Gauge-equivariant Hilbert bimodules and crossed products by endomorphisms

Gauge-equiv arian t Hilb ert bim o dules and crossed pro ducts b y endomorphis ms Ezio V asselli Dip a rt imento di Matematic a University of R ome ”L a Sap ienza” P.le Aldo Mor o, 2 - 00185 R oma - Italy vasselli@mat.uniroma2.it Septem ber 22, 2021 Abstract C* -endomorphisms ari sing from sup ersel ection structures with non- trivial cen tre deﬁne a ’rank’ and a ’ﬁrst Chern cla ss’. Crossed produ cts by such endomorphisms inv olve the Cun tz- Pimsner algebra of a vector bun dle having the ab o ve-men tioned rank and ﬁrst Chern class, and can b e used to construct a d u al ity for abstract (nonsymmetric) tensor catego ries vs. group bundles acting on (nonsy mmetric) Hilbert bimodu le s. Existence and u nicit y of the dual ob ject (i.e., the ’gauge’ group bundle) are not ensured: w e give a description of this phen omenon in terms of a certain mo duli space associated with the giv en endomorphism. The ab o ve-men tioned Hilb ert bimo dules are noncomm utative analogues of gauge-equiv ariant vector bu n dles in the sense of Nistor-T roitsky . AMS Subj. Class.: 46L05, 46L08, 22D35. Keywor ds: Pimsner algebras; Crossed Produ ct s; C 0 ( X ) -algebras; T enso r C* -categories; V ector bundles. 1 In tro duction. One of the novelties in tro duced in the algebra ic approach to quantum ﬁeld theory , in pa rticular in the ca se o f lo calized quantu m charges, is the way in which sup erselection s e ctors are rega rded. In fact, whilst they are usually deﬁned as representations of the obs e rv able algebra , in the alge br aic approach they b e come endomorphisms of the observ able algebra. The main adv antage of this point of view is that the phenomenon of comp osition of charges is co n v eniently describ ed in terms of comp osition of endomo rphisms. Of cour se not all the endomorphisms of the obser v able algebra , that w e denote by A , hav e a ph ysical in terpretatio n. One of the features c haracterizing the ones of ph ysica l in terest is the symmetry , which des cribes the statistical pr operties of the sector in terms of a unitary represe n tation ε of the p erm utation group in A (see [7, 10, 11] and related references). Now, if ρ ∈ end A is an endomorphis m asso ciated with a sector , and ( ρ r , ρ s ) := { t ∈ A : tρ r ( a ) = ρ s ( a ) t , a ∈ A} , r , s ∈ N , (1.1) 1 are the intert winer spa ces, then the Doplicher-Rob erts theor y allows o ne to cons truct a cr ossed pro duct of A by ρ , in terms of a C* -algebra F generated by A a nd orthogo na l pa rtial isometries ψ 1 , . . . , ψ d , d ∈ N , such that ρ ( a ) = X i ψ i aψ ∗ i , a ∈ A (1.2) (see [9 , 10, 11]). A rema rk able prop ert y is that F comes equipp ed with a co mpact gr oup a ction G → aut F with ﬁxed-p oint a lgebra A , such that eac h ( ρ r , ρ s ) , r, s ∈ N , is interpreted as the space of G -inv ariant op erators betw een tenso r p o wers of a Hilb ert space of dimension d . In physical terms, F plays the role of the ﬁeld algebra , and G is the gauge group de s cribing the sup erselection structure of sectors ρ r , r ∈ N . The pair ( F , G ) is uniquely determined by the C* -dynamical sy stem ( A , ρ ) . A t the mathematical level, o ne of the crucial pr operties required for the construction o f F is triviality of the centre of A . In recent times, the more gener al situation in whic h A has a non trivial centre Z has b een consider ed, s o metimes coupled with a bra ided sy mmet ry (in the context of low- dimensional quantum ﬁeld theory , [17]), o ther times in the pres ence of a w eak form of permutation symmetry , that we call p erm utation quasi-symmetry (see [2, 3, 2 3 ], or § 6.1 of the pr esen t pa p er): roughly sp eaking, with this we mean that not all the elements of ( ρ r , ρ s ) , r , s ∈ N , neces sarily fulﬁll p erm utation symmetry . In the pr esen t paper we present a theory for endomorphisms with permutation qua si-symmetry , following the research line of [22, 23, 24]. Diﬀeren tly from former works, we do not make a ny assumption o n the structure of the spaces ( ρ r , ρ s ) (these w ere supp osed to b e free Z -bimo dules in [2, 3], and loc a lly trivial ﬁelds of Banach spaces in [25]). Our main cons tr uction yields a crossed pro duct F generated by A and a Hilb ert Z -bimo dule M , whose elemen ts play a role analog ous to (1 .2 ). The bimo dule M is generally no t free – and this implies that ρ do es no t fulﬁll the sp e cial c onjugate pr op erty in the sense o f [10] – , but also non- symmetric, in the se nse that the left a nd right Z -ac tio ns may not co