Equivalence of Fell Systems and their Reduced C*-Algebras

EQUIV ALENCE OF FELL SYSTEMS AND THEIR REDUCED C ∗ -ALGEBRAS EL-KA ¨ IOUM M. MOUTUOU AND JEAN-LOUIS TU A bstract . This paper is aimed at investigating links between Fell bundles over Morita equivalent groupoids and t h eir corresponding reduced C ∗ -algebras. Mainly , we review the notion of Fell pa irs over a Morita equiv ale nce of groupoids, and give the analogue of the Renault’s E quivalence Theorem for the reduced C ∗ -algebras of equivalent Fell systems. Eventually , we will use this theo rem to co nnect the reduced C ∗ -algebra of an S 1 -central groupoid exten sion to that of its as s ociated Dixmier-Douady bund le. I ntroduction A Fell system consists of a pair ( G , E ), where E is a Fell bundle over the groupoid G . The notion of (Morita) equivalence of Fell systems was first intr oduced by S. Y amagami in [22 ] , and by then it was studied by M uhly in [9] and very recently by Muhly and W illiams in [12] where the auth ors prove that if ( Γ , F ) and ( G , E ) are equivalent, then their full C ∗ -algebras C ∗ ( Γ ; F ) and C ∗ ( G ; E ) are Morita equivalent (see [9, Theorem 11], and [12, Theorem 6.4]). Ho wever , it has not bee n k nown so far whether an equivalence of Fell systems gives rise to a Morita equi valence between the associated reduced C ∗ -algebras. The first motivation of o ur work came from twisted K -theory: to every groupoi d G and every cocycle α ∈ ˇ C 2 ( G • , T ) is associated a Fell sys tem ( Γ α , L α ), and the twisted K -groups K ∗ α ( G ) are defined as the C ∗ -algebraic K -groups of the reduced C ∗ -algebra C ∗ r ( Γ α , L α ) (cf. [21]). Moreover , i t is k nown that when α ∼ β , then not only Γ α is Morita e quivalent to Γ β but also the ass ociated reduced C ∗ -algebras C ∗ r ( Γ α , L α ) and C ∗ r ( Γ β , L β ) are Mor ita equivalent (see [21, Proposition 3.3]); so that K ∗ α ( G )  K ∗ β ( G ). This has led us to a generalisation of the s o-called Renault’s equivalence T heorem f or reduced g roupoid C ∗ -algebras ( [18, Theorem 13]) to Fell sys tems. W e recall from [21] some concepts related to groupoid s such as generalized homomorphisms and Di xmier-Douady bundles in § 1, and we review the basics of Fell systems and their reduced C ∗ -algebras from [5] and [21 ] in § 2. In § 3, we discuss the notion of equivalence of Fell systems of [9] and [ 12] from another for m ali sm that better suits with the construction of the l i nking Fell systems introduced in § 4. The equivalence theorem for the reduced C ∗ -algebras of Fell s ystems is proved in § 5, and then, in § 6, we apply this theorem to link the C ∗ -algebra associated to an S 1 -central extension of a groupoid G to the reduced cross-p roduct A ⋊ r G , where A is some Dixmier-Douady bundle. 1. P rel iminaries Although we assume that t he reader is famili ar with the language of groupoids (see for instance [16]), we recall so me of their basics used substantially throughout this paper . All the groupoids we are working with are supposed to be Hausdo r ff , locally compact, second countable, and are equipped with Haar s ystems. They are also assumed to have finite-dimentional base spaces, in the s ense of [4]. 1.1. Given two g roupoids G / / r s / / G (0) and Γ / / r s / / Γ (0) , a generalized morphism Z : Γ − → G consists of a (locally compact Hausdor ff ) space Z , two maps Γ (0) Z r o o s / / G (0) ( s and r are called generalized source map and generalized range map of Z , respectively), a l eft action of Γ on Z with respect to r , a right action of G on Z with The first author is supported by the German Research Foundation (DFG) and the Universit ´ e franco-allemande (DFH-UF A) via the International Research T ra ining Group 1133 ”Geo metry and Analysis o f Symmetries”. 1 2 EL-KA ¨ IOUM M. MOUTUOU AND JEAN-LOUIS TU respect to s , s uch that the two actions commute, and Z − → Γ (0) is a rig ht G -pr i ncipal bundle. S uch a m o rphism is a Morita eq uivalence if in add ition, Z − → G (0) is a left Γ -principal bundle; in this case, we say that Γ and G are Morita equivalent , and we write Γ ∼ Z G . T he terminolog y of generalized morphism is justified by the fact that any strict groupoid homomorphism (see [16]) f : Γ − → G induces a generali zed o ne Z f : Γ − → G , where Z f : = Γ (0) × f , G (0) , r G , with generalized source and range s ( y , g ) : = s ( g ) and r ( y , g ) : = y , while the actions are γ · ( s ( γ ) , g ) : = ( r ( γ ) , f ( γ ) g ), and ( y , g ) · g ′ : = ( y , g g ′ ). 1.2. If Γ ∼ Z G , then G ∼ Z − 1 Γ , where Z − 1 is Z as topological space, and if ♭ : Z − → Z − 1 is the identity map, the generalized source an d range are s ♭ ( ♭ ( z )) : = r ( z ) an d r ♭ ( ♭ ( z )) : = s ( z ). The left G -action on Z − 1 is given by g · ♭ ( z ) : = ♭ ( z g − 1 ) for ( z , g − 1 ) ∈ Z ∗ G , and the right Γ -action is ♭ ( z ) · γ : = ♭ ( γ − 1 z ) w henever ( γ − 1 , z ) ∈ Γ ∗ Z . If Γ ∼ Z 1 G 1 ∼ Z 2 G , then Γ ∼ Z 1 × G 1 Z 2 G , where Z 1 × G 1 Z 2 is the quotient of the fibre product space Z 1 × s 1 , G (0) 1 , r 2 Z 2 by the equivalence relation ( z 1 , z 2 ) ∼ ( z 1 g 1 , g − 1 1 z 2 ). 1.3. A Dixmier-Douady bundle A over G is a locally trivial bundle A − → G (0) with fibre the C ∗ -algebra K of compact operators on the separable infinite-dimensional Hilber t space H = l 2 ( N ), together with an action α by automorphisms of G on A ; that i s, a continuous family of iso mo rphisms of C ∗ -algebras α g : A s ( g ) − → A r ( g ) such that α gh = α g ◦ α h whenever g and h are compo sable, and α g − 1 = α − 1 g . Such a bundle is represented by the triple ( A , G , α ). Let H G : = L 2 ( G ) ⊗ H , where L 2 ( G ) i s the G -e quivariant C 0 ( G (0) )-Hilbert modul e obtained by completing C c ( G ) with resp e ct to the scalar product h ξ, η i ( x ) = R G x ξ ( g ) η ( g ) d µ x G ( g ). W e say that two Dix mier-Douady bundles A and B are Morita equivalent , and wri te A ∼ B , if A ⊗ K ( H G )  B ⊗ K ( H G ). The se t o f Morita equivalence classes of Dix mier-Douady bundles forms an abelian group Br ( G ) called the Brauer group o f G . W e refer to [6], [20], o r [8] for more details about the structures of Br ( G ). 2. F ell systems and their reduced C ∗ - algebras If p : E − → G is a Banach bundle, we set E [2] : = n ( e 1 , e 2 ) ∈ E × E | ( p ( e 1 ) , p ( e 2 )) ∈ G (2) o . Let m : G (2) − → G d enote the par tial multipl ication of the groupoid G / / r s / / G (0) . Then m ∗ E − → G (2) is a B anach bundle. Definition 2.1. (cf. [5] , [21, Appendix A] ) . A multiplicat ion on E consi st s of a continuous map E [2] ∋ ( e 1 , e 2 ) 7− →  ( p ( e 1 ) , p ( e 2 )) , e 1 e 2  ∈ m ∗ E satisfyin g the followin g properties: (i) th e i nduced map E g × E h − → E gh is bili near for all ( g , h ) ∈ G (2) ; (ii) ( associativity ) ( e 1 e 2 ) e 3 = e 1 ( e 2 e 3 ) whenever the multiplication makes sense; and (iii) k e 1 e 2 k ≤ k e 1 kk e 2 k , f or every ( e 1 , e 2 ) ∈ E [2] . A ∗ - involution on E is a conti n uous 2 -periodic map ∗ : E ∋ e 7− → e ∗ ∈ E such that (iv) p ( e ∗ ) = p ( e ) − 1 , and (v) for all g ∈ G , the induced map ∗ : E g − → E g − 1 is conjugate linear . Finally, we say that p : E − → G is a Fel l bun dle if in addition the followin g conditions hold: (vi) ( e 1 e 2 ) ∗ = e ∗ 2 e ∗ 1 , ∀ ( e 1 , e 2 ) ∈ E [2] ; (vii) k e ∗ e k = k e k 2 , ∀ e ∈ E ; i n particular , E x is a C ∗ -algebra, for x ∈ G (0) ; (viii) e ∗ e ≥ 0 , ∀ e ∈ E ; an d (ix) ( fullness ) the image of the map E g × E h − → E gh spans a dense subspace of E gh , for all ( g , h ) ∈ G (2) . If we are given such a Fell bu ndle, we say that ( G , E ) is a Fell system . Example 2.2. If A is a Dixmier-Douady bundle over G with action α , we get a Fell system ( G , s ∗ A ) , wher e s ∗ A is the C ∗ -bundle over G obtained b y pul l ing back A through the source map s : G − → G (0) . The multi plication on s ∗ A is given b y A s ( g ) × A s ( h ) ∋ ( a , b ) 7− → α h − 1 ( a ) b ∈ A s ( g h ) = A s ( h ) , and the ∗ -involution is A s ( g ) ∋ a 7− → α g ( a ) ∗ ∈ A s ( g − 1 ) = A r ( g ) . EQUIV ALENCE OF FELL SYSTEMS AND THEIR REDUCED C ∗ -ALGEBRAS 3 Given a Fell system ( G , E ), we turn the s pace C c ( G ; E ) of compactly suppor ted continuous sections of E into a convolution algebra by setting ( ξ ∗ η )( g ) : = Z G r ( g ) ξ ( γ ) η ( γ − 1 g ) d µ r ( g ) G ( γ ) , and ξ ∗ ( g ) : = ξ ( g − 1 ) ∗ , for ξ, η ∈ C c ( G ; E ) , g ∈ G . (1) Let k ξ k 1 : = sup x ∈ G (0) R G x k ξ ( g ) k d µ x G ( g ). W e next define the I-norm k · k I by k ξ k I : = max {k ξ k 1 , k ξ ∗ k 1 } . Then, the completion L 1 ( G ; E ) of C c ( G ; E ) with respect to k · k I is a Banach ∗ -algebra. Its envelopping C ∗ -algebra C ∗ ( G ; E ) is called the fu ll C ∗ -algebra of ( G , E ). Note that we have a C ∗ -bundle over the base G (0) , defined as the pul l-back of E along the identity map G (0) ֒ → G ; we denote it by E (0) . W e can vie w E (0) as the restriction E | G (0) , once we have ide ntified G (0) with a subset of G . Moreover , equipped with the pointwise norm, A : = C 0 ( G (0) ; E (0) ) is a C ∗ -algebra. W e will usually write A x for the C ∗ -algebra which is the fibre o f E (0) over x ∈ G (0) . The fo llowing proposition is proved, for instance, in [8], (and in [5] in the case o f p roper groupoids), so we omit the proof. Propositio n 2.3. C c ( G ; E ) is a pre-Hilbert (lef t) A-module under the operations ( f · ξ )( g ) : = f ( r ( g )) ξ ( g ) , for f ∈ A , ξ ∈ C c ( G ; E ) , a ∈ G , an d (2) A h ξ, η i ( x ) : = Z G x ξ ( g ) η ( g ) ∗ d µ x G ( g ) , for ξ, η ∈ C c ( G ; E ) , x ∈ G (0) . (3) Let L 2 ( G ; E ) be the Hilbert A -module o btained by completing C c ( G ; E ) with respect to the norm k ξ k 2 : = k A h ξ, ξ i k 1 / 2 , for ξ ∈ C c ( G ; E ) . Then, left multiplication by an e lement of C c ( G ; E ) (i.e. the map π l ( ξ ) : η 7− → ξ ∗ η, ξ, η ∈ C c ( G ; E )) is a bounded A -linear operator with respect to the norm k · k 2 which is adjointable (see [8]). Hence π l extends to a ∗ -monorphism π l : C c ( G ; E ) − → L ( L 2 ( G ; E )) : = L A ( L 2 ( G ; E )) . The e x tension of π l to L 1 ( G ; E ) is known as the lef t regular representation of L 1 ( G ; E ). Definition 2.4. (cf. [21, A.3] ). Under t he ab ove n otations, the closure of the i mage π l ( C c ( G ; E )) in L ( L 2 ( G ; E )) with respect to the operator norm is called the reduced C ∗ -algebra of the Fell system ( G , E ) , and is denoted b y C ∗ r ( G ; E ) ; i.e. C ∗ r ( G ; E ) : = π l ( C c ( G ; E )) ⊂ L ( L 2 ( G ; E ))) . Remark 2.5. On e can think of C ∗ r ( G ; E ) as the completion of the convolution ∗ -algebra C c ( G ; E ) with respect to the reduced norm k · k r given b y k ξ k r : = sup {k π l ( ξ ) η k 2 | η ∈ C c ( G ; E ) , k η k 2 ≤ 1 } . Remark 2.6. If ( A , G , α ) is a Dixmier-Douady bun dle, then the reduced C ∗ -algebra C ∗ r ( G ; s ∗ A ) associated to the F ell bundle s ∗ A , denoted by A ⋊ r G , is called the reduced crossed product of ( A , G , α ) ; it plays an important role in twi st ed K -theory of groupoids ( [ 21] , [20] ). Alternatively , we will sometimes use another definition of the reduced norm, which i s a generalis ation of that of [18]. Suppose we are g iven a rig ht Fel l sys tem ( G , E ). Then, f or all x ∈ G (0) , consider the inclusion i x : G x − → G . Then, as in [21, A.3], we define the ( right) Hilbert A x -module L 2 ( G x ; E ) as the completion of C c ( G x ; i ∗ x E ) with respect to the inner product h ξ, η i A x : = R G x ξ ( g ) ∗ η ( g ) d µ x ( g ) (the rig ht action being ( ξ · a ) : G x ∋ g 7− → ξ ( g ) · a ∈ E g ). The following l emma i s very e asy to prove. Lemma 2.7. Let ( G , E ) be as above. Then, for all x ∈ G (0) , left multiplication by elements of C c ( G ; E ) gives a ∗ -repr esentation π G x : C c ( G ; E ) − → L A x ( L 2 ( G x ; E )) . Moreover , we have k ξ k C ∗ ( G ; E ) : = k ξ k r = sup x ∈ G (0) {k π G x ( ξ ) k , ∀ ξ ∈ C c ( G ; E ) 4 EL-KA ¨ IOUM M. MOUTUOU AND JEAN-LOUIS TU 3. E quiv alence of F ell systems In this s ection, we are presenting the notion of equivalences of Fell bundles over Mor ita equivalent groupoids . Our de finitions are s l ight modifications of those g iven by P . Muhly and D. W illiams in [12, § .6]. Suppose that Z is a (rig ht) principal G -s p ace; that is, there is a principal G -action α : Z ∗ G − → Z . If π : X − → Z is a Banach bundle, and i f p : E − → G is a Fell bundle, we set X ∗ E : = { ( u , e ) ∈ X × E | ( π ( u ) , p ( e )) ∈ Z ∗ G } . Definition 3.1. A right Fell G -pair over the principal G -space Z i s a pair ( X , E ) consisti n g of a Fell bundle E over G , a Banach b u ndle π : X − → Z, and a continuous map X ∗ E ∋ ( u , e ) 7− → ue ∈ α ∗ X , su ch that (i) ( bilinearity ) for all ( z , g ) ∈ Z ∗ G , the induced map X z × E g − → X zg is bilinear , and is compatible with the scalar multiplication; i.e. ( λ u ) e = u ( λ e ) = λ ( u e ) , ∀ λ ∈ C , ( u , e ) ∈ X z × E g ; (ii) ( associativity ) i f ( z , g ) ∈ Z ∗ G and ( g , h ) ∈ G (2) , one has u ( e 1 e 2 ) = ( ue 1 ) e 2 , ∀ ( u , e 1 , e 2 ) ∈ X z × E g × E h ; (iii) k ue k = k u kk e k , ∀ ( u , e ) ∈ X z × E g ; (iv) ( f aithfulln ess ) the induced map X z × X g − → X zg spans a dense subspace of X zg . We also say that ( G , E ) acts on X on the right over Z. Likewise, on e defines a left F ell G -pair ( E , X ) over a principal left G -space Z . Remark 3.2. Notice that if ( X , E ) is a right Fell G -pair over Z, then for every z ∈ Z, X z is a right E s ( z ) -module. Now suppose that Γ ∼ Z G . Then there are a continuous Γ -valued inner product Γ < · , · > : Z × G (0) Z − 1 − → Γ , and a continuous G -valued inner product < · , · > G : Z − 1 × Γ (0) Z − → G , defined as follows • for ( z , ♭ ( z ′ )) ∈ Z × Γ (0) Z − 1 , Γ < z , z ′ > i s the unique element of Γ such that z = Γ < z , z ′ > · z ′ ; • for ( ♭ ( z ) , z ′ ) ∈ Z − 1 × Γ (0) Z , < z , z ′ > G is the unique element of G such that z ′ = z · < z , z ′ > G . Observe that these functions are well defined, fo r Z − → Γ (0) is a G -principal bundle, and Z − → G (0) is a Γ -principal bundle. Fur thermore, they satisfy the following eq uali tie s (cf. [13, § .6.1]): Γ < z , z ′ > − 1 = Γ < z ′ , z >, ∀ ( z , ♭ ( z ′ )) ∈ Z × G (0) Z − 1 , (4) < z , z ′ > − 1 G = < z ′ , z > G , ∀ ( ♭ ( z ) , z ′ ) ∈ Z − 1 × Γ (0) Z , and (5) z · < z ′ , z ” > G = Γ < z , z ′ > · z ” , ∀ ( z , ♭ ( z ′ ) , z ”) × Z × G (0) Z − 1 × Γ (0) Z . (6) Lemma 3.3. Let Γ ∼ Z G . Then, any ri ght Fell G -pair ( X , E ) over Z gives rise to the l eft Fell G -pair ( E , X ) over the inverse Z − 1 , where X is defined as the conju gate bundle of X . A similar statement holds for a left Γ -pair over Z. Proof. By definition X is X as sp ace. If ♭ : X − → X d e notes the identity map, we define the projection ¯ π : X − → Z − 1 by ¯ π ( ♭ ( u )) : = ♭ ( π ( u )). The fibre X ♭ ( z ) is the conjugate Banach space of X z ; the left G -action on X is g · ♭ ( u ) : = ♭ ( u · g − 1 ), while the left action of E on X is given by E g × X ♭ ( z ) ∋ ( e , ♭ ( u )) 7− → ♭ ( u · e ∗ ) ∈ X g · ♭ ( u ) .  