) if t > 0 is a C ∞ map. Pr o of. Since the re s ult w ill hold if and o nly if it is true in each co ordinate, it is enough to pr o ve that if we hav e F : U → R a C ∞ map with F ( x, 0 ) = 0, then the map ˜ F : Ω U V → R given by ˜ F ( x, ξ , t ) =  ∂ F ∂ ξ ( x, 0) · ξ if t = 0 1 t F ( x, tξ ) if t > 0 is a C ∞ map. F o r that, we write F ( x, ξ ) = ∂ F ∂ ξ ( x, 0) · ξ + h ( x, ξ ) · ξ with h : U → R q a C ∞ map such that h ( x, 0) = 0. Then 1 t F ( x, tξ ) = ∂ F ∂ ξ ( x, 0) · ξ + h ( x, tξ ) · ξ from which we immediately get the r esult. Definition 3.3 (DNC). Let X ⊂ M b e as ab o ve. The s et D M X provided with the C ∞ structure with bo r der induced by the atla s describ ed in the last prop osition is called ‘ The deformation to normal c one asso ciate d to X ⊂ M ‘. W e will o ft en wr ite DNC ins tea d of Deformation to the normal co ne. Remark 3.4. F ollowing the s ame steps, it is p ossible to define a de fo rmation to the normal cone asso ciated to a n injective immersio n X ֒ → M . Examples 3 .5. Let us mention some ba sic examples of DCN manifolds D M X : 1. Consider the case when X = ∅ . W e have that D M ∅ = M × (0 , 1] with the usual C ∞ structure on M × (0 , 1]. W e used this fact implicitly for cov er D M X as in (4). 2. Consider the case when X ⊂ M is an o pen subset. Then w e do no t ha ve any defor mation at zer o a nd w e immediately see by definition that D M X is just the op en subset of M × [0 , 1] c o nsisting in the union o f X × [0 , 1] and M × (0 , 1]. The most imp ortant featur e ab out the DNC construction is that it is in so me sense functorial. Mo r e explicitly , let ( M , X ) and ( M ′ , X ′ ) b e C ∞ -couples a s a bov e and let F : ( M , X ) → ( M ′ , X ′ ) be a co uple morphis m, i.e., a C ∞ map F : M → M ′ , 8 P aulo Carrillo Rouse with F ( X ) ⊂ X ′ . W e define D ( F ) : D M X → D M ′ X ′ by the following formulas: D ( F )( x, ξ , 0) = ( F ( x ) , d N F x ( ξ ) , 0) and D ( F )( m, t ) = ( F ( m ) , t ) for t 6 = 0, where d N F x is by definition the map ( N M X ) x d N F x − → ( N M ′ X ′ ) F ( x ) induced b y T x M dF x − → T F ( x ) M ′ . W e hav e the following pr oposition, which is also an immediate c o nsequence of the lemma ab o ve. Prop osition 3. 6. The map D ( F ) : D M X → D M ′ X ′ is C ∞ . Remark 3 . 7. If we consider the catego ry C ∞ 2 of C ∞ pairs given by a C ∞ manifold and a C ∞ submanifold and pair mo rphisms as ab ove, we can refor mulate the prop osition and say that we hav e a functor D : C ∞ 2 → C ∞ where C ∞ denote the ca tegory of C ∞ manifolds w ith b order. 