incide; indeed, these coincide if and only if ρ has symmetry in the usual sense. Moreov er, the relation A ′ ∩ F = Z (1.3) is fulﬁlled, in acco rd with the principle sta ted in [17] in the setting of low dimensiona l quantum ﬁeld theory . In this case, we say that F is a Hilb ert exten s ion of ( A , ρ ) . Some facts hav e to b e r emark ed. Fir s t, if we denote the ﬁxed-p oin t a lgebra of Z with resp ect to the ρ -action by C ( X ρ ) , then we ﬁnd that C ( X ρ ) is contained in the centre of A and F . This implies that A and F hav e a natural structure of C ( X ρ ) -a lgebra in the se ns e of Ka sparov ([1 5 ]), with the cons e quence that we may rega rd them as bundles of C* -algebras over X ρ . Secondly , instead of a compact group w e obtain a group bundle G → X ρ playing the role of the gauge group; the action of G on F is deﬁned in terms of to the notion of ﬁbr e d action in tro duced in [24] (ro ughly sp eaking, each ﬁbre of G acts on the corr esponding ﬁbre of F ; in the present paper we will use the expression gauge action instead of the one of ﬁbr e d action ). Finally , a crucia l fact is that existence and unicity of ( F , G ) are not ensur ed; w e give a complete description of this pheno menon in terms of the space of s ections of a certain bundle of homogene o us spaces as s ociated w ith ρ . The pr esen t pap er is organized as follows. In § 3 w e study so me prop erties of gauge actions on the Cuntz-Pimsner alg ebra of a vector bundle; this will yield our mo del for symmetric endomor phisms. In § 4 w e intro duce the notion of gauge-e quivariant Hilb ert bimo dule . In our a ppr oac h to the con- struction of F , ga uge-equiv aria n t Hilb ert bimo dules sha ll substitute the Hilb ert spaces g e nerated 2 by the ab ov e-men tioned par tia l is ometries ψ i , i = 1 , . . . , d . Moreov er, we introduce the notio n o f gauge-eq uiv ariant Kaspa ro v mo dule, g eneralizing the ones o f g auge-equiv ar ian t vector bundle a nd gauge-inv ar ian t F redholm op erator in the se nse of [19]. In § 5 we analyze the way in which gauge actio ns interact with dual actions . If E → X is a vector bundle and G → X is a bundle of unitary automorphisms of E , then a dual a ction on ( A , ρ ) is a functor µ : b G → b ρ , where b G is the catego ry with ob jects the tens o r p o wers E r , r ∈ N , and a rrows the spaces ( E r , E s ) G of G -in v ariant morphisms from E r to E s , a nd b ρ is the ca tegory with ob jects ρ r , r ∈ N , and ar ro ws ( ρ r , ρ s ) . Applying a v ar ian t of the constr uction in [23, § 3], we co nstruct a cross ed pro duct C* -algebra A ⋊ µ b G equipped with a gaug e G -action and a g a uge- equiv a rian t Hilb ert Z - bimodule M ⊂ A ⋊ µ b G . This bimo dule is generally non-symmetric, and may be r egarded as a tenso r pro duct of the type b E ⊗ C ( X ) Z , where b E is the module of s ections of E . The pair ( A ⋊ µ b G , G ) is our mo del for Hilbert extens ions of ( A , ρ ) . In § 6 we prov e our main re s ults. The starting po in t is the fact that if ρ is qua si-symmetric and fulﬁlls a t wisted version of the specia l conjugate prope rt y ( § 6.2), then every vector bundle E → X ρ with suitable rank and ﬁr st Chern class induces a dual action µ : [ S U E → b ρ , where S U E is the bundle of sp ecial unitaries of E . The sp ectrum o f the cen tre of A ⋊ µ [ S U E deﬁnes by Gel’fand duality a bundle Ω E → X ρ . In Theore m 6.5, we show that the space of sections of the t yp e s : X ρ ֒ → Ω E is in o ne-to-one corresp ondence with the set of Hilb ert extensions ( A ⋊ ν b G , G ) , G ⊆ S U E . In Theor e m 6.7, we consider Hilb ert extensio ns ( A ⋊ ν b G , G ) , ( A ⋊ ν ′ b G ′ , G ′ ) o f ( A , ρ ) and give a necessary and suﬃcient condition to get a n isomo rphism G ≃ G ′ . The third Theorem 6.10 yields a co mplete cla ssiﬁcation of Hilbe r t extensions of ( A , ρ ) at v arying o f E . Finally , in Theorem 6.1 4 a dua lit y is prov ed, c haracter izing each ( ρ r , ρ s ) as the space ( M r , M s ) G of G - inv ariant ope r ators b et w een tensor powers of M ; in par ticular, we show that the spaces ( ρ r , ρ s ) ε of in tertwiners that fulﬁll per m utation s y mmet ry are isomorphic to the spaces ( E r , E s ) G . Exa mples of non-existence and non-unicity of the Hilb ert extensio n are given in § 6.3.1 and § 6.3.2. 