Let us fix some notat ions that will be used in the se quel. Suppose Γ ∼ Z G . If ( X , E ) is a Fell G -pair , we d e fine the topological spaces X ∗ X : = { ( u , ♭ ( u ′ )) ∈ X × X | ( π ( u ) , ¯ π ( ♭ ( u ′ ))) ∈ Z × G (0) Z − 1 } and X ∗ X : = { ( ♭ ( u ) , u ′ ) ∈ X × X | ( ¯ π ( ♭ ( u )) , π ( u ′ )) ∈ Z − 1 × Γ (0) Z } . Observe that the space Z × G (0) Z − 1 is a locally compact groupoid with base Z as follows: the product is ( z , ♭ ( z ′ )) · ( z ′ , ♭ ( z ”)) : = ( z , ♭ ( z ”)), the source o f ( z , ♭ ( z ′ )) is z ′ , its r ange is z , and its inverse is ( z ′ , ♭ ( z )). Similarl y Z − 1 × Γ (0) Z is a locally compact g roupoid with base Z − 1 . If Γ ∼ Z G , and if ( G , E ) and ( Γ , F ) are Fell s ystems, we denote by E > G and F Γ < the Fel l bundles over Z − 1 × Γ (0) Z and Z × G (0) Z − 1 , respectively , obtained by pul ling back E − → G along the continuous map < · , · > G , and F − → Γ along the continuous map Γ < · , · > , resp ectively . Note that, fo r instance, the fiber of E > G over ( ♭ ( z ) , z ′ ) is isomorphic to E < z , z ′ > G . Definition 3.4. Assume Γ ∼ Z G and ( X , E ) is a Fell G -pair over Z. An E - valued inner product on X is a continuous map h· , ·i E : X ∗ X − → E > G , ( ♭ ( u ) , u ′ ) 7− → h u , u ′ i E , such that EQUIV ALENCE OF FELL SYSTEMS AND THEIR REDUCED C ∗ -ALGEBRAS 5 (i) for ( ♭ ( z ) , z ′ ) ∈ Z − 1 × Γ (0) Z, the induced map h · , ·i E : X ♭ ( z ) × X z ′ − → E < z , z ′ > G is linear in b oth the first and the second variable; (ii) ( E - linearity ) if ( ♭ ( z ) , z ′ ) ∈ Z − 1 × Γ (0) Z and ( z , g ) ∈ Z ∗ G , then h u , u ′ i E · e = h u , u ′ · e i E , ∀ ( ♭ ( u ) , u ′ , e ) ∈ X ♭ ( z ) × X z ′ × E g ; (iii) h u , u ′ i ∗ E = h u ′ , u i E ∈ E < z , z ′ > − 1 G = E < z ′ , z > G ; (iv) ( p ositivity ) for all z ∈ Z and u ∈ X z , h u , u i E ≥ 0 in E < z , z > G = E s ( z ) ; and the equality h u , u i E = 0 implies u = 0 . In this case, we say that X is a right ( G , E )-inner product mod ule over Z. Likewise, if ( F , X ) is a left Fel l Γ -pair , one defines an F -valued in ner product F h· , · i : X ∗ X − → F Γ < , all the actions being considered on the l ef t. Remark 3.5. Observe that conditi on s (ii) and (iii ) of t he definition imply that h u · e , u ′ i E = e ∗ · h u , u ′ i E , whenever the multiplications an d the inn er prod uct are defined. Mor eover , for all z ∈ Z, X z is a pre-Hilbert A s ( z ) -module. Definition 3.6. An equivalence between t wo F el l systems ( Γ , F ) and ( G , E ) is a pair ( Z , X ) such that Γ ∼ Z G , X is a left ( F , Γ ) -inn er product module an d a right ( G , E ) -inner product module over Z, with the followin g properties (i) ( eq uivaria nce ) for all ( γ, z , g ) ∈ Γ ∗ Z ∗ G , the multiplication F γ × X z × E g − → X γ zg is associative; i. e. f · ( u · e ) = ( f · u ) · e , ∀ ( f , u , e ) ∈ F γ × X z × E g ; (ii) ( compatibility ) for all ( z , ♭ ( z ′ ) , z ”) ∈ Z × G (0) Z − 1 × Γ (0) Z and ( u , ♭ ( u ′ ) , u ”) ∈ X z × X ♭ ( z ′ ) × X z ” , F h u , u ′ i · u ” = u · h u ′ , u ” i E in X z · < z ′ , z ” > G = X Γ < z , z ′ > · z ” ; (iii) the F -valued inner product i s full ; i.e. , the image of the induced map X z × X ♭ ( z ′ ) − → F Γ < z , z ′ > spans a dense subspace of F Γ < z , z ′ > ; (iv) t he E -valued inner product is fu ll. In this case, we write ( Γ , F ) ∼ ( Z , X ) ( G , E ) . Remark 3.7. It follows from Definition 3.6 and Lemma 3.3 that if ( Γ , F ) ∼ ( Z , X ) ( G , E ) , then ( G , E ) ∼ ( Z − 1 , X ) ( Γ , F ) . Furthermore, it is starightforward that f or all z ∈ Z, (the completion with respec t to the in ner products of) X z is an imprimitivity B r ( z ) -A s ( z ) -bimodule. Example 3.8. If ( G , E ) is a Fell system, then ( G , E ) ∼ ( Z G , E ) ( G , E ) , where Z G is the space of morphisms G (1) (with which we identify G ). Indeed, Z G implements a Morita equivalence G ∼ Z G G , the generalized source and range map s bein g the source and range maps s and r of G / / r s / / G (0) , together wi th the canonical left and right actions given by partial multiplications. Notice that Z − 1 G = { g − 1 | g ∈ G } . It is easy to see that the inner products G < · , · > : Z G × G (0) Z − 1 G − → G and < · , · > G : Z − 1 G × G (0) Z G − → G are G < g , h > = gh − 1 and < g , h > G = g − 1 h, respectiv ely . E acts on itself over Z G by definition of a Fell bu ndle. Now , the conjugate bundle E − → Z − 1 G is given fibrewise by E g − 1 = { e ∗ | e ∈ E g } . The inner products are E g × E h − 1 ∋ ( e 1 , e ∗ 2 ) 7− → e 1 e ∗ 2 ∈ E gh − 1 , and E g − 1 × E h ∋ ( e ∗ 1 , e 2 ) 7− → e ∗ 1 e 2 ∈ E g − 1 h . It is straightforward that all the conditions of Defini t i on 3.6 are satisfied. By virtue of Remark 3.7 and Example 3.8, equivalence of F ell sy stems is sy mmetric and reflexive. Also, it is not hard to show that it i s transitive, so that it defines an equivalence relation among the collection o f Fe ll systems (cf. [8]). In the seq uel, we will need the following result. 6 EL-KA ¨ IOUM M. MOUTUOU AND JEAN-LOUIS TU Propositio n 3.9. If ( Γ , F ) ∼ ( Z , X ) ( G , E ) , C c ( Z ; X ) is a full pre-inner product C c ( Γ ; F ) - C c ( G ; E ) -bimodule with respect to the inductive limit topologies 1 under the following operations: ( ξ · φ )( z ) : = Z Γ r ( z ) ξ ( γ ) φ ( γ − 1 · z ) d µ r ( z ) Γ ( γ ) , (7) ( φ · η )( z ) : = Z G s ( z ) φ ( z · g ) η ( g − 1 ) d µ s ( z ) G ( g ) , (8) C c ( Γ ; F ) h φ, ψ i ( γ ) : = Z G s ( z ) F D φ ( z · g ) , ψ ( γ − 1 · z · g ) E d µ s ( z ) G ( g ) , where r ( z ) = r ( γ ) , and (9) h φ, ψ i C c ( G ; E ) ( g ) : = Z Γ r ( z ) D φ ( γ − 1 · z ) , ψ ( γ − 1 · z · g ) E E d µ r ( z ) Γ ( γ ) , where s ( z ) = r ( g ) , (10) Proof. See [12] or [8].  W e will ado pt the fo llowing notations. Notations 3.10. 1. For the sake of simplicity , w e wil l sometimes write ⋆ h· , ·i for C c ( Γ ; F ) h· , · i and h· , ·i ⋆ for h · , · i C c ( G ; E ) . 2. As in [18] , if ξ ∈ C c ( G ; E ) , η ∈ C c ( Γ ; F ) , and φ, ψ ∈ C c ( Z − 1 ; X ) , we will write ξ : φ an d φ : η for the l eft and ri ght actions of C c ( G ; E ) and C c ( Γ ; F ) on C c ( Z − 1 ; X ) , respectively , and we will write ⋆ h h φ , ψ i i for C c ( G ; E ) h φ, ψ i and h h φ , ψ i i ⋆ for h φ , ψ i C c ( Γ ; F ) . Remark 3.11. We should note that the proof of Proposition 3.9 is mostly based on the crucial result proved in [12, Proposition 6.10] that guarantees the existence of a net { f λ } λ ∈ Λ in C c ( Γ ; F ) of the form f λ = P n λ i = 1 ⋆ h φ λ i , φ λ i i , with each φ λ i ∈ C c ( Z ; X ) , which is an appro ximate identity with r espect to the inductive limit topology for both the left ac tion of C c ( Γ ; F ) on itself and on C c ( Z ; X ) . By symmetry , a si mi l ar statement holds for ( G , E ) . In particular , b y Example 3.8, the same result shows that for any Fell system ( G , E ) , C c ( G ; E ) admits an approximate ident i ty for the inductive limit topology . 