3.1. The tangen t group oid. Definition 3.8 (T angent group oid). Let G ⇒ G (0) be a Lie groupo id. The tangent gr oup oid associa ted to G is the group oid that has D G G (0) as the set of arrows and G (0) × [0 , 1] a s the units, with: · s T ( x, η , 0) = ( x, 0) a nd r T ( x, η , 0) = ( x, 0) a t t = 0. · s T ( γ , t ) = ( s ( γ ) , t ) a nd r T ( γ , t ) = ( r ( γ ) , t ) at t 6 = 0. · The pr oduct is given by m T (( x, η , 0) , ( x, ξ , 0 )) = ( x, η + ξ, 0) and m T (( γ , t ) , ( β , t )) = ( m ( γ , β ) , t ) if t 6 = 0 and if r ( β ) = s ( γ ). · The unit map u T : G (0) → G T is given by u T ( x, 0) = ( x, 0) and u T ( x, t ) = ( u ( x ) , t ) for t 6 = 0 . W e denote G T := D G G (0) . As we hav e seen ab ov e G T can b e cons idered as a C ∞ manifold with b order. As a consequence of the functoriality o f the DNC construc tio n we ca n show that the tangent group oid is in fact a L ie g roupoid. Indeed, it is easy to chec k tha t if we identify in a canonical w ay D G (2) G (0) with ( G T ) (2) , then m T = D ( m ) , s T = D ( s ) , r T = D ( r ) , u T = D ( u ) A Sch w artz typ e algebra for the T angent groupoid 9 where w e are considering the following pair morphisms: m : (( G ) (2) , G (0) ) → ( G , G (0) ) , s, r : ( G , G (0) ) → ( G (0) , G (0) ) , u : ( G (0) , G (0) ) → ( G , G (0) ) . Finally , if { µ x } is a s mooth Haar system o n G , then, setting • µ ( x, 0) := µ x at ( G T ) ( x, 0) = T x G x and • µ ( x,t ) := t − q · µ x at ( G T ) ( x,t ) = G x for t 6 = 0 , where q = dim G x , one obtains a smo oth Haa r system fo r the T angent gro upoid (details ma y b e found in [22]). Examples 3 .9. W e finish this section with some interesting examples of group oids and their tangent group oids. ( i ) The tangent gr oup oid of a gr oup . Let G be a Lie gro up considered as a Lie group oid, G := G ⇒ { e } . In this case the normal bundle to the inclus io n { e } ֒ → G is of cour se identified with the Lie a lgebra of the Group. Hence, the tangent group oid is a deformation of the g roup in its Lie algebra: G T = g × { 0 } F G × (0 , 1]. ( ii ) The tangent gr oup oid of a smo oth ve ctor bund le . Let E p → X b e a smo oth vec- tor bundle ov er a C ∞ manifold X (connexe). W e can co ns ider the Lie group oid E ⇒ X induced by the vector str ucture of the fib ers, i . e . , s ( ξ ) = p ( ξ ) = r ( ξ ) and the comp osition is given by the vector sum ξ ◦ η = ξ + η . In this case the nor mal vector bundle a ssocia ted to the zero section can b e identified to E itself. Hence, as a set the tangent gr oupoid is E × [0 , 1 ] but the C ∞ -structure at zero is given lo cally as in (3). ( iii ) The t angent gr oup oid of a C ∞ − manifold . Let M a C ∞ -manifold. W e can consider the pro duct gr oupoid G M := M × M ⇒ M . The tangent gr oupoid in this case takes the following form G T M = T M × { 0 } F M × M × (0 , 1]. This is called the tangent g roupoid to M and it was intro duced by Co nnes for giving a very conceptual pro of of the A tiyah-Singer index theore m (see [8] and [10]). 