2 Keyw ords and Notation. F or every set S , we denote the co rrespo nding iden tity map by id S . Abo ut C* -categ ories and (semi)tensor C* -catego r ies, w e refer the reader to [9, 6]. If X is a compact Hausdorﬀ space, then we denote the C* -algebra of contin uous functions from X to C by C ( X ) . F or ea ch x ∈ X , we denote the closed ideal of functions v anishing o n x b y C x ( X ) . F or every op en U ⊂ X , we denote the ideal o f functions v anishing on X − U by C 0 ( U ) ⊂ C ( X ) . If { X i } is an op en cover of X , then we write X ij := X i ∩ X j . A bund le is given by a s urjectiv e map of lo cally compac t Hausdorﬀ spaces p : Y → X . The ﬁbred pro duct with a bundle p ′ : Y ′ → X is deﬁned a s the space Y × X Y ′ := { ( y , y ′ ) ∈ Y × Y ′ : p ( y ) = p ′ ( y ′ ) } , which b ecomes a bundle when endow ed w ith the natural pro jection on X . The ﬁbr e of Y ov er x ∈ X is given by Y x := p − 1 ( x ) . A se ction of Y is given by a co n tin uous map s : X → Y such that p ◦ s = id X . The set of sections of Y is denoted by S X ( Y ) , and is endow ed with the top ology such that each map of the type c ∗ : S X ( Y ) → C 0 ( X ) , c ∗ ( s ) := c ◦ s , c ∈ C 0 ( Y ) , is con tinu ous. F or basic pro perties of ve ctor bund les , w e refer the reader to [1, 14]. In the present pap er we assume 3 that every v ector bundle is endow ed with a Hermitian structure. In particular, we will make us e o f the notion of e quivariant ve ctor bund le ([1 , § 1.6 ]), and gauge-e quivariant ve ct o r bund le ([19, § 3]). Let ρ ∈ end A , ρ ′ ∈ end A ′ be C* -endomorphisms. A C* -morphism η : A → A ′ such that η ◦ ρ = ρ ′ ◦ η is denoted b y η : ( A , ρ ) → ( A ′ , ρ ′ ) . W e denote the ide ntit y automor phism by ι ∈ end A , a nd us e the conven tion ρ 0 := ι , 0 ∈ N . Let X b e a compact Hausdor ﬀ space. A unital C ( X ) - algebr a A is a unital C* -algebra endo wed with a unital mor phism C ( X ) → A ′ ∩ A , called the C ( X ) - action (se e [15]). W e ass ume tha t the C ( X ) -action is als o injective, th us ele ments of C ( X ) ar e rega rded as element s of A . F or every x ∈ X , the quotient π x : A → A x := A / ( C x ( X ) A ) is called the ﬁbr e epimorphism . F or ev ery op en U ⊂ X , we deﬁne the r estriction A U as the closed span of elements o f the type f a , f ∈ C 0 ( U ) , a ∈ A . Note that A U is a closed ideal o f A . C* -morphisms b et ween C ( X ) -algebras equiv ar ian t with res pect to the C ( X ) -actions ar e called C ( X ) - morp hisms . T he set of C ( X ) -automorphisms (resp. C ( X ) -endomorphisms) of A is denoted by aut X A (resp. end X A ). The category of C ( X ) -algebras is equiv alent to the one with ob jects certain top ological ob jects called C* - bund les . A C* -bundle with ba se space X is given by a sur jective co n tin uous map Q : Σ → X , where Σ is a Hausdor ﬀ space such that: (1) Every Σ x := Q − 1 ( x ) , x ∈ X , is homeomor phic to a unital C* -algebra ; (2) Σ is ful l , i.e. fo r every v ∈ Σ there is a section a : X → Σ , Q ◦ a = id X , such that a ( x ) = v ; (3) The algebr aic op erations (+ , · , ∗ ) are contin uous o n each ﬁbre Σ x . The equiv a lence with the categor y o f C ( X ) -algebras is realized by recogniz ing that the set S X (Σ) of sections of a C* -bundle Σ is a C ( X ) -algebra; on the other side, every C ( X ) -algebra A deﬁnes a C* -bundle b A given by the disjoint union b A := ˙ ∪ x A x endow ed with a suitable top ology , in such a way that A is isomorphic to S X ( b A ) . If η : A → A is a C ( X ) -morphism, then we denote the asso ciated morphis m of C* -bundles by b η : b A → c A ′ ; b η is deter mined by the relatio ns b η ◦ π x ( a ) = π ′ x ◦ η ( a ) , wher e a ∈ A , x ∈ X , a nd π ′ x : A ′ → A ′ x is the ﬁbre epimo rphism. In particular, contin uous bundles of C* -algebras are C ( X ) -algebras ; the ass ociated C* -bundles are characterized by the prop erty that the pro jection Q is o p en. F or details on this topics, see [13]. If A is a C* -algebra and M a right Hilber t A - module ([4]), then we denote the group of unitar y , right A - module o perator s by U M , and the C* -algebra of co mpact, right A -module op erators by K ( M ) . Moreov er, we denote the in ternal tensor pro duct of Hilbert A -bimo dules by ⊗ A . In the present pap er we shall make use of the following c o nstruction. Let φ : C → B b e a nondegener ate C* - morphism and N a right Hilbert C -mo dule. The algebraic tens o r pro duct N ⊙ C B with co eﬃcien ts in C is endow ed with a natural B -v alued scalar pro duct; we denote the right Hilb ert B -mo dule obtained by the corresp onding completion by N ⊗ C B . In the s equel, we sha ll apply this cons tr uction to the following c ases: (1) C = C ( X ) , and B is a unital C ( X ) -algebra ; (2) C is a C ( X ) -algebra, and B = C x , x ∈ X , is the image of C with r espect to a ﬁbre epimorphism. Abo ut Cuntz-Pimsner algebr as ([20]) we follow the catego rical approach of [6]. W e recall that given d ∈ N the Cuntz algebra O d is deﬁned as the univ ersal C* -algebra gener ated by a set { ψ i } d i =1 of o rthogonal isometries with total suppo rt the identit y ([5]); for every closed subgr oup G ⊆ U ( d ) , there is an a utomorphic actio n G → aut O d , g 7→ b g : b g ( ψ i ) := X j g ij ψ j , (2.1) 4 where { g ij } is the set of matrix co eﬃcien ts of g (see [7]). 