4. T he linking F ell system In this se ction, we use some constructions from [13, Chapter 6] and [10, § .2]. If Γ ∼ Z G , then for m the Linking groupoid M / / / / M (0) by s etting: M : = Γ ⊔ Z ⊔ Z − 1 ⊔ G , and M (0) : = Γ (0) ⊔ G (0) , the source and r ange maps s M and r M being the obvious ones. The partial multiplication of M is given by M (2) − → M ,                                    ( γ 1 , γ 2 ) ∈ Γ (2) : γ 1 γ 2 ∈ Γ ( γ, z ) ∈ Γ ∗ Z : γ. z ∈ Z ( z , ♭ ( z ′ )) ∈ Z × G (0) Z − 1 : z .♭ ( z ′ ) : = Γ h z , z ′ i ∈ Γ ( z , g ) ∈ Z ∗ G : z . g ∈ Z ( ♭ ( z ) , z ′ ) ∈ Z − 1 × Γ (0) Z : ♭ ( z ) . z ′ : = h z , z ′ i G ∈ G ( ♭ ( z ) , γ ) ∈ Z − 1 ∗ Γ : ♭ ( z ) .γ : = ♭ ( γ − 1 . z ) ∈ Z − 1 ( g , ♭ ( z )) ∈ G ∗ Z − 1 : g .♭ ( z ) : = ♭ ( z g − 1 ) ∈ Z − 1 ( g 1 , g 2 ) ∈ G (2) : g 1 g 2 ∈ G                                    , 1 Since we are dealing with Banach ∗ -algebras, the only properties we take into acc ount here are the co ntinuity of th e actions and the pre-inner products with respect to the i n ductive limit topologies, the compatibility between the action s and the pre-inner products, and the fullness of the latters. EQUIV ALENCE OF FELL SYSTEMS AND THEIR REDUCED C ∗ -ALGEBRAS 7 so that M (2) = Γ (2) ⊔ Γ ∗ Z ⊔ Z × G (0) Z − 1 ⊔ Z ∗ G ⊔ Z − 1 ∗ Γ ⊔ Z − 1 × Γ (0) Z ⊔ G ∗ Z − 1 ⊔ G (2) . Finally , the inversion in M is defined by M − → M ,                Γ ∋ γ 7− → γ − 1 ∈ Γ Z ∋ z 7− → ♭ ( z ) ∈ Z − 1 Z − 1 ∋ ♭ ( z ) 7− → z ∈ Z G ∋ g 7− → g − 1 ∈ G                . W ith these s tructures, M / / / / M (0) is a locally compact Hausdor ff groupoi d with ope n s ource and range maps ( [13, Propositio n 6.2.2]). Now , let µ Γ and µ G be left Haar systems on Γ and G , respectively . Then, if Γ ∼ Z G , there exists a full r -sy stem 2 µ Z = { µ y Z } y ∈ Γ (0) of Radon measures o n Z determined by µ y Z ( φ ) : = Z G s ( z ) φ ( z · g ) d µ s ( z ) G ( g ) , (11) for all y ∈ Γ (0) and φ ∈ C c ( Z ), where z is some arbitrar y e l ement of the fibre Z y = r − 1 ( y ). Furthermore, µ Z is a left Haar sy stem on Z f or the left action of Γ ; tha t is , for all γ ∈ Γ and φ ∈ C c ( Z ), we h ave R Z r ( γ ) φ ( z ) d µ r ( γ ) Z ( z ) = R Z s ( γ ) φ ( γ. z ) d µ s ( γ ) Z ( z ) (see [13, § .6.4], [18] ). Simi larly , consid e ring the inverse Z − 1 : G − → Γ , the Haar system µ Γ induces a l e ft Haar sys tem µ Z − 1 = { µ x Z − 1 } x ∈ G (0) on Z − 1 for le ft action of G . Note that we have supp µ x Z − 1 = ( r ♭ ) − 1 ( x ) = Z − 1 x , and that for φ ∈ C c ( Z − 1 ) and ♭ ( z ) ∈ Z − 1 x , we have µ x Z − 1 ( φ ) : = Z Γ s ♭ ( ♭ ( z )) =Γ r ( z ) φ ( ♭ ( γ − 1 z )) d µ r ( z ) Γ ( γ ) . (12) Moreover , µ Γ , µ G , µ Z , and µ Z − 1 induces a left Haar s y stem µ M on M as it is shown in the following proposi tio n. Propositio n 4.1. Assume Γ ∼ Z G , and µ Γ and µ G are left Haar systems on Γ and G , respectively . T hen, under the above constructions, there is a left Haar system µ M = { µ ω M } ω ∈ M (0) on the linkin g groupoid M determined b y µ ω M ( F ) : =        µ ω Γ ( F | Γ ) + µ ω Z ( F | Z ) , if ω ∈ Γ (0) , and µ ω Z − 1 ( F | Z − 1 ) + µ ω G ( F | G ) , if ω ∈ G (0) , (13) for all ω ∈ M (0) and F ∈ C c ( M ) . Proof. See [13, Proposi tion 6.4.5], or [18, Lemma 4].  Propositio n 4.2. Suppose ( Γ , F ) ∼ ( Z , X ) ( G , E ) . Then we define a Banach bu ndle L over the linking groupoid M , where L is the topological space L : = F ⊔ X ⊔ X ⊔ E , the projection is given by p L : L − → M ,                F ∋ f 7− → p F ( f ) ∈ Γ X ∋ u 7− → π ( u ) ∈ Z X ∋ ♭ ( v ) 7− → ♭ ( π ( v )) ∈ Z − 1 E ∋ e 7− → p E ( e ) ∈ G                . (14) 2 See for i n stance [17 ] for the definition. 8 EL-KA ¨ IOUM M. MOUTUOU AND JEAN-LOUIS TU Moreover , p L : L − → M is a Fell bu ndle with respect to the multiplication L [2] − → m ∗ L and involution ( ∗ ) : L − → L respective ly given by                                      F γ 1 × F γ 2 ∋ ( f 1 , f 2 ) 7− → f 1 f 2 ∈ F γ 1 γ 2 , for ( γ 1 , g 2 ) ∈ Γ (2) F γ × X z ∋ ( f , u ) 7− → f · u ∈ X γ · z , for ( γ, z ) ∈ Γ ∗ Z X z 1 × X ♭ ( z 2 ) ∋ ( u , ♭ ( v )) 7− → F h u , v i ∈ F Γ < z 1 , z 2 > , for ( z 1 , ♭ ( z 2 )) ∈ Z × G (0) Z − 1 X z × E g ∋ ( u , e ) 7− → u · e ∈ X zg , for ( z , g ) ∈ Z ∗ G X ♭ ( z ) × F γ ∋ ( ♭ ( u ) , f ) 7− → ♭ ( f ∗ · u ) ∈ X ♭ ( γ − 1 z ) , for ( ♭ ( z ) , γ ) × Z − 1 ∗ Γ X ♭ ( z 1 ) × X z 2 ∋ ( ♭ ( u ) , v ) 7− → h u , v i E ∈ E < z 1 , z 2 > G , for ( ♭ ( z 1 ) , z 2 ) ∈ Z − 1 × Γ (0) Z E g × X ♭ ( z ) ∋ ( e , ♭ ( u )) 7− → ♭ ( u · e ∗ ) ∈ X ♭ ( z g − 1 ) , for ( g , ♭ ( z )) ∈ G ∗ Z − 1 E g × E h ∋ ( e 1 , e 2 ) 7− → e 1 e 2 ∈ E gh , for ( g , h ) ∈ G (2)                                      (15) and ( ∗ ) : L → L ,                F γ ∋ f 7− → f ∗ ∈ F γ − 1 , for γ ∈ Γ X z ∋ u 7− → ♭ ( u ) ∈ X ♭ ( z ) , for z ∈ Z X ♭ ( z ) ∋ ♭ ( v ) 7− → v ∈ X z , for ♭ ( z ) ∈ Z − 1 E g ∋ e 7− → e ∗ ∈ E g − 1 , for g ∈ G                . (16) L is called the linking Fell bundle , and ( M , L ) is the linking Fell s ystem . Proof. It is clear that p L : L − → M is a Banach bundle. Next, observe that all of the conditions of Definition 2.1 are verified by the operations (15) an d (16) b y merely apply ing Definition 3 .6 to th e equivalences ( Z , X ) and ( Z − 1 , X ).  At this point, we can do integration on M with values on the the linking F ell bundle L . W e then can for m the conv olution ∗ -algebra C c ( M ; L ). Note t hat we have an isomorphism of conv olution ∗ -algebras C c ( M ; L )  C c ( Γ ; F ) ⊕ C c ( Z ; X ) ⊕ C c ( Z − 1 ; X ) ⊕ C c ( G ; E ); so that an el ement ξ ∈ C c ( M ; L ) can be written as a matrix ξ =       ξ 11 ξ 12 ξ 21 ξ 22       , where ξ 11 : = ξ | Γ ∈ C c ( Γ ; F ) , ξ 12 : = ξ | Z ∈ C c ( Z ; X ) , ξ 21 : = ξ | Z − 1 ∈ C c ( Z − 1 ; X ), and ξ 22 : = ξ | G ∈ C c ( G ; E ). W ith respe ct to this de composition, the involution in C c ( M ; L ) is g iven by ξ ∗ =       ξ ∗ 11 ξ ∗ 21 ξ ∗ 12 ξ ∗ 22       =       ξ ∗ 11 ♭ ◦ ξ 21 ◦ ♭ ♭ ◦ ξ 12 ◦ ♭ ξ ∗ 22       , where ξ ∗ 11 and ξ ∗ 22 are the images of ξ 11 and ξ 22 under the standard involutions in C c ( Γ ; F ) and C c ( G ; E ), respectively . Furthermore, routine calculations (cf. [8 ]) show that the convolution in C c ( M ; L ) is g iven by           ξ 11 ξ 12 ξ 21 ξ 22           ∗           η 11 η 12 η 21 η 22           =            ξ 11 ∗ η 11 + ⋆ h ξ 12 , η ∗ 21 i ξ 11 · η 12 + ξ 12 · η 22 ( η ∗ 11 · ξ ∗ 21 ) ∗ + ( η ∗ 21 · ξ ∗ 22 ) ∗ h ξ ∗ 21 , η 12 i ⋆ + ξ 22 ∗ η 22            . (17) Suppose ( Γ , F ) ∼ ( Z , X ) ( G , E ). For x ∈ G (0) , we also denote by X − → Z x the pull-back of X − → Z along t he inclusion Z x ֒ → Z . Then, L 2 ( Z x ; X ) is the completion of C c ( Z x ; X ) with r espect to the A x -valued inner product h φ , ψ i ⋆ ( x ) = R Γ r ( z ) h φ ( γ − 1 · z ) , ψ ( γ − 1 · z ) i E d µ r ( z ) Γ ( γ ), wh ere s ( z ) = x , and the right A x -action ( φ · a )( z ) : = φ ( z ) a , for φ ∈ C c ( Z x ; X ) , a ∈ A x . Thus, L 2 ( Z x ; X ) is a Hilbert A x -module. Similarly , for all y ∈ Γ (0) , one can form the Hilber t B y -module L 2 ( Z − 1 y ; X ). The fo llowing proposi tion will be crucial i n the proof of the equivalence theorem (Theorem 5.5). EQUIV ALENCE OF FELL SYSTEMS AND THEIR REDUCED C ∗ -ALGEBRAS 9 Propositio n 4.3. Su ppose ( Γ , F ) ∼ ( Z , X ) ( G , E ) . For x ∈ G (0) , the left action of C c ( Γ ; F ) on C c ( Z x ; X ) induces a ∗ - repr esentation R Γ x : C c ( Γ ; F ) − → L A x ( L 2 ( Z x ; X )) that factors through the