4. An alge bra for the T angen t g roup oid In this sectio n w e will show how to c o nstruct an algebra for the tangent gro upoid which consis t of C ∞ functions tha t satisfy a rapid decay condition at zero while out o f zero they sa tisfy a compact support condition. This alg e bra is the main construction in this work. 10 P aulo Carrillo Rouse 4.1. Sch w artz type spaces for Deformation to the normal cone manifolds. Our algebra fo r the T angent group oid will b e a particular case o f a construction asso ciated to an y deformation to the normal cone. W e s tart by defining a space for DNCs as sociated to op en subsets of R p × R q . Definition 4.1. Let p, q ∈ N and U ⊂ R p × R q an op en subset, and let V = U ∩ ( R p × { 0 } ). (1) Let K ⊂ U × [0 , 1 ] b e a compact subset. W e say tha t K is a conic co mpact subset o f U × [0 , 1 ] relative to V if K 0 = K ∩ ( U × { 0 } ) ⊂ V (2) Let g ∈ C ∞ (Ω U V ). W e say that f has compact conic supp o rt K , if there exists a conic c o mpact K of U × [0 , 1 ] re la tiv e to V such that if t 6 = 0 and ( x, tξ , t ) / ∈ K then g ( x, ξ , t ) = 0. (3) W e denote by S r,c (Ω U V ) the set of functions g ∈ C ∞ (Ω U V ) that hav e compact conic suppo rt and that satisfy the following condition: ( s 1 ) ∀ k , m ∈ N , l ∈ N p and α ∈ N q it exists C ( k,m,l,α ) > 0 such that (1 + k ξ k 2 ) k k ∂ l x ∂ α ξ ∂ m t g ( x, ξ , t ) k ≤ C ( k,m,l,α ) Now, the spa ces S r,c (Ω U V ) ar e inv ariant under diffeomorphisms. More precisely if F : U → U ′ is a C ∞ diffeomorphism as in lemma 3.2 then we can prove the next result. Prop osition 4. 2. L et g ∈ S r,c (Ω U ′ V ′ ) , then ˜ g := g ◦ ˜ F ∈ S r,c (Ω U V ) . Pr o of. The first o bserv ation is that ˜ g ∈ C ∞ (Ω U V ), thanks to lemma 3.2. Let us chec k tha t it has co mpact conic supp ort. F o r that, let K ′ ⊂ U ′ × [0 , 1 ] the conic compact suppo rt of g . W e let K = ( F − 1 × id [0 , 1] ) ⊂ U × [0 , 1 ] , which is a conic compact s ubset of U × [0 , 1] relative to V , and it is immediate by definition that ˜ g ( x, ξ , t ) = 0 if t 6 = 0 and ( x, t · ξ , t ) / ∈ K , that is, ˜ g ha s compact conic suppo rt K . W e now check the rapid decay pr operty ( s 1 ): F or simplify the pro of w e first int ro duce some useful notation. W riting F = ( F 1 , F 2 ) as in the lemma 3 .2, w e denote F 1 ( x, ξ ) = ( A 1 ( x, ξ ) , ..., A p ( x, ξ )) and F 2 ( x, ξ ) = ( B 1 ( x, ξ ) , ..., B q ( x, ξ )). W e denote also w = w ( x, ξ , t ) = ( A 1 ( x, tξ ) , ..., A p ( x, tξ )) a nd η = η ( x, ξ , t ) = ( ˜ B 1 ( x, ξ , t ) , ..., ˜ B q ( x, ξ , t )) wher e ˜ B j is a lso as ab o ve, i . e . , ˜ B j ( x, ξ , t ) =    ∂ B j ∂ ξ ( x, 0) · ξ if t = 0 1 t B j ( x, tξ ) if t 6 = 0 A Sch w artz typ e algebra for the T angent groupoid 11 In particular by definition w e hav e ˜ F ( x, ξ , t ) = ( w, η , t ). W e also write z = ( x, ξ , t ) and u = ( ω , η , t ). Hence, what we would like is to find b ounds for expr essions of the following type k ξ k k k ∂ α z ˜ g ( z ) k , for arbitra ry k ∈ N a nd α ∈ N p × N q × N . A simple calc ulation shows that the deriv a tes ∂ α z ˜ g ( z ) a re of the following form ∂ α z ˜ g ( z ) = X | β |≤| α | P β ( z ) ∂ β u g ( u ) where P β ( z ) is a finite s um of pr oducts of the form ∂ γ z ω i ( z ) · ∂ δ z η j ( z ) . W e a re only interested in se e what happ ens in the set K Ω := { z = ( x, ξ , t ) ∈ Ω : ( x, t · ξ , t ) ∈ K } since out of this set w e hav e that g and all its