3 Group bundles and Cun tz-P im sner algebras. In the present section we give a diﬀerent version o f some results pr o ved in [22]. Instea d of fo cusing our attention to (noncompact) section g roups acting o n C* -alg ebras, w e consider the underlying group bundles. In par ticular, we make use of the no tion of ﬁbr e d action of a g roup bundle on a C ( X ) -algebr a introduced in [24]. There a r e several re a sons for this change of scenario. First, it was kept in evide nce in a previous w ork that the rele v ant pro perties of the C* -dynamica l s ystems ( O G , σ G ) in which we a re interested dep end on a g roup bundle rather than the group itself (see [22, Deﬁnition 4.3]). Secondly , ﬁbred actio ns need less technicalities, and have b een used to prov e a re sult that s hall play an imp ortant r ole in the prese n t pa p er ([24, Theorem 6.1]). Our approa c h is also motiv a ted by a r ecen t work o f V. Nistor and E. T roitsky: the no tions of ﬁbred action on a C ( X ) -algebra and ga uge-equiv aria n t Hilbert bimodule (in the sense of the following § 4) ma y b e regar ded as noncommutativ e versions of the gauge actions int ro duced in [19, § 3]. F or this r eason, but also to emphasize the role that group bundles will play in the pr esen t pa per, w e will use the term gauge action instead o f ﬁbr e d action . Let X b e a compact Hausdorﬀ space. A gr oup bund le is g iven by a bundle p : G → X s uch that each ﬁbre G x := p − 1 ( x ) is homeomorphic to a loc a lly co mpa ct gro up. In g eneral, w e do not assume that G is lo cally trivial, thus the isomor phism class of the ﬁbres may v ar y at v a rying of x in X . W e note that the C* -alg ebra C 0 ( G ) is a C ( X ) -algebra in a na tu ral wa y , and rec a ll from [24] that an invariant C ( X ) -functional is a p ositive C ( X ) -mo dule map ϕ : C 0 ( G ) → C ( X ) such that ϕz ( x ) = R G x z ( y ) dµ x ( y ) , z ∈ C 0 ( G ) , wher e µ x is a Haar measure of G x . The set S X ( G ) of sections of G is endow ed with a natural s tructure of top ological g roup. A se ct i on gr oup G is a subgro up o f S X ( G ) s uc h that for every y ∈ G there is g ∈ G with y = g ◦ p ( y ) . F or exa mple, if G 0 is a lo cally compact group and G is the triv ia l bundle X × G 0 , then the group C ( X , G 0 ) of contin uous maps from X to G 0 is a section group; anyw ay , also the group of constant G 0 -v a lued maps is a section group for G . Let G b e a gro up bundle, a nd A is a C ( X ) -algebr a. A gauge action of G o n A is a family of strongly contin uous ac tio ns { α x : G x → aut A x } x ∈ X such that the map α : G × X b A → b A , α ( y , v ) := α x y ( v ) (3.1) is contin uous. A C ( X ) -algebr a endow ed with a gauge action is calle d G - C ( X ) - al gebr a . The ﬁxed- po in t algebra A α is given by the C* -subalgebr a of thos e a ∈ A such that α x y ◦ π x ( a ) = π x ( a ) for all y ∈ G , x := p ( y ) . If G is endow ed with an inv ariant C ( X ) -functional ϕ , then a n inv ariant mean m ϕ : A → A α is na turally deﬁned. F or each section gro up G ⊆ S X ( G ) ther e is an action α G : G → aut X A (3.2) such that π x ◦ α G g ( a ) = α x g ( x ) ◦ π x ( a ) for every a ∈ A , g ∈ G , x := p ( y ) . In particular, usual strongly co n tin uous actions α 0 : G 0 → aut X A corresp ond to gauge actio ns of the tr iv ial bundles X × G 0 . The following class of ex amples shall play an impo r tan t role in the prese nt pap er. Let us consider a rank d vector bundle E → X with as sociated U ( d ) -coc ycle ( { X i } , { u ij } ) ∈ H 1 ( X, U ( d )) (see [14, I.3.5]). F or the rest of the present pap er, b E will denote the Hilber t C ( X ) -bimodule of sections of E , and { ψ l } a (ﬁnite) s et of gener ators of b E . The Cuntz-Pimsner algebra O E asso ciated with 5 b E (see [20]) c an b e descr ib ed in ter ms of ge ne r ators and relations: ψ ∗ l ψ m = h ψ l , ψ m i , f ψ l = ψ l f , n X l ψ l ψ ∗ l = 1 , (3.3) where f ∈ C ( X ) , h · , ·i is the C ( X ) -v alued scalar pro duct of b E , and 1 is the identit y . Let L := { l 1 , . . . , l r } be a multi-index of length r ∈ N (in such a ca se, w e write | L | = r ); deﬁning ψ L := ψ l 1 · · · ψ l r , (3.4) we obtain an