C ∗ -algebra C ∗ r ( Γ ; F ) . Similarly , for all y ∈ Γ (0) , we get a representation R G y : C ∗ r ( G ; E ) − → L B y ( L 2 ( Z − 1 y ; X )) . Proof. Let ξ ∈ C c ( Γ ; F ); then for φ, ψ ∈ C c ( Z x ; X ), simple calculations give h ξ · φ , ψ i ⋆ ( x ) = h φ , ξ ∗ · ψ i ⋆ ( x ). It follows that the A x -linear operator C c ( Z x ; X ) ∋ φ 7− → ξ · φ ∈ C c ( Z x ; X ) is adjointable, and then bounded with respect to the norm k · k L 2 ( Z x ; X ) , which g ives the ∗ -representation R Γ x : C c ( Γ ; F ) − → L A x ( L 2 ( Z x ; X )) , ξ − → ( R Γ x ( ξ ) : φ 7− → ξ · φ ). Now , let z 0 ∈ Z x , and let y : = r ( z 0 ). Then, to comple te t he proof it su ffi ces to check tha t for all ξ ∈ C c ( Γ ; F ), k R Γ x ( ξ ) k ≤ k π Γ y ( ξ ) k , where π Γ y : C c ( Γ ; F ) − → L B y ( L 2 ( Γ y ; F )) is the representation d efined in Lemma 2.7. Consider the (left) Hilber t B y -module X z 0 , and for m the interior tensor product L 2 ( Γ y ; F ) ⊗ B y X z 0 which is a ri g ht Hilbert A x -module under the operations defined on simple tensor s by: ( ξ ⊗ u ) · a : = ξ ⊗ ( ua ), and h ξ ⊗ u , η ⊗ v i : = h u , h ξ, η i B y · v i A x . Then, the map u z 0 : L 2 ( Γ y ; F ) ⊗ B y X z 0 − → L 2 ( Z x ; X ) , X i ξ i ⊗ u i 7− → X i ξ i · u i , (18) where for ξ ∈ C c ( Γ y ; F ) and u ∈ X z 0 , ( ξ · u )( z ) : = ξ ( Γ < z , z 0 > ) · u ∈ X Γ < z , z 0 > · z 0 , is an isomorphism o f Hilber t A x -modules. The map (18) is clearly A x -linear and injective. T o see tha t it i s surjective, first notice that the well defined map Z x ∋ z 7− → Γ < z , z 0 > ∈ Γ y , is a homeomorphism of Γ -spaces (its inverse being Γ y ∋ γ 7− → γ · z 0 ∈ Z x ). Next, for all z ∈ Z x , the linear span o f the image of F Γ < z , z 0 > × X z 0 ∋ ( f , u ) 7− → f · u ∈ X Γ < z , z 0 > · z 0 is de nse i n X Γ < z , z 0 > · z 0 by definition of a Fell pair; s o that, using the W eiers trass theorem, span n η · u : Z x ∋ z 7− → η ( Γ < z , z 0 > ) · u ∈ X Γ < z , z 0 > · z 0 | η ∈ C c ( Γ y ; F ) , u ∈ X z 0 o is d ense in C c ( Z x ; X ) in the inductive limit topology . It follo ws that any φ ∈ C c ( Z x ; X ) is the inductive limit of some P i η i · u i = u z 0 ( P i η i ⊗ u i ). W e then have an isomor phism of C ∗ -algebras ˜ u z 0 : L A x ( L 2 ( Γ y ; F ) ⊗ B y X z 0 ) − → L A x ( L 2 ( Z x ; X )) such that ˜ u z 0 ( T ) ( P i ξ i · u i ) : = u z 0 ( T ( P i ξ i ⊗ u i )) , for all T ∈ L A x ( L 2 ( Γ y ; F ) ⊗ B y X z 0 ). Fur themore, the fol lowing diagram i s commutative C c ( Γ ; F ) π Γ y   R Γ x / / L A x ( L 2 ( Z x ; X )) L B y ( L 2 ( Γ y ; F )) / / L A x ( L 2 ( Γ y ; F ) ⊗ B y X z 0 ) ˜ u z 0 O O where the lower horizontal arrow is the map T 7− → T ⊗ id (cf. for instance [7, p.50]). Indeed, l e t ξ ∈ C c ( Γ ; F ), and φ ∈ C c ( Z x ; X ). W ithout loss of generality , we can suppo s e that φ = η · u ; then, ˜ u z 0 ( π Γ y ( ξ ) ⊗ id) φ = ( π Γ y ( ξ ) ⊗ id)( η ⊗ u ) = ( π Γ y ( ξ ) η ) ⊗ u = ( ξ ∗ η ) · u = ξ · ( η · u ) = R Γ x ( ξ )( η ⊗ u ) = R Γ x ( ξ ) φ, which compl etes the proof si nce u z 0 is an isomorp his m and k π Γ y ( ξ ) ⊗ id k ≤ k π Γ y ( ξ ) k (s ee [7, p.50]).  5. T he equiv a lence theorem for reduced C ∗ - algebras of F ell systems W e start this section by the foll owing observations. Let ( G , E ) be a Fell s ystem and let A : = C 0 ( G (0) ; E (0) ) be as usual. Suppose we are given a bounded continuous section f ∈ C b ( G (0) ; E (0) ). Then, for ξ ∈ C c ( G ; E ), we de fine an element L f ξ = : f ξ ∈ C c ( G ; E ) by se tting: L f ξ ( g ) : = f ( r ( g )) ξ ( g ) ∈ E g , for all g ∈ G . (19) Also, we de fine an element ξ f ∈ C c ( G ; E ) by G ∋ g 7− → ξ f ( g ) : = ξ ( g ) f ( s ( g )) ∈ E g . (20) Notice that C b ( G (0) ; E | G (0) ) is a C ∗ -algebra under pointwise o p erations and the supremum norm ( [1 , Le mma 3.2]). 10 EL-KA ¨ IOUM M. MOUTUOU AND JEAN-LOUIS TU Lemma 5.1. For all f ∈ C b ( G (0) ; E (0) ) , we have L f ∈ L ( L 2 ( G ; E )) , where L f is the element defined by (19) . M oreover , t he map L : C b ( G (0) ; E (0) ) ∋ f 7− → L f ∈ L A ( L 2 ( G ; E )) i s a ∗ -homomorphism. Proof. L f is clearl y continuous; also it is bounded since f is a bounded section (it i s straig htforward that k L f k op ≤ k f k , where k · k op is the operator norm in L ( L 2 ( G ; E ))). If ξ, η ∈ C c ( G ; E ) and x ∈ G (0) , then A h L f ξ, η i ( x ) = Z G x L f ξ ( g − 1 ) ∗ η ( g − 1 ) d µ x G ( g ) = Z G x ξ ∗ ( g ) f ( r ( g − 1 )) ∗ η ( g − 1 ) d µ x G ( g ) = A h ξ, L f ∗ η i ( x ); hence, L f is adjointable with adjoint L ∗ f : = L f ∗ . Moreover , L f 1 f 2 ( ξ ) = L f 1 ( L f 2 ( ξ )) , ∀ ξ ∈ C c ( G ; E ); thus L f 1 f 2 = L f 1 L f 2 , ∀ f 1 , f 2 ∈ C b ( G (0) ; E (0) ).  Propositio n 5.2. Let ( G , E ) be as ab ove. Then, C b ( G (0) ; E (0) ) is a C ∗ -subalgebra of M ( C ∗ r ( G ; E )) . Proof. If π l ( ξ ) ∈ C ∗ r ( G ; E ) and f ∈ C b ( G (0) ; E (0) ), we put L f ( π l ξ ) : = π l ( L f ξ ) = π l ( f ξ ) , and R f ( π l ξ ) : = π l ( ξ f ) . (21) W e ver ify that with these formulas, we obtain a double centralizer ( L f , R f ) ∈ M ( C ∗ r ( G ; E )). T o see this, observe that for ξ, η ∈ C c ( G ; E ) and g ∈ G , one has ( f ( ξ ∗ η ))( g ) = f ( r ( g )) Z G r ( g ) ξ ( h ) η ( h − 1 g ) d µ r ( g ) G ( h ) = Z G r ( g ) f ( r ( h )) ξ ( h ) η ( h − 1 g ) d µ r ( g ) G ( h ) = Z G r ( g ) ( f ξ )( h ) η ( h − 1 g ) d µ r ( g ) G ( h ) = f ξ ∗ η ; and sim i larly one shows that ( ξ ∗ η ) f = ξ ∗ η f . Moreover , we have ( ξ f ∗ η )( g ) = Z G r ( g ) ξ ( h ) f ( s ( h )) η ( h − 1 g ) d µ r ( g ) G ( h ) = Z G r ( g ) ξ ( h ) f ( r ( h − 1 g )) η ( h − 1 g ) d µ r ( g ) G ( h ) = Z G r ( g ) ξ ( h )( f η )( h − 1 g ) d µ r ( g ) G ( h ) = ( ξ ∗ f η )( g ); so that R f ( π l ( ξ )) π l ( η ) = π l ( ξ ) L f ( π l ( η )), and by continuity , for every f ∈ C b ( G (0) ; E (0) ), the pair ( L f , R f ) verifies R f ( a ) b = aL f ( b ) for all a , b ∈ C ∗ r ( G ; E ); i.e. ( L f , R f ) ∈ M ( C ∗ r ( G ; E )).  In what follows, we identify the double centralizer ( L f , R f ), and hence the element f ∈ C b ( G (0) ; E (0) ), with L f ∈ L ( L 2 ( G ; E )), by consider i ng L f as a multiplier of C ∗ r ( G ; E ) under the formulas: L f π l ( ξ ) : = π l ( L f ξ ) = π l ( f ξ ), and π l ( ξ ) L f : = R f ( π l ( ξ )) = π l ( ξ f ). Now , consider the field o f C ∗ -algebras M ( E ) : = a x ∈ G (0) M ( E x ) over G (0) . Then, denote by C str b ( G (0) ; M ( E )) the unital C ∗ -algebra (under pointwise oper ations and the supremum norm) consisting of all the bounded strictly continu ous sections of M ( E ) over G (0) (see [1 , p.7] for details). Note that EQUIV ALENCE OF FELL SYSTEMS AND THEIR REDUCED C ∗ -ALGEBRAS 11 the unit 1 ∈ C str b ( G (0) ; M ( E )) is the section given by 1 : G (0) ∋ x 7− → (id E x , id E x ) ∈ M ( E x ), where id E x : E x − → E x is the identity map. From Propositio n 5.2 we obtain the f ollowing corollar y . Corollary 5.3. Let ( G , E ) be as above. Then C str b ( G (0) ; M ( E )) is a uni tal C ∗ -subalgebra of M ( C ∗ r ( G ; E )) . Proof. The map C b ( G (0) ; E G (0) ) ∋ f 7− → L f ∈ M ( C ∗ r ( G ; E )) is non-degenerate; indeed, by considering the left Fe ll G -pair ( E , E ) determined by th e full maps E g × E h − → E gh , we se e that for f ∈ C b ( G (0) ; E (0) ) ⊂ C 0 ( G (0) ; E (0) ) and ξ ∈ C c ( G ; E ), the element L f ξ ∈ C c ( G ; E ) is nothing but the canonical action of C 0 ( G (0) ; E (0) ) on C c ( G ; E ) defined by the formula ( f · ξ ) ( g ) : = f ( r ( g )) ξ ( g ). It follows that i f { a i } i ∈ I is an approximate identity of C b ( G (0) ; E (0) ), then thanks to [11, Lemma 6.1], for all ξ ∈ C c ( G ; E ), a i · ξ − → ξ in C c ( G ; E ) with respect to the i nductive limit topology . Thus L a i π l ( ξ ) = π l ( a i · ξ ) − → π l ( ξ ) in C ∗ r ( G ; E ). Whence, L ( C b ( G (0) ; E (0) )) C ∗ r ( G ; E ) is dense in C ∗ r ( G ; E ). Now , from [ 14, § .3.12.10 and § .3.12.12], the map L ex tends to a unital strictly continuous ∗ -homomorphism M ( C b ( G (0) ; E (0) )) − → M ( C ∗ r ( G ; E )); this map is again denoted by L . Furthermore, from [1, Lemma 3.1], we have that M ( C 0 ( G (0) ; E (0) )) = C str b ( G (0) ; M ( E )), which s e ttles the result.  