deriv ates v anis h (( x, tξ , t ) ∈ K iff ( w, tη , t ) ∈ K ′ ). F o r a p oin t z = ( x, ξ , t ) ∈ K Ω we hav e that ( x, t · ξ ) is in a compac t set and then it follows that the expressio ns k ∂ γ z ω i ( z ) k are b ounded in K Ω . F or the expressions k ∂ δ z η j ( z ) k , we pr oceed firs t by dev eloping as in lemma 3.2, that is, η j ( x, ξ , t ) =  ∂ B j ∂ ξ ( x, 0) · ξ + h j ( x, tξ )  · ξ Now, s ince we a re only considering p oin ts in K Ω , it is immedia te that we ca n find constants C j > 0 such that k ∂ δ z η j ( z ) k ≤ C j · k ξ k m δ . In the same way (remember F is a diffeomorphism) we can hav e constants C i > 0 such that k ξ i ( ω , η , t ) k ≤ C i · k η k . Putting a ll together, and using the prop ert y ( s 1 ) for g , we get b ounds C > 0 such that k ξ k k k ∂ α z ˜ g ( z ) k ≤ C, and this co ncludes the pro of. Remark 4 .3. W e can r esume the last inv ariance result as follows: If ( U , V ) is a C ∞ pair diffeomorphic to ( U, V ) with U ⊂ E , an op e n subset of a vector space E , and V = U ∩ E , then S r,c ( D U V ) is well defined and do es not depend on the pa ir diffeomorphism. With the last co mpatibilit y result in hand we are rea dy to give the main defi- nition in this work. 12 P aulo Carrillo Rouse Definition 4.4. Let g ∈ C ∞ ( D M X ). (a) W e s ay that g has compact co nic supp ort K , if ther e exists a co mpact subset K ⊂ M × [0 , 1 ] with K 0 := K ∩ ( M × { 0 } ) ⊂ X (conic co mpact relative to X ) such that if t 6 = 0 a nd ( m, t ) / ∈ K then g ( m, t ) = 0 . (b) W e say that g is rapidly decaying at zero if for every ( U , φ ) X -slice chart and for every χ ∈ C ∞ c ( U × [0 , 1 ]), the ma p g χ ∈ C ∞ (Ω U V ) given by g χ ( x, ξ , t ) = ( g ◦ ϕ − 1 )( x, ξ , t ) · ( χ ◦ p ◦ ϕ − 1 )( x, ξ , t ) is in S r,c (Ω U V ), where p is the pro jection p : D M X → M × [0 , 1 ] given by ( x, ξ , 0) 7→ ( x, 0), and ( m, t ) 7→ ( m, t ) for t 6 = 0. Finally , we denote by S r,c ( D M X ) the set o f functions g ∈ C ∞ ( D M X ) that are rapidly decaying at zero with compact conic s upport. Remark 4 .5. (a) By definition o f S r,c ( D M X ) w e see that C ∞ c ( D M X ) is con tained as a vector subspa ce. (b) It is clear that C ∞ c ( M × (0 , 1]) can be considere d as a subspace of S r,c ( D M X ) by extending by zero the functions at N M X . F ollowing the lines o f the la st remark we ar e going to precise a p ossible de- comp osition o f our space S r,c ( D M X ) that will be very useful in the sequel. L e t { ( U α , φ α ) } α ∈ ∆ a family of X − slices cov ering X . Consider the open cover o f M × [0 , 1] co nsisting in { ( U α × [0 , 1] , φ α ) } α ∈ ∆ union with M × (0 , 1]. W e can take a pa rtition of the unity sub ordinated to the last cover, { χ α , λ } α ∈ ∆ . That is, we hav e the following pro p erties: · 0 ≤ χ α , λ ≤ 1 · supp χ α ⊂ U α × [0 , 1] a nd supp λ ⊂ M × (0 , 1]. · P α χ α + P λ = 1 Let f ∈ S r,c ( D M X ), w e denote f α := f | D U α V α · ( χ α ◦ p ) ∈ C ∞ ( D U α V α ) and f λ := f | M × (0 , 1] · ( λ ◦ p ) ∈ C ∞ ( M × (0 , 1]) , then w e obtain the following decomp osition: f = X α f α + f λ Now, since f is conic co mpactly supp o rted we can supp ose, without los t of gener - ality , that A Sch w artz typ e algebra for the T angent groupoid 13 · f λ ∈ C ∞ c ( M × (0 , 1]), and · that χ α is compactly supp orted in U α × [0 , 1]. What we conclude of all this, is that we can decompo se our space S r,c ( D M X ) as follows S r,c ( D M X ) = X α ∈ Λ S r,c ( D U α V α ) + C ∞ c ( M × (0 , 1]) . (5) As we ment ioned in the intro duction, w e w ant to see the space S r,c ( D M X ) as a field of vector spaces ov er the int erv al [0 , 1], wher e at zero we ta lk ed ab out Sch wartz spaces. In o ur case we are interested in Sch wartz functions on the vector bundle N M X . Let us first r e call the notion of the Sc hw artz s pace asso ciated to a v ector bundle. Definition 4.6. Let ( E , p, X ) be a smo oth vector bundle over a C ∞ manifold X . W e define the Sch wartz space S ( E ) as the set of C ∞ functions g ∈ C ∞ ( E ) such tha t g is a Sch wartz function at each fib er (uniformly) and g has compact suppo rt in the direction o f X , i . e . , if there exists a compa c t subset K ⊂ X such that g ( E x ) = 0 for x / ∈ K . The vector space S ( E ) is an as sociative alg ebra with the pr oduct given as follows: for f , g ∈ S ( E ), we put ( f ∗ g )( ξ ) = Z E p ( ξ ) f ( ξ − η ) g ( η ) dµ p ( ξ ) ( η ) , (6) where µ ξ is a smo oth Haa r system of the Lie group oid E ⇒ X . A classic al F ourier argument can b e applied to show that the last alg e br a is isomor phic to ( S ( E ∗ ) , · ) (punctual pro duct). In pa rticular this implies that K 0 ( S ( E )) ∼ = K 0 ( E ∗ ). In the ca se we are interested, w e have a couple ( M , X ) and a v ector bundle asso ciated to it, tha t is , the normal bundle ov er X , N M X . The rea son why we gav e the last definition is bec ause we get ev a luation linear ma ps e 0 : S r,c ( D M X ) → S ( N M X ) , (7) and e t : S r,c ( D M X ) → C ∞ c ( M ) (8) for t 6 = 0 . C o nsequen tly , we hav e that the v ector space S r,c ( D M X ) is a field of vector spaces o ver the closed in terv al [0 , 1], whic h fibers spaces are: S ( N M X ) at t = 0 and C ∞ c ( M ) for t 6 = 0. Examples 4.7. Let us finish this subsec tio n by giving the examples of spaces S r,c ( D M X ) corresp onding to the DCN ma nifolds seen at 3.5 ab o ve. 1. F or X = ∅ , we have that S r,c ( D M ∅ ) ∼ = C ∞ c ( M × (0 , 1]). 2. F or X ⊂ M a n o pen subset we have that S r,c ( D M X ) ∼ = C ∞ c ( W ) where W ⊂ M × [0 , 1] is the op en subse t consisting of the union of X × [0 , 1 ] and M × (0 , 1]. 14 P aulo Carrillo Rouse 4.2. Sch w artz t yp e a lgebra for the T angen t groupoid. In this section we define a n a lgebra structure on S r,c ( G T ). W e start by defining a function m r,c : S r,c ( D G (2) G (0) ) → S r,c ( D G G (0) ) b y the following formulas: F or F ∈ S r,c ( D G (2) G (0) ), w e let m r,c ( F )( x, ξ , 0) = Z T x G x F ( x, ξ − η , η , 0) dµ x ( η ) and m r,c ( F )( γ , t ) = Z G s ( γ ) F ( γ ◦ δ − 1 , δ, t ) t − q dµ s ( γ ) ( δ ) If we canonica lly identify D G (2) G (0) with ( G T ) (2) , the map a bov e is no thin g else tha t the in tegration along the fiber s of m T : ( G T ) (2) → G T . W e hav e the following prop osition: Prop osition 4. 