element of O E which can b e regar ded as an element of the C ( X ) - bimodule tensor pro duct b E r . Now, O E is a cont inuous bundle over X with ﬁbre the Cuntz algebra O d (see [22, 23]). The C* -bundle b O E → X may b e describ ed a s the clutc hing of the trivial bundles X i × O d with resp ect to the transitio n maps ( { X i } , { b u ij } ) ∈ H 1 ( X, aut O d ) deﬁned using (2.1). Let p : U E → X denote the group bundle of unitary endomorphisms of E (see [14, I.4.8(c)]), and G ⊆ U E a compa c t group bundle. Then ther e is a natural action G × X E → E (3.5) in the sens e of [19, § 3], in fact each ﬁbre U E x ≃ U ( d ) acts by unitary op erators on the co rrespo nding ﬁbre E x ≃ C d . By universality of the Cuntz-Pimsner alg ebra, the action (3.5) extends to a g auge action G × X b O E → b O E , ( y , ξ ) 7→ b y ( ξ ) , (3.6) which, ﬁb erwise, behav es like (2.1). If there is a s ection group G ⊆ S X ( G ) then we ha ve an automorphic action G → aut X O E . An impo r tan t example is the bundle S U E of sp ecial unita ry endomorphisms of E , which induces the strongly contin uous a c tion SU E → aut X O E , where SU E := S X ( S U E ) (see [2 2]) . W e denote the ﬁxed-p oint algebra of O E with re s pect to the gauge action (3.6) by O G . F or future refer ence, we also introduce the canonical endomorphism σ E ( t ) := X l ψ l tψ ∗ l , t ∈ O E ; (3.7) which fulﬁlles the relations ( E r , E s ) = ( σ r E , σ s E ) , r, s ∈ N . If G ⊆ U E then σ E restricts to an endomorphism σ G ∈ end X O G such that ( E r , E s ) G = ( σ r G , σ s G ) , r , s ∈ N (see [22, § 4]). Let us now denote the tensor catego ry with ob jects E r , r ∈ N , ident ity ob ject ι := E 0 := X × C , and a r ro ws ( E r , E s ) , r , s ∈ N by E ⊗ (so that, ( ι, ι ) = C ( X ) ). By the Ser re-Sw an equiv a lence, ( E r , E s ) is the set of C ( X ) -mo dule opera to rs from b E r to b E s , and can be interpreted as the mo dule of sectio ns of a vector bundle E r,s → X (see [14, I.4.8(c)]); in this way , every t ∈ ( E r , E s ) can b e r egarded as a map t : X ֒ → E r s . By using the notation (3.4), we also ﬁnd ( E r , E s ) = span { f ψ L ψ ∗ M , f ∈ C ( X ) , | L | = s, | M | = r } ; in fact, ψ L ψ ∗ M can b e naturally identiﬁed with the o perator θ LM ϕ := ψ L h ψ M , ϕ i , ϕ ∈ b E r . Since each ψ L belo ngs to O E , every ( E r , E s ) can b e reg arded as a subspace of O E . Let us now consider the gauge action (3.5), and the s pa ces of in v ariant morphisms ( E r , E s ) G := { t ∈ ( E r , E s ) : y s · t ( x ) = t ( x ) · y r , y ∈ G , x := p ( y ) } , (3.8) 6 where y r denotes the r -fold tensor pow er of y as a line a r op erator on the ﬁbre E x ≃ C d . Then we can deﬁne a tensor c a tegory b G with ob jects E r , r ∈ N , and arr o ws ( E r , E s ) G . Note that ( ι, ι ) G = C ( X ) , a nd that b G is symmetric in the sense of [9]; in fact, the s ymmetry op erator θ ∈ ( E 2 , E 2 ) , θψ ψ ′ := ψ ′ ψ , ψ , ψ ∈ b E , b elongs to ( E 2 , E 2 ) G for every G ⊆ U E . When X reduces to a p oint , O E is the Cuntz alg e br a O d and b G is the category o f tensor powers of the deﬁning repres en tation of a co mpact Lie gro up G ⊆ U ( d ) (se e [7]). The following Lemma 3 .1 can b e prov ed using a standard ar gumen t based on the mea n m ϕ : O E → O G induced by the inv ar ian t functional ϕ : C ( G ) → C ( X ) , whilst the successive Lemma 3.2 is just a ”ﬁbred v ersio n” of [22, Cor .4.9]; so that, in bo th the cases we omit the pro of. Lemma 3.1. L et ϕ : C ( G ) → C ( X ) b e an invariant functional. Then the set 0 O G := ∪ r,s ( E r , E s ) G is dense in O G . Lemma 3. 2. L et G , G ′ ⊆ U E b e c omp act gr oup bund les, and u ∈ S X ( U E ) . If b u ∈ aut X O E r estricts t o an isomorphi sm fr om O G onto O G ′ , then G ′ = u G u ∗ := { u ( x ) G x u ( x ) ∗ , x ∈ X } . On the c onverse, if G ′ = u G u ∗ , then b u ∈ aut X O E r estricts to an isomorphism O G ′ → O G . Now, O G is a co ntin uous bundle ov er X with ﬁbres ( O G ) x , x ∈ X ; we deﬁne S G :=  u ∈ U E : b u ( t ) = t , t ∈ ( O G ) p ( u )  . It is clear that G ⊆ S G . The bundle S G is called the sp e ctr al bund le as sociated with G , and may b e regarded as a ’r egularization’ of G (see the example b elow). F o r every r , s ∈ N w e hav e ( E r , E s ) S G = ( E r , E s ) G (see [22, Lemma 4 .10]); th us, at the level of the ca tegory b G it is not p ossible to dis tin guish G from S G . In the sequel o f the present paper we wil l always assume t ha t G = S G . As an example, take X = [0 , 1] , E = X × C d , G = { ( x, u ) ∈ X × SU ( d ) : x = 1 / 2 ⇒ u = 1 } ; we ﬁnd O E = C ( X ) ⊗ O d and O G = C ( X ) ⊗ O SU ( d ) , so that G is strictly contained in S G = X × SU ( d ) . 