Propositio n 5.4. Suppose ( Γ , F ) ∼ ( Z , X ) ( G , E ) . Let χ Γ (0) and χ G (0) be the characteristic functions of Γ (0) and G (0) respective ly . Then we get two elements χ Γ (0) 1 and χ G (0) 1 of C str b ( M (0) ; M ( L )) , where 1 ∈ C str b ( M (0) ; M ( L )) , defined by scalar mul tiplication. Now define p Γ : = L χ Γ (0) 1 , and p G : = L χ G (0) 1 ∈ M ( C ∗ r ( M ; L )) . Then p Γ and p G are complement ary fu ll projections 3 in M ( C ∗ r ( M ; L )) . Proof. It is straightforward that χ Γ (0) 1 and χ G (0) 1 are complementary projections of C str b ( M (0) ; M ( E )). Hence, their images p Γ and p G are complementary projections of M ( C ∗ r ( M ; L )), by virtue of Corollary 5.3. Now , let ξ, η ∈ C c ( M ; L ). Th en π M l ( ξ ) p Γ π M l ( η ) = π M l ( ξ ∗ p Γ η ) = π M l ( ξ p Γ ∗ η ) = π M l            ξ 11 ∗ η 11 ξ 11 · η 12 ξ 21 : η 11 h ξ ∗ 21 , η 12 i ⋆            . So, to check that p Γ is full, we just have to s how that span        π M l       ξ 11 ∗ η 11 ξ 11 · η 12 ξ 21 : η 11 h ξ ∗ 21 , η 12 i ⋆       | ξ 11 ∈ C c ( Γ ; F ) , ξ 21 ∈ C c ( Z − 1 ; X ) , η 12 ∈ C c ( Z ; X ) , η 11 ∈ C c ( Γ ; E )  (22) is dense in C ∗ r ( M ; L ). But this i s not hard to verify , by using the previous results. Indeed, the existence of an approximate id entity in C c ( Γ ; F ) for both the left actions of C c ( Γ ; F ) on itself and on C c ( Z ; X ) shows that el ements of the form ξ 11 ∗ η 11 , fo r ξ 11 , η 11 ∈ C c ( Γ ; F ) span a dense subspace of C c ( Γ ; F ) and that elements o f the form ξ 11 · η 12 , for η 12 ∈ C c ( Z ; X ) , span a dense subspace of C c ( Z ; X ). Also, that el ements of the form ξ 21 : η 11 , where ξ 21 ∈ C c ( Z − 1 ; X ) , η 11 ∈ C c ( Γ ; F ), span a dense subspace of C c ( Z − 1 ; X ) fol l ows from the existence of an approximate identity in C c ( Γ ; F ) fo r th e right action of C c ( Γ ; F ) on C c ( Z − 1 ; X ) (cf. Remark 3.11). F inally , since C c ( Z ; X ) is a full pre-inner product C c ( Γ ; F )- C c ( G ; E )-bimodule (Pr oposition 3.9), the image of h· , ·i ⋆ is a dense subspace of C c ( G ; E ). W e th en have shown that C ∗ r ( M ; L ) p Γ C ∗ r ( M ; L ) i s dense in C ∗ r ( M ; L ). I n a similar fashion, we get t hat C ∗ r ( M ; L ) p G C ∗ r ( M ; L ) is d ense in C ∗ r ( M ; L ), which completes the proof.  3 Recall from [2] that a projection p ∈ M ( A ) is said to be full if pAp is no t c o ntained in any proper clos e d two-sided ideal of A ; th at is , span { ApA } is dens e in A (see for instance [3] or [15 , p.50]). In th is case, we say t h at pAp is a full c o rner of A . T wo projection s p , q ∈ M ( A ) are c omplementary if p + q = 1, in which case pAq is a pAp - qAq -imprimitivity bimodule; i. e. pAp and qAq are Mo r i ta eq uivalent. Conversely , two C ∗ -algebras A and B are Mo rita eq uivalent if and o nly if th ere is a C ∗ -algebra C with complementary full c orners isomorphic to A and B , respectively (cf. [3, T heorem 1.1], [15, Theorem 3.19]). 12 EL-KA ¨ IOUM M. MOUTUOU AND JEAN-LOUIS TU Theorem 5.5. Let Γ and G b e locally comp act Hausdor ff groupoids. Suppose ( Γ , F ) ∼ ( Z , X ) ( G , E ) . Then the isomorphisms of convolut ion ∗ -algebras C c ( Γ ; F ) ∋ ξ 11 7− →       ξ 11 0 0 0       ∈ p Γ C c ( M ; L ) p Γ , (23) and C c ( G ; E ) ∋ η 22 7− →       0 0 0 η 22       ∈ p G C c ( M ; L ) p G (24) extend to two isomorphisms of C ∗ -algebras C ∗ r ( Γ ; F ) − → p Γ C ∗ r ( M ; L ) p Γ , and C ∗ r ( G ; E ) − → p G C ∗ r ( M ; L ) p G . (25) In particular , C ∗ r ( Γ ; F ) and C ∗ r ( G ; E ) are Morita equi valent wi th imprimitivity bi module p Γ C ∗ r ( M ; L ) p G which is isometrically isomorphic to the completion X r of C c ( Z ; X ) i n the norm k φ k E : = kh φ , φ i ⋆ k 1 / 2 C ∗ r ( G ; E ) , for φ ∈ C c ( Z ; X ) . Proof. That the maps defined by (23) and (24) are is omorphisms of convolutions ∗ -algebras is obvious. As previously , let us put B : = C 0 ( Γ (0) ; F (0) ) and A : = C 0 ( G (0) ; E (0) ). Then C 0 ( M (0) ; L (0) )  B ⊕ A , as C ∗ -algebras. Now , with respect to this decomp o sition, simple calculations s how that B ⊕ A h ξ, η i =  B h ξ 11 , η 11 i + ⋆ h ξ ∗ 21 , η ∗ 21 i | Γ (0)  ⊕  h ξ 12 , η 12 i ⋆ | G (0) + A h ξ 22 , η 22 i  , (26) for all ξ =       ξ 11 ξ 12 ξ 21 ξ 22       and η =       η 11 η 12 η 21 η 22       in C c ( M ; L ). In particular , suppos e that ξ =       ξ 11 0 0 0       ∈ p Γ C c ( M ; L ) p Γ , then B ⊕ A *       ξ 11 0 0 0       ,       ξ 11 0 0 0       + = B h ξ 11 , ξ 11 i ⊕ 0 , so that             ξ 11 0 0 0             L 2 ( M ; L ) = k ξ 11 k L 2 ( Γ ; F ) ; (27) thus, (23 ) extends to an isometric B -li near map u Γ of B -mod ules u Γ : L 2 ( Γ ; F ) − → p Γ L 2 ( M ; L ) p Γ , where p Γ L 2 ( M ; L ) p Γ is the completion of p Γ C c ( M ; L ) p Γ with respect to the norm of L 2 ( M ; L ). S imilarly , fo r ξ 22 ∈ C c ( G ; E ), we g et B ⊕ A *       0 0 0 ξ 22       ,       0 0 0 ξ 22       + = 0 ⊕ A h ξ 22 , ξ 22 i , and hence             0 0 0 ξ 22             L 2 ( M ; L ) = k ξ 22 k L 2 ( G ; E ) ; (2 8) so that (24) extends to an iso me tric A -linear map u G of A -modules u G : L 2 ( G ; E ) − → p G L 2 ( M ; L ) p G . Furthermore, s ince u Γ and u G are surjective, then from [7, Theorem 3.5], they are unitaries in L B ( L 2 ( Γ ; F ) , p Γ L 2 ( M ; L ) p Γ ) and L A ( L 2 ( G ; E ) , p G L 2 ( M ; L ) p G ), respectively ; in other words, L 2 ( Γ ; F ) ≈ p Γ L 2 ( M ; L ) p Γ EQUIV ALENCE OF FELL SYSTEMS AND THEIR REDUCED C ∗ -ALGEBRAS 13 as Hil bert B -mod ul es, and L 2 ( G ; E ) ≈ p G L 2 ( M ; L ) p G as Hilbert A -modules, here the s i gn ” ≈ ” stands fo r unitarily equivalent . M o reover , it i s very easy to see tha t the following d iagrams commute: C c ( Γ ; F ) π Γ l    / / p Γ C c ( M ; L ) p Γ π M l   L  L 2 ( Γ ; F )   / / L  p Γ L 2 ( M ; L ) p Γ  C c ( G ; E ) π G l    / / p G C C ( M ; L ) p G π M l   L  L 2 ( G ; E )   / / L  p G L 2 ( M ; L ) p G  (29) It then only remains to check that for ξ =       ξ 11 0 0 0       and η =       0 0 0 η 22       , we have k ξ k C ∗ r ( M ; L ) = k ξ 11 k C ∗ r ( Γ ; F ) and k η k C ∗ r ( M ; L ) = k η 22 k C ∗ r ( G ; E ) which will lead to the desired isomorphisms of C ∗ -algebras (25) since p Γ and p G are complementary (cf. Proposition 5.4). However , by symmetry i t su ffi ces to check one of the latter equalities. T o this end, we will use the constructions of Lemma 2.7. Note that we have C c ( M ω ; L ) =      C c ( Γ y ; F ) ⊕ C c ( Z − 1 y ; X ) , if ω = y ∈ Γ (0) ; C c ( Z x ; X ) ⊕ C c ( G x ; E ) , if ω = x ∈ G (0) In other