8. m r,c : S r,c (( G T ) (2) ) → S r,c ( G T ) is a wel l define d line ar map. The interesting part of the pro position is that the map is well defined sinc e it will e viden tly b e linear. Let us supp ose for the moment that the la st pr oposition is true. Under this assumption, we will de fine the pr oduct in S r,c ( G T ). Definition 4.9. Let f , g ∈ S r,c ( G T ), w e define a function f ∗ g in G T by ( f ∗ g )( x, ξ , 0) = Z T x G x f ( x, ξ − η , 0) g ( x, η , 0) dµ x ( η ) and ( f ∗ g )( γ , t ) = Z G s ( γ ) f ( γ ◦ δ − 1 , t ) g ( δ, t ) t − q dµ s ( γ ) ( δ ) for t 6 = 0 . W e can enounce our main result. Theorem 4.10. ∗ defines an asso ciative pr o duct on S r,c ( G T ) . Pr o of. Remem b er w e are assuming for the moment the prop osition 4.8. Let f , g ∈ S r,c ( G T ). W e let F := ( f , g ) the function in ( G T ) (2) defined b y: ( f , g )( x, ξ , η , 0) = f ( x, ξ , 0 ) · g ( x, η , 0 ) and ( f , g )(( γ , t ) , ( δ, t )) = f ( γ , t ) · g ( δ, t ) for t 6 = 0. Now, from the Leibnitz formula for the deriv ate of a pro duct it is immediate that ( f , g ) ∈ S r,c (( G T ) (2) ). Finally , by definition we hav e that m r,c (( f , g )) = f ∗ g , hence, thanks to prop osition 4.8, f ∗ g is a well defined element in S r,c ( G T ). A Sch w artz typ e algebra for the T angent groupoid 15 F or the asso ciativit y of the pro duct, let us rema r k that when one r estrict the pro duct to C ∞ c ( G T ), this co incides with the pro duct classically considere d on C ∞ c ( G T ) (which is asso ciative, see for example [22]). The asso ciativit y for S r,c ( G T ) is prov ed exac tly in the same wa y that for C ∞ c ( G T ). W e hav e then to prov e prop osition 4.8. W e ar e going to star t loc ally . Let U ∈ R p × R q × R q be an ope n set and V = U ∩ R p × { 0 } × { 0 } . Let P : R p × R q × R q → R p × R q the canonical pr o jection ( x, η , ξ ) 7→ ( x, η ). W e set U ′ = P ( U ) ∈ R p × R q , then U ′ is a lso an open subset, V ∼ = U ′ ∩ R p × { 0 } and P | V = I d V . W e denote a lso b y P the restriction P : U → U ′ . W e hav e as in lemma 3.2 a C ∞ map ˜ P : Ω U V → Ω U ′ V , which in this ca se is e xplicitly written by ˜ P ( x, η , ξ , t ) = ( x, η , t ) W e define ˜ P r,c : S r,c (Ω U V ) → S r,c (Ω U ′ V ) as fo llows ˜ P r,c ( F )( x, η , t ) = Z { ξ ∈ R q :( x,η ,ξ,t ) ∈ Ω U V } F ( x, η , ξ , t ) dξ . Let us pr ove the following lemma. Lemma 4.11. ˜ P r,c : S r,c (Ω U V ) → S r,c (Ω U ′ V ) is wel l define d. Pr o of. The first observ ation is that the integral in the definition o f ˜ P r,c is always well defined. Indeed, we deduce it fro m the next tw o p oin ts: · F or t = 0, ξ 7→ F ( x, η , ξ , 0) ∈ S ( R q ). · F or t 6 = 0, ξ 7→ F ( x, η , ξ , t ) ∈ C ∞ c ( R q ). Once we ca n deriv ate under the integral sy m b ol, we obtain that ˜ P r,c ( F ) ∈ C ∞ (Ω U ′ V ). Then, we just have to show that ˜ P r,c ( F ) verifies the tw o conditions of the definition 4.1. F or the first, if K ⊂ U × [0 , 1] is the co mpact conic s upport of F , then it is enough to put K ′ = ( P × id [0 , 1] )( K ) in or der to obtain a conic compact subset of U ′ × [0 , 1] relative to V and to chec k that K ′ is the co mpact conic supp ort of ˜ P r,c ( F ) . Let us now verify the co ndition ( s 1 ). Let k , m ∈ N , l ∈ N p and β ∈ N q . W e wan t to find C ( k,m,l,β ) > 0 such that (1 + k η k 2 ) k k ∂ l x ∂ β η ∂ m t ˜ P r,c ( F )( x, η , t ) k ≤ C ( k,m,l,α ) F or k ′ ≥ k + q 2 and α = (0 , β ) ∈ R q × R q we hav e by hypo thesis that it exists C ′ ( k ′ ,m,l,α ) > 0 such that k ∂ l x ∂ β η ∂ m t F ( x, η , ξ , t ) k ≤ C ′ 1 (1 + k ( η , ξ ) k 2 ) k ′ Then, we also hav e that k ∂ l x ∂ β η ∂ m t ˜ P r,c ( F )( x, η , t ) k ≤ C ′ Z { ξ ∈ R q :( x,η ,ξ,t ) ∈ Ω U V } 1 (1 + k ( η , ξ ) k 2 ) k ′ dξ 16 P aulo Carrillo Rouse ≤ C ′ 1 (1 + k η k 2 ) q 2 − k ′ Z { ξ ∈ R q } 1 (1 + k ξ k 2 ) k ′ dξ ≤ C 1 (1 + k η k 2 ) k with C = C ′ · Z { ξ ∈ R q } 1 (1 + k ξ k 2 ) k ′ dξ . W e can now give the pro of of the prop osition 4 .8. Pr o of of 4.8. Let us first fix some notation. W e supp ose dim G = p + q and dim G (0) = p , in particular this implies that dim G (2) = p + q + q . Let ( U , φ ) and ( U ′ , φ ′ ) be G (0) -slices in G (2) and G resp ectiv ely s uc h that the following diag ram commutes U m φ U ′ φ ′ U P U ′ , where P : R p × R q × R q → R p × R q is the ca nonical pro jection (as above) and P ( U ) = U ′ . This is pos sible since m is a surjectiv e submersion. Now, we apply the DNC construction to the diagr am a b ov e to obtain, thanks to the functoriality of the construction, the following commut ative diag ram D U V D ( m ) ˜ φ D U ′ V ′ ˜ φ ′ D U V D ( P ) D U ′ V ′ Ω φ Ψ φ ˜ P Ω φ ′ , Ψ φ ′ where Ψ φ and ˜ φ a re as in section 3. Let g ∈ S r,c ( D U V ), w e define P r,c ( g )( x, η , t ) =    R R q g ( x, η , ξ , 0) dξ if t = 0 R { ξ ∈ R q :( x,η ,ξ,t ) ∈ Ω U V } g ( x, η , ξ , t ) t − q dξ if t 6 = 0 Then, from the la st commutativ e diagram, w e get that P r,c ( g ) = ˜ P r,c ( g ◦ Ψ φ ) ◦ (Ψ φ ′ ) − 1 , hence, thanks to 4.11, we ca n conclude that we hav e a well defined linear map P r,c : S r,c ( D U V ) → S r,c ( D U ′ V ′ ) . A Sch w artz typ e algebra for the T angent groupoid 17 W e now use the prop osition 4 .2 to write S r,c ( D U V ) = { h ∈ C ∞ ( D U V ) : h ◦ ˜ φ − 1 ∈ S r,c ( D U V ) } , and so for h ∈ S r,c ( D U V ), w e see that P r,c ( h ◦ ˜ φ − 1 ) ◦ ˜ φ ′ ∈ S r,c ( D U ′ V ′ ) . W e use again the last commutativ e dia gram to se e that m r,c ( h ) = P r,c ( h ◦ ˜ φ − 1 ) ◦ ˜ φ ′ . W e then hav e a well defined linea r map m r,c : S r,c ( D U V ) → S r,c ( D U ′ V ′ ) . T o pass to the g lobal case we only have to use the decomp osition of S r,c ( D G (2) G (0) ) and of S r,c ( D G G (0) ) as in (5), and of course