4 Gauge-equiv arian t Hilb ert bimo dules. In the present section we discuss some basic pr operties of the catego ry of C ( X ) -Hilb ert bimo dules in the sense of Kaspa ro v ([15]), and intro duce the notion o f g auge-equiv ar ian t Hilb ert bimo dule. This class o f bimodules will yield the mo del catego ry for the dualit y that we shall pr o ve in § 6.4. Since in the sequel we shall make use o f unital C* -algebra s, to simplify the e xposition we dis cuss only the ca se in w hich the co eﬃcient alg ebra o f our bimodules is unital; the non-unital case will b e approached in a future pap er. Let A , B b e unital C ( X ) -alg e br as. W e denote the ﬁbre epimorphisms of A , B r espectively b y π x : A → A x , π ′ x : B → B x , x ∈ X , and the iden tity of B by 1 . A C ( X ) - Hilb ert A - B - bimo dule is giv en by a Hilbe r t A - B -bimodule such that ψ f = f ψ , f ∈ C ( X ) , ψ ∈ M . W e denote the category of C ( X ) - Hilbert A - B -bimo dules by bmo d X ( A , B ) , with arrows the sets ( M , M ′ ) o f a djoin table, (b ounded) right A -module o perator s T : M → M ′ , M , M ′ ∈ bmo d X ( A , B ) . In particula r , the space s of compact op erators are deno ted by K ( M , M ′ ) . By deﬁnition of C ( X ) -Hilbert bimo dule, every T ∈ ( M , M ′ ) fulﬁlles the relations T ( f ψ ) = T ( ψ f ) = T ( ψ ) f = f T ( ψ ) , ψ ∈ M , f ∈ C ( X ) . This implies that ( M , M ) is a C ( X ) -algebra, a nd the same is true fo r the ideal of compact op erators K ( M ) ⊆ ( M , M ) . The prop erty of M being a C ( X ) - Hilbert bimo dule is tra nslated as the fact 7 that the left A -module a ction φ : A → ( M , M ) is a C ( X ) - morphism. T o b e concise, in the sequel we will write aψ ≡ φ ( a ) ψ , a ∈ A , ψ ∈ M ; on the o ther s ide, the ope r ator φ ( a ) ∈ ( M , M ) shall not be confused with a ∈ A . If b L M → X is the C* -bundle ass ociated with ( M , M ) , then by general prop erties of C ( X ) -algebr as there is a mo rphism b φ : b A → b L M . (4.1) If w e denote the ﬁbre epimorphisms of ( M , M ) by δ x : ( M , M ) → L M ,x , x ∈ X , (4.2) then b φ is determined by the r elations b φ ◦ δ x = δ x ◦ φ . F o r every x ∈ X , we consider the rig h t Hilber t B x -mo dule M x := M ⊗ B B x and the asso ciated map η x : M → M x , η x ( ψ ) := ψ ⊗ η x (1) . (4.3) Note that (4.3) is surjective: in fa c t ψ ⊗ w = η x ( ψ b ) , where ψ ∈ M , w ∈ B x , a nd b ∈ π − 1 x ( w ) . Lemma 4.1. F or every x ∈ X , the sp ac e M x has the fol lowing structur e of Hilb ert A x - B x - bimo dule:    ( ψ ⊗ w ) w ′ := ψ ⊗ ( w w ′ ) h ψ ⊗ w , ψ ′ ⊗ w ′ i x := π x ( h ψ , ψ ′ i ) w ∗ w ′ v ( ψ ⊗ w ) := ( aψ ) ⊗ w , π x ( a ) = v , ψ , ψ ′ ∈ M , v ∈ b A , w , w ′ ∈ b B , a ∈ A . Mor e over, for every x ∈ X ther e is a natu r al isomorphism L M ,x ≃ ( M x , M x ) . Pr o of. The fact that the r igh t B x -mo dule action and the sca lar pro duct are well-deﬁned follows by general prop erties of internal tenso r pro ducts o f Hilber t bimo dules. Th us, it remains to verify only that the left A x -mo dule action is well-deﬁned. T o this end, let a, a ′ ∈ A x such that v = π x ( a ) = π x ( a ′ ) . Then ther e ar e f ∈ C x ( X ) , a ′′ ∈ A such that f a ′′ = a − a ′ . In this way , we ﬁnd ( aψ ) ⊗ w = [( a ′ + f a ′′ ) ψ ] ⊗ w = ( a ′ ψ ) ⊗ w + f ( x )( a ′′ ψ ) ⊗ w = ( a ′ ψ ) ⊗ w . This implies that our deﬁnition of left A x -mo dule action does not dep end on the c hoice of a ∈ π − 1 x ( v ) . Finally , if δ x , η x are as by (4.2,4.3), then an isomorphism β x : L M ,x → ( M x , M x ) is deﬁned by [ β x ◦ δ x ( t )] [ η x ( ψ )] := η x ( tψ ) , t ∈ ( M , M ) , ψ ∈ M (we leave to the reader the task to v erify that β x is a ctually an is omorphism). As for C ( X ) -algebr as, we endow the disjoint union c M := ˙ ∪ x M x with the natura l pro jection P : c M → X , and the top ology genera ted by the base T U,ε,ψ :=  ξ ∈ P − 1 ( U ) :   ξ − η P ( ξ ) ( ψ )   < ε  , (4.4) where U ⊆ X is op en, ψ ∈ M , and ε > 0 . F o r the notion of Banach bund le , s e e [12, 1 3 ] (anyw ay , it is analogo us to the one of C* -bundle). Lemma 4. 2 . The map P : c M → X deﬁnes a ful l Banach bun dle, and M c oincides with the set S X ( c M ) of se ctions of c M . 8 Pr o of. By [13, Theorem 6.2], to prove the Lemma it suﬃces to verify that M is a lo cally co n v ex Banach C ( X ) -bimo dule, i.e. that for every f ∈ C ( X ) , 0 ≤ f ≤ 1 , ψ 1 , ψ 2 ∈ M , k ψ 1 k , k ψ 2 k ≤ 1 , it turns out k f ψ 1 + (1 − f ) ψ 2 k ≤ 1 . This can be easily done b y using the B -v alued scalar pr oduct, and estimating h f ψ 1 + (1 − f ) ψ 2 , f ψ 1 + (1 − f ) ψ 2 i ≤ f 2 + (1 − f ) 2 ≤ 1 (4.5) (note that h f ψ 1 , (1 − f ) ψ 2 i = 0 ; for the last inequality , w e us ed the rela tions f 2 ≤ f , (1 − f ) 2 ≤ 1 − f ). T aking the norms of the terms of (4.5), we conclude that M is lo cally conv ex. The bundle c M may b e endow ed with further str uc tur e: in fac t, the A - B -bimo dule structure and the B -v alued scalar pro duct of M induce contin uous maps b A × X c M → c M , c M × X b B → c M , c M × X c M → b B . The cor respondence M 7→ c M established in the previous Lemma has a functor ial nature. If T ∈ ( M , M ′ ) , then there is an as sociated bundle mo rphism b T : c M → c M ′ , deter mined by the prop ert y b T ◦ η x ( ψ ) = η ′ x ◦ T ( ψ ) , where ψ ∈ M , and η x , η ′ x denote the ev alua tions of M , M ′ on x ∈ X . Example 4.1. L et q : Ω → X b e a c omp act bund le (i.e., C (Ω) is a unital C ( X ) -algebr a), and E → Ω a ve ctor bund le. F or every x ∈ X , we c onsider the ve ctor bund le obtaine d as t he r estriction E | Ω x → Ω x , Ω x := q − 1 ( x ) . We denote the set of se ctions of E by M ; cle arly, M is a C ( X ) - Hilb ert C (Ω) -bimo dule. A standar d ar gument (t he Tietze ext ensio n the or em for ve ct or bund les, se e [1, 1.6.3]) al lows one to c onclude that for every x ∈ X ther e is a natur al isomorphism V : M x := M ⊗ C (Ω) C (Ω x ) → S Ω x ( E | Ω x ) , V ( ψ ⊗ z ) := ψ | Ω x z , wher e ψ : Ω → E b elongs to M , z ∈ C (Ω x ) , and S Ω x ( E | Ω x ) is the Hilb ert C (Ω x ) - bimo dule of se ctions of E | Ω x . Thus, the bund le c M → X has ﬁbr es S Ω x ( E | Ω x ) , x ∈ X . Example 4. 2. L et A b e a C*-algebr a, X a c omp act Hausdorﬀ s p ac e, and E → X a ve ctor A - bund le in the sense of [18]. Then the set M of se ctions o f E is a C ( X ) -Hilb ert C ( X ) - ( C ( X ) ⊗ A ) - bimo dule, and c M = E . Deﬁnition 4.3. L et p : G → X denote a gr oup bund le with gauge actions α : G × X b A → b A , β : G × X b B → b B , and M a C ( X ) -Hilb ert A - B -bimo dule. A gauge action of G on M is given by a family { U x : G x × M x → M x } x of actions by isometric line ar op er ators making e ach M x a G x -Hilb ert A x - B x -bimo dule, such that the map U : G × X c M → c M , U ( y , ξ ) := U x y ξ , is c ontinuous. In such a c ase, we say that M is a G -Hilb ert A - B - bimo dule . In explicit ter ms, a G -Hilbe r t A - B -bimo dule M is characterized b y the relations   U x y ( ξ ) , U x y ( ξ ′ )  x = β x y ( h ξ , ξ ′ i x ) U x y · b φ ( v ) = b φ ◦ α x y ( v ) · U x y , (4.6) y ∈ G , x := p ( y ) , ξ , ξ ′ ∈ c M , v ∈ b A . If M , M ′ are G -Hilb ert A - B - bimodules, we deﬁne the set of G -equiv ar ian t op erators ( M , M ′ ) G := n T ∈ ( M , M ′ ) : b T ◦ U ( y , ξ ) = U ( y , b T ξ ) , y ∈ G , ξ ∈ c M o . The pro of of the following result is analo gous to [24, Pr oposition 3.3], th us it is omitted. 9 Prop osition 4. 4 . L et M b e a C ( X ) -Hilb ert A -bimo dule with a gauge action U : G × X c M → c M . Then for every se ction gr oup G ⊆ S X ( G ) ther e is an action U G : G × M → M , such that η x ◦ U G g = U x g ( x ) ◦ η x , x ∈ X , (4.7) wher e η x is deﬁne d by (4.3 ). On the c onverse, every action U G : G × M → M such that t he r e is a set { U x : G x × M x → M x } x fulﬁl ling (4.7) deﬁnes a gauge action U that do es not dep en d on G . In par ticular, if G ≃ X × G 0 is a trivial bundle, then we may take G = G 0 and o btain a G -a ction on M in the usua l sense ([4, VII I.20]). R emark 4.1 . In t he pr esent p ap er we shal l make use of actions U : G × X c M → c M c ouple d with trivial actions on the c o eﬃcient algebr as, in such a way that (  U x y ( ξ ) , U x y ( ξ ′ )  x = h ξ , ξ ′ i x h U x y , b φ ( v ) i = 0 , (4.8) y ∈ G , x := p ( y ) , ξ , ξ ∈ c M , v ∈ b A . In this c ase, U x y ∈ U M x , x ∈ X , and we may expr ess (4.8(2)) in the fol lowing, mor e c oncise way: φ ( a ) ∈ ( M , M ) G , a ∈ A . The following Lemma generalizes (3.6): Lemma 4. 