words, ele ments of C c ( M y ; L ), for y ∈ Γ (0) , are of the form       η 11 0 η 21 0       with η 11 ∈ C c ( Γ y ; F ) and η 21 ∈ C c ( Z − 1 y ; X ), while elements o f C c ( M x ; L ), for x ∈ G (0) , ar e of the fo rm       0 η 12 0 η 22       with η 12 ∈ C c ( Z x ; X ) and η 22 ∈ C c ( G x ; E ). Then, for all y ∈ Γ (0) , and η, ζ ∈ C c ( M y ; L ), one has h η, ζ i B y = Z Γ y η 11 ( γ ) ∗ ζ 11 ( γ ) ∗ d ( µ Γ ) y ( γ ) + Z Z − 1 y F h η 21 ( ♭ ( z )) , ζ 21 ( ♭ ( z )) i d ( µ Z − 1 ) y ( ♭ ( z )) , where ( µ Z − 1 ) y is the Radon measure o n Z − 1 with suppo rt Z − 1 y , which is the image of µ y on Z under the ”inversion” Z − 1 − → Z , ♭ ( z ) 7− → z ; it is then g iven by ( µ Z − 1 ) y ( φ ) = Z G r ♭ ( ♭ ( z )) φ ( φ ( g − 1 · ♭ ( z ))) d µ r ♭ ( ♭ ( z )) G ( g ) , for φ ∈ C c (Z − 1 ) . So, by using Notations 3.10, we g e t h ξ, η i B y = h η 11 , ζ 11 i B y + h h η 21 , ζ 21 i i ⋆ ( y ); hence L 2 ( M y ; L ) = L 2 ( Γ ; F ) ⊕ L 2 ( Z − 1 y ; X ). In the same way , we veri fy that L 2 ( M x ; L ) = L 2 ( Z x ; X ) ⊕ L 2 ( G x ; E ). Thus, for all ξ ∈ C c ( M ; L ), we have k ξ k C ∗ r ( M ; L ) = max        sup y ∈ Γ (0) k π M y ( ξ ) k , s up x ∈ G (0) k π M x ( ξ ) k        . In particular , if ξ =       ξ 11 0 0 0       ∈ C c ( M ; L ), and y ∈ Γ (0) , then π M y ( ξ ) = π Γ y ( ξ 11 ) ⊕ 0, so that k ξ k C ∗ r ( M ; L ) = max        k ξ 11 k C ∗ r ( Γ ; F ) , s up x ∈ G (0) k π M x ( ξ ) k        . (3 0) Now , let x ∈ G (0) , and suppose η ∈ C c ( M x ; L ) is such that k η k L 2 ( M x ; L ) ≤ 1; i.e . max n k η 12 k L 2 ( Z x ; X ) , k η 22 k L 2 ( G ; E ) o ≤ 1. Then, from a s imple calculation we obtain h π M x ( ξ ) η, π M x ( ξ ) η i A x = h ξ 11 · η 12 , ξ 11 · η 12 i ⋆ ( x ) = h R Γ x ( ξ 11 ) η 12 , R Γ x ( ξ 11 ) η 12 i ⋆ ( x ); hence, by applyi ng Proposition 4.3, we ge t k π M x ( ξ ) η k L 2 ( M x ; L ) = k R Γ x ( ξ 11 ) η 12 k L 2 ( Z x ; X ) ≤ k ξ 11 k C ∗ r ( Γ ; F ) . Ther efore, from (30), we get k ξ k C ∗ r ( M ; L ) = k ξ 11 k C ∗ r ( Γ ; F ) .  14 EL-KA ¨ IOUM M. MOUTUOU AND JEAN-LOUIS TU Recall ( [ 6, p.14]) that if A is a Dix mier-Douady bundle over G , and Γ ∼ Z G , we de fine the pull- b ack A Z over Γ as the quotient space of the pull-back s ∗ A : = { ( z , a ) ∈ Z × A | s ( z ) = p ( a ) } by G , where G acts on s ∗ A (on the right) by ( z , a ) · g : = ( z · g , α − 1 g ( a )). Corollary 5.6. Assume that ( A , G , α ) is a Dixmier-Douady bu ndle, and that Γ ∼ Z G . Then A ⋊ r G ∼ Morita A Z ⋊ r Γ . Proof. Obser ve that for γ ∈ Γ , the fibre ( s ∗ Γ A Z ) γ = A Z s Γ ( γ ) is i dentified with ( Z s Γ ( γ ) × G (0) A ) / G . Conside r the C ∗ -bundle s ∗ A − → Z . Then, the F ell system ( Γ , s ∗ Γ A Z ) acts on ( Z , s ∗ A ) on the left via Z s Γ ( γ ) × G (0) A G × A s ( z ) ∋ ([ z , a ] , b ) 7− → ab ∈ A s ( γ z ) = A s ( z ) , (31) where ( γ, z ) ∈ Γ ∗ Z . Also, we have a rig ht Fel l G -pair ( s ∗ G A , s ∗ A ) over Z determined by the rig ht action A s ( z ) × A s G ( g ) ∋ ( a , b ) 7− → α − 1 g ( a ) b ∈ A s ( zg ) = A s G ( g ) . (32) Next, define the inner products in the obvious way: if ( z , ♭ ( z ′ )) ∈ Z × G (0) Z − 1 , we set A s ( z ) × A s ( z ) ∋ ( a , ♭ ( b )) 7− → [ z , ab ∗ ] ∈ ( s ∗ Γ A Z ) Γ < z , z ′ > = Z s Γ ( Γ < z , z ′ > ) × G (0) A G , (33) and if ( ♭ ( z ) , z ′ ) ∈ Z − 1 × Γ (0) Z , we put A s ( z ) × A s ( z ′ ) ∋ ( ♭ ( a ) , b ) 7− → α − 1 < z , z ′ > G ( a ) ∗ b ∈ ( s ∗ G A ) < z , z ′ > G = A s G ( < z , z ′ > G ) = A s ( z ′ ) . (34) It is not hard to check that the se ttings ( 31), ( 32), ( 33) , and ( 34 ) give an equivalence of Fell s ystems ( Γ , s ∗ Γ A Z ) ∼ ( Z , s ∗ A ) ( G , s ∗ G A ). W e thus comp l ete the proof by applying T heorem 5.5 .  Remark 5.7. In particular , it results f rom the last corollary that twisted K-theory ( [21] ) is in variant un der Morita equivalences of locally compact Hausdor ff groupoids; i .e. i f A ∈ Br ( G ) , and Γ ∼ Z G , one has K ∗ A ( G )  K ∗ A Z ( Γ ) . 6. T he reduced C ∗ - algebra of an S 1 - central extension Let G be groupoid. Recall that ( [21], [6], [20]) an S 1 -central extension of G is a pair ( S 1 / / e Γ π / / Γ , P ), where S 1 / / e Γ π / / Γ is a central groupoid extensio n, and Γ ∼ P G ; that is e Γ (0) = Γ (0) , S 1 acts contin uously on e Γ , and π : e Γ − → Γ i s an S 1 -principal bundle. Such an object is symbolized as ( e Γ , P ) if there is no risk of confusion. W e say that ( e Γ 1 , P 1 ) and ( e Γ 2 , P 2 ) are Morita equivalent if there exists an S 1 -equivariant Morita equivalence Z : e Γ 1 − → e Γ 2 4 such that the following d iagrams commute (in terms of g eneralized morphisms ): Γ 1 Z / S 1 / / P 1   @ @ @ @ @ @ @ @ Γ 2 P 2   G and Γ 1 Z / S 1 / / Z δ 1 A A A A A A A A Γ 2 Z δ 2   Z 2 The set of Morita eq uivalence classes of S 1 -central ex tensio ns of G is an Abelian group denoted by Ext ( G , S 1 ). Note that the inverse of a class [ E ] in Ext ( G , S 1 ) is the class of the opposite E op defined as fo l lows: if E = ( e Γ , P ), then E op : = ( e Γ op , P ), where e Γ op is e Γ as a topolog i cal groupoid but the S 1 -action is the conjugate o ne; i.e. t · ˜ γ op : = ( t − 1 ˜ γ ) op . It is known ( [6], [19], [8], [21]) that elements of Ext ( G , S 1 ) are in bijection to those of of the 2-cohomology group ˇ H 2 ( G • , S 1 ); mo re precisely , there is an is morphism of abelian groups Ext ( G , S 1 )  ˇ H 2 ( G • , S 1 ); we refer to [6], [19], [ 8] 4 Let π i : e Γ i − → Γ i , i = 1 , 2 be an S 1 -principal bundle. A generalised morphi sm Z : e Γ 1 − → e Γ 2 is said to be S 1 -equivariant if there is an action of S 1 on Z such that ( λ ˜ γ 1 ) · z · ˜ γ 2 = ˜ γ 1 · ( λ z ) · ˜ γ 2 = ˜ γ 1 · z · ( λ ˜ γ 2 ), for any ( λ, ˜ γ 1 , z , ˜ γ 2 ) ∈ S 1 × e Γ 1 × Z × e Γ 2 such that thes e products make sense. Is is shown ( [21]) that such a morphism induces a gen e ralized morphism Z / S 1 : Γ 1 − → Γ 2 . EQUIV ALENCE OF FELL SYSTEMS AND THEIR REDUCED C ∗ -ALGEBRAS 15 for more details. Given an S 1 -central ex tension E = ( e Γ , P ) o f G , we form the F ell system ( L , Γ ) as foll ows: L : = e Γ × S 1 C : = ( e Γ × C ) / ( ˜ γ, t ) ∼ ( λ · ˜ γ,λ − 1 t ) ,λ ∈ S 1 . If [ ˜ γ, t ] denote the class of ( ˜ γ, t ) ∈ e Γ × C in L , then we get a line bundle over Γ by setting L ∋ [ ˜ γ, t ] 7− → π ( ˜ γ ) ∈ Γ . Next, we define the multipli cation and the ∗ -involution on L as L γ 1 × L γ 2 ∋ ([ ˜ γ 1 , t 1 ] , [ ˜ γ 2 , t 2 ]) 7− → [ ˜ γ 1 ˜ γ 2 , t 1 t 2 ] ∈ L γ 1 γ 2 , for ( γ 1 , g 2 ) ∈ Γ (2) , and L γ ∋ [ ˜ γ , t ] 7− → [ ˜ γ − 1 , t − 1 ] ∈ L γ − 1 , respectively . Definition 6.1. The reduced C ∗ -algebra C ∗ r ( E ) of E is defined as the reduced C ∗ -algebra of the Fell system ( L , Γ ) ; i.e. C ∗ r ( E ) : = C ∗ r ( Γ ; L ) . Propositio n 6.2. (Compare with [21, Proposi tion 3.3] ). Suppose E 1 ∼ E 2 ; i.e. [ E 1 ] = [ E 2 ] in Ext ( G , S 1 ) . Then C ∗ r ( E 1 ) ∼ Morita C ∗ r ( E 2 ) . Proof. Suppose E i = ( S 1 / / e Γ i π i / / Γ i , δ i , P i ), and L i : = e Γ 1 × S 1 C . If Z : e Γ 1 − → e Γ 2 is an S 1 -equivariant Morita equivalence, then take X : = Z × S 1 C = Z × C / ( z , t ) ∼ ( λ z ,λ − 1 t ) . Then, X is a line bundle over Z / S 1 , the projection being the map [ z , t ] 7− → [ z ], where [ z ] is the class of z in the quotient space Z / S 1 . Furthermore, it is easy to verify that ( Z / S 1 , X ) imp l ements an equivalence of Fell systems ( Γ 1 , L 1 ) ∼ ( Γ 2 , L 2 ) . Therefore, o ur asser tion follows from Theorem 5.5.  