the in v ar iance under diffeomor phisms (prop osition 4.2). Let us recall that we hav e well defined ev aluatio n morphisms as in (7) and (8). In the case of a the tangent gr oupoid they a re by definitio n morphisms of a lgebras. Hence, the algebr a S r,c ( G T ) is field of algebras ov er the clo sed interv al [0 , 1], with asso ciated fib er alg ebras, S ( A G ), at t = 0, a nd C ∞ c ( G ) for t 6 = 0. It is very in teresting to see what this means in the ex amples given in 3 .9. 5. F urther dev elop emen ts Let G ⇒ G (0) be a Lie gr oupoid. In index theo ry for Lie gro upoids the tangent group oid has b een us ed to de fine the a na lytic index asso ciated to the gr oup, as a morphism K 0 ( A ∗ G ) → K 0 ( C ∗ r ( G )) (see [19]) or as a K K -element in K K ( C 0 ( A ∗ G ) , C ∗ r ( G )) (see [13]), this can b e done because one has the following short exact sequence of C ∗ -algebra s 0 → C ∗ r ( G × (0 , 1 ]) − → C ∗ r ( G T ) e 0 − → C 0 ( A ∗ G ) − → 0 , (9) and b ecause of the fact that the K -gro ups of the a lgebra C ∗ r ( G × (0 , 1]) v anish (homotopy inv aria nce). The index defined a t the C ∗ -level ha s proven to b e very useful (see for example [9 ]) but extracting numerical inv aria n ts from it, with the existent to ols, is v ery difficult. In non comm utative geo metr y , a nd also in classical geometry , the to ols for o btain more explicit inv a rian ts ar e more develop ed for the ’smo oth o b jects’; in our case this means the conv olution a lg ebra C ∞ c ( G ), where we 18 P aulo Carrillo Rouse can for exa mple apply Chern-W eil-Connes theory . Hence, in some wa y , the indices defined in K 0 ( C ∞ c ( G )) are mor e refined ob jects a nd for some cases it would be preferable to work with them. Unfortunately this indices a re not go o d enough, since fo r exa mple they are not homotopy inv a rian ts; in [8] Alain Connes dis c usses this and also other reaso ns why it is not enough to keep with the C ∞ c -indices. The main r eason to construct the a lgebra S r,c ( G T ) is that it gives an in termediate way betw een the C ∞ c -level a nd the C ∗ -level a nd will allow us in [5] to define another analytic index morphism asso ciated to the group oid, with the adv a n tage that this index will take v alue s in a group that allows to do pairings with c yclic co cycles and in general to a pply Chern- Connes theory to it. The w ay we are go ing to define our index is by obtaining first a s hort exact sequence analogue to (9), that is, a sequence of the following kind 0 → J − → S r,c ( G T ) e 0 − → S ( A ∗ G )) − → 0 . (10) The problem her e will b e that we do not disp ose of the adv antages of the K − theory for C ∗ -algebra s, since the alg ebras we are cons idering a re not of this type (we do not hav e for example homoto p y inv a riance). 6. References [1] Aastrup, J., Melo, S.T., Month ub ert, B. Schrohe, E. Boutet de Monv el’s Calculus and Group oi ds I. Pr eprint arxiv:math.KT/0611336 . 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