5. L et M b e a G -Hilb ert A -bimo dule with trivial gauge G -actions on A . Then the Cuntz-Pimsner algebr a O M is a C ( X ) -algebr a with asso ciate d C*-bund le b O M , and the G -action on M ex tends to a gauge action G × X b O M → b O M . Pr o of. The Cun tz-Pimsner a lg ebra O M may be describ ed as the one genera ted by the spaces K ( M r , M s ) , r, s ∈ N , a s in [9, § 4], [6, § 3]. Since f t = tf , t ∈ K ( M r , M s ) , f ∈ C ( X ) , we conclude that C ( X ) is co n tained in the cent re of O M , i.e. O M is a C ( X ) -algebra. W e denote the ﬁbr es of O M by ( O M ) x , x ∈ X , and the so- obtained C* -bundle by b O M → X . By [6, § 3], it follows that for every x ∈ X there is a strongly contin uous action G x → aut ( O M ) x , y 7→ b y . In this w ay , we obtain a map G × X b O M → b O M , ( y , ξ ) 7→ b y ( ξ ) , whose contin uit y can b e easily prov ed using contin uit y of the a ction U : G × X c M → c M . Example 4.3 (Equiv ariant vector bundles , [1, 21]) . L et G b e a c omp act gr oup, and Ω a c omp act Hausdorﬀ G -sp ac e. We denote the orbit sp ac e by X := Ω /G , and c onsider the natura l pr oje ction q : Ω → X . If E → Ω is a G -e quivariant ve ct o r bu nd le, t he n the H i lb ert C (Ω) -bimo dule M of se ctions of E is natur al ly endowe d with a G -actio n. In C*-algebr aic terms, we have a str ongly c ont inuous action α : G → C (Ω) with ﬁ x e d-p oint algebr a (isomorphic to) C ( X ) , so that C (Ω) is a C ( X ) - al gebr a with asso ciate d C*-bund le b A Ω → X a nd ﬁbr es A Ω ,x ≃ C (Ω x ) , Ω x := q − 1 ( x ) , x ∈ X . As mentione d in § 3, α may b e r e gar de d as a gauge action G × X b A Ω → b A Ω , wher e G := X × G . N o te that e ach g ∈ G acts as a home omorph ism on t he r estriction Ω x , in fac t q ( g ω ) = q ( ω ) for every ω ∈ Ω ; this implies that the G -action on E r estricts to G -actions G × E | Ω x → E | Ω x , x ∈ X . By Example 4. 1, we ﬁnd M x = S Ω x ( E | Ω x ) , x ∈ X . Thus, we c onclu de t ha t M is a G -Hilb ert C (Ω) -bimo dule. Example 4.4 (Gauge-equiv ariant vector bundles, [19]) . L et p : G → X b e a gr oup bund le, and q : Ω → X a c omp act bu nd le c arrying an action G × X Ω → Ω . In other terms, the C*-algebr a C (Ω) is a C ( X ) -algebr a endowe d with a gauge action α : G × X b A Ω → b A Ω , wher e b A Ω → X is the C*-bund le asso ciate d with C (Ω) . L et π : E → Ω b e a ve ctor bund le. As in the pr evious example, we have that the s et M of se ctions of E is endowe d with a st ructur e of C ( X ) -Hilb ert C (Ω) -bimo dule. 10 Note that E is a bund le over X with r esp e ct t o the map q ◦ π , with ﬁbr es ( q ◦ π ) − 1 ( x ) = E | Ω x , x ∈ X ; t hus, it m a kes sense to c onsider the ﬁ br e d pr o duct G × X E . The ve ct or bun dle E is said to b e G -e quivariant if ther e is an action by home omorphisms G × X E → E deﬁne d in such a way that e ach E | Ω x → Ω x , x ∈ X , is a G x -e quivaria nt ve ctor bund le in the s ense of the pr evious example. This implies t h at M is a G -Hilb ert C (Ω) - bi mo dule. In the pr esent p ap er we shal l make use of t he c ase in which Ω = X is endowe d with the trivial G -action, as in § 3. A notion of Kaspa ro v cycle in the s etting of ga uge-equiv aria n t Hilbert bimodules can be in tro- duced. Let X b e a compact Hausdorﬀ s pace, G → X a gr oup bundle, a nd A , B unital, separ able Z 2 -graded G - C ( X ) -algebras. A Kasp ar ov cycle is a pair ( M , F ) , where M is a countably g ener- ated, Z 2 -graded G -Hilb ert A - B -bimodule with left A -action φ : A → ( M , M ) , a nd F = F ∗ ∈ ( M , M ) is a n op erator with degree one suc h that [ φ ( a ) , F ] , [ F 2 − 1 , φ ( a )] ∈ K ( M ) for ev ery a ∈ A ; mo r eo ver, the following ”qua si- G -equiv a riance” is required: [ b F , U x y ] ∈ K ( M x ) , y ∈ G , x := p ( y ) . The notions of homotopy , e q uiv a lence and dir ect sum a re exactly the s ame as in [15]. In this wa y , we can deﬁne the gauge-e qu i variant K K - gr oup K K G X ( A , B ) . When G = X × G is trivial we obtain the usual Kaspa ro v g roup R K K G ( X ; A , B ) ([15]). F ur ther prop erties of the bifunctor K K G X ( − , − ) (in particular, details on the non-unital case) will be discus sed in a future reference . Example 4.5. L et G → X b e a c omp act Lie gr oup bun dle acting on Hilb ert bund les H k → X , k = 0 , 1 , and F :=  F x : H 0 x → H 1 x  x a c ontinuous family of G -invariant F r e dhol m op er ators (se e [19, § 4]. Then the sets of se ctions b H k , k = 0 , 1 , ar e G -Hilb ert C ( X ) -bimo dules, and F may b e r e gar de d as an element of ( b H 0 , b H 1 ) G . L et now M := b H 0 ⊕ b H 1 , and e F :=  0 F ∗ F 0  ; then, we obtain a p air ( M , e F ) , deﬁning an element of K K G X ( C ( X ) , C ( X )) . Let M , M ′ ∈ bmo d X ( A , B ) ; we consider the space of right B -mo dule operato rs that comm ute with the left A -action: ( M , M ′ ) A := { T ∈ ( M , M ′ ) : T a = aT , a ∈ A} . (4.9) W e now fo calise our atten tion to the case A = B , and deﬁne Z := A ′ ∩ A . The class bmo d X A := bmo d X ( A , A ) beco mes a C* -categor y if equipped with the sets of a r ro ws ( M , M ′ ) , M , M ′ ∈ bmo d X A . Let us endow bmo d X A with the in ternal tensor pr oduct ⊗ A . It is well-known that at the level of arrows the tensor pro duct T ⊗ A