Let us now recall some construct ions that we will need in the next result (see for instance [6, 20, 8] for more details). F rom an S 1 -central extension E = ( e Γ , P ) of G , one constructs a D ixmier-Douady bundle ( A E , G , α E ) in the following way . Let µ e Γ = { µ y e Γ } y ∈ Γ (0) be a Haar system on e Γ . For any y ∈ Γ (0) , de fine the space C c ( e Γ y ; H ) S 1 of compactly s upported continuous H -valued S 1 - equivariant functions on e Γ x as C c ( e Γ y ; H ) S 1 : = n ξ ∈ C c ( e Γ y ; H ) | ξ ( t · ˜ γ ) = t − 1 ξ ( ˜ γ ) , ∀ t ∈ S 1 , ˜ γ ∈ e Γ y o . Next, define a scalar product h· , ·i y on C c ( e Γ y ; H ) S 1 by h ξ, ζ i y : = R e Γ y h ξ ( ˜ γ ) , ζ ( ˜ γ ) i d µ y e Γ ( ˜ γ ). Denote by H e Γ y : = L 2 ( e Γ y ; H ) S 1 the Hilbert space obtained by completing C c ( e Γ y ; H ) S 1 with respect to h· , ·i y , and let H e Γ : = ` y ∈ Γ (0) H e Γ y . Then H e Γ − → Γ (0) , being a counta bly generated continuous field of i nfinite-dimensional Hilbert spaces over the finite dimensional locally compact space Γ (0) , is a locally trivial Hilber t bundle (cf. [4, Th ´ eor ` eme 5]). Moreover , e Γ acts continuously and by unitaries on H e Γ under the operation: C c ( e Γ s ( γ ) ; H ) S 1 ∋ ξ 7− →  ˜ γ · ξ : e Γ r ( γ ) ∋ ˜ h 7− → ξ ( ˜ γ − 1 ˜ h ) ∈ H  ∈ C c ( e Γ r ( γ ) ; H ) S 1 , for ˜ γ ∈ e Γ . Consider the continuous fiel d of elementary C ∗ -algebras K e Γ : = ` y ∈ Γ (0) K ( H e Γ y ). Then K e Γ − → Γ (0) is a lo cally trivial C ∗ -bundle wit h fibre K , according t o [4, Th ´ eor ` eme 8] (by comparing th e field H e Γ with th e trivial Hilbert bundle). Furthermore, there is a continu ous action α of Γ by automorphisms on K e Γ given by: K e Γ s ( γ ) ∋ T 7− → ˜ γ − 1 T ˜ γ ∈ K e Γ r ( γ ) , where ˜ γ is any lift of γ on e Γ , which gives us an element ( K e Γ , Γ , α ) i n Br ( Γ ). F inally , we define A E over G as the pull- back of K e Γ through the Mor ita equivalence G ∼ P − 1 Γ ; i .e. A E : = ( K e Γ ) P − 1 . T his construction g ives a homomorphism o f abelian groups Ψ : E xt ( G , S 1 ) − → B r ( G ). Convers ely , f rom a Dixmie r- Douady bundle over G , it is not hard to build an S 1 -central extension of G , and then to construct a homomorphism Br ( G ) − → Ext ( G , S 1 ) which is inverse to Ψ ( [6], [21]). Theorem 6.3. L et E ∈ Ext ( G , S 1 ) , an d let ( A E , G , α E ) in Br ( G ) be its corresponding Dixmier-Douady bun dle over G . Then, under the ab ove constructions and notations, we have A E ⋊ r G ∼ Morita C ∗ r ( E op ) . Proof. W ri te E = ( S 1 / / e Γ π / / Γ , Z ), where Z : G − → Γ is a Morita equi valence. For the sake of simplicity , we wil l denote A : = K e Γ . From Coroll ary 5.6 , we have A E ⋊ r G ∼ Morita A ⋊ r Γ : = C ∗ r ( Γ ; s ∗ K e Γ ) . 16 EL-KA ¨ IOUM M. MOUTUOU AND JEAN-LOUIS TU Thus, we only have to s how that C ∗ r ( Γ ; s ∗ A ) ∼ Morita C ∗ r ( Γ ; L ) = : C ∗ r ( E op ) , where L : = e Γ op × S 1 C . (3 5) However , again in view of the Renault’s equivalence Theorem 5.5, it su ffi ce s to build an equivalence between the Fell systems ( Γ , s ∗ A ) and ( Γ , L ). Consider the Banach bundle X : = s ∗ H e Γ over Γ defined as the pull-back of the Hilbert e Γ -bundle H e Γ − → Γ (0) through the source map o f Γ . W e claim that X implements the d esired equivalence over Γ ; that is, that ( Γ , s ∗ A ) ∼ ( Γ , X ) ( Γ , L ) . (36) From the e Γ -action on H e Γ defined in the discus s ion before the theorem, we g e t a left action of s ∗ A on X given by K ( L 2 ( e Γ s ( γ 1 ) , H ) S 1 ) × L 2 ( e Γ s ( γ 2 ) , H ) S 1 − → L 2 ( e Γ s ( γ 1 γ 2 ) , H ) S 1 ( T , ξ ) 7− → T · ξ : = ˜ γ − 1 2 T ( ˜ γ 2 ξ ) , (37) and a rig ht action o f L on X L 2 ( e Γ s ( γ 2 ) , H ) S 1 ×  e Γ op γ 3 × S 1 C  − → L 2 ( e Γ s ( γ 2 γ 3 ) , H ) S 1 ( ξ , [ ˜ γ, λ ]) 7− → ξ · [ ˜ γ, λ ] : = ˜ γ − 1 3 · λξ (38) where ˜ γ 3 is any lift of γ 3 in e Γ . T he maps ( 37) and ( 38) are continuous since the Γ -actions are contin uous. Also, they are full si nce the actions are, in fact, iso morphisms. W e now construct the s ∗ A -valued and L -valued inner products X ∗ X − → s ∗ A Γ < and X × X − → L > Γ , respectively . Note t hat, as in Example 3.8, Γ − 1 = { γ − 1 | γ ∈ Γ } , if ( γ, ♭ ( γ ′ )) ∈ Γ × Γ (0) Γ − 1 (in oth er words, s ( γ ) = s ( γ ′ )), then Γ < γ, γ ′ > = γγ ′− 1 , and if ( ♭ ( γ ′ ) , γ ”) ∈ Γ − 1 × Γ (0) Γ (i. e . r ( γ ′ ) = r ( γ ”)), then < γ ′ , γ ” > Γ = γ ′− 1 γ ”. W e then define these inner prod ucts as X γ × X γ ′ − → A s ( γγ ′− 1 ) = K ( L 2 ( e Γ r ( γ ′ ) , H ) S 1 ) ( ξ, ♭ ( η )) 7− → s ∗ A h ξ, η i : = θ ˜ γ ′ ξ, ˜ γ ′ η (39) where for ζ , ζ ′ ∈ L 2 ( e Γ y , H ) S 1 , θ ζ,ζ ′ ∈ K ( L 2 ( e Γ y , H ) S 1 ) is the rank one operator L 2 ( e Γ y , H ) S 1 ∋ ζ ” 7− → ( h ζ ′ , ζ ” i x ) ζ ∈ L 2 ( e Γ y , H ) S 1 , y ∈ Γ (0) ; and X γ ′ × X γ ” − → L γ ′− 1 γ ” = e Γ op γ ′− 1 γ ” × S 1 C ( ♭ ( ξ ) , η ) 7− → h ξ, η i L : = h ˜ γ ′ − 1 ˜ γ ” , h ˜ γ ′ ξ, ˜ γ ” η i r ( γ ′ ) i (40) where, as usual, ˜ γ ′ and ˜ γ ” are any l ifts of γ ′ and γ ”, respectively . Recall that for y ∈ Γ (0) , the scalar product h· , ·i ( y ) on H e Γ y = X y is de fined as h ξ, η i ( y ) = Z e Γ y h ξ ( ˜ h ) , η ( ˜ h ) i d µ y e Γ ( ˜ h ) ∈ C . The algebraic properties o f these maps are easy to check. The map ( 39) is full, for span n θ ζ,ζ ′ | ζ , ζ ′ ∈ L 2 ( e Γ r ( γ ′ ) , H ) S 1 o is the ideal of finite-rank operators on L 2 ( e Γ r ( γ ′ ) , H ) S 1 and the map L 2 ( e Γ s ( γ ′ ) , H ) S 1 − → L 2 ( e Γ r ( γ ′ ) , H ) S 1 given by the e Γ -action is an iso morphism of Hil bert sp aces. The map (40 ) is clearly surjective. Thus, it only remains to verify that the compatibility condition (cf. D efinition 3.6 (ii)) holds ; that is, for any triple ( γ, γ ′− 1 , γ ”) ∈ Γ × Γ (0) Γ − 1 × Γ (0) Γ , ξ · h ξ ′ , ξ ” i L = s ∗ A h ξ, ξ ′ i · ξ ” , ∀ ( ξ, ♭ ( ξ ′ ) , ξ ”) ∈ X γ × X γ ′ × X γ ” . (41) EQUIV ALENCE OF FELL SYSTEMS AND THEIR REDUCED C ∗ -ALGEBRAS 17 One has ξ · h ξ ′ , ξ ” i L = ξ · h ˜ γ ′− 1 ˜ γ ” , h ˜ γ ′ ξ ′ , ˜ γ ” ξ ” i ( r ( γ ′ )) i = ˜ γ ” − 1 ˜ γ ′ · ( h ˜ γ ′ ξ ′ , ˜ γ ” ξ ” i ( r ( γ ′ ))) ξ = ˜ γ ” − 1 · ( h ˜ γ ′ ξ ′ , ˜ γ ” ξ ” i ( r ( γ ′ )))( ˜ γ ′ ξ ) = ˜ γ ” − 1 · θ ˜ γ ′ ξ, ˜ γ ′ ξ ′ ( ˜ γ ” ξ ”) = s ∗ A h ξ, ξ ′ i · ξ ” , which compl etes the proof.  R eferences [1] Akemann, C.A., Pedersen, G.K., T omiyama, J., Multipliers of C ∗ -Algebras . Journal of Functional Analysis, 13 , 277-301 (1973). [2] Brown, L., Stable Isomorphis m of He reditary Subalgebras of C ∗ -Algebras . Pacific Journal of Mathe matics , vol. 71 , No. 2 (1977), 335-348. [3] Brown, L., Green, P ., Rie ff el. M. , Stable Isomorphism and Strong M orita Equivalence of C ∗ -Algebras . Pacific Journal of Mathematics, V ol. 71 , No. 2 (1977). [4] Dixmier , J., Douady , A., Champs continus d’espaces hilbertiens et de C ∗ -alg` ebres . B ull. So c. Math. 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D ep artment of M a thema ti cs , P aderborn U niversity , W arburger S tr . 100, D-33098 P ad erborn , G ermany , and U niversit ´ e P aul V erlaine - M etz , L MAM - CNRS UMR 7122, B ˆ atiment A, I le du S aulcy , 57000 M etz , F rance E-mail address : mout uou@m ath.upb.de U niversit ´ e P aul V erlaine - M etz , L MAM - CNRS UMR 7122, B ˆ atiment A, I le du S aulcy , 57000 M etz , F rance E-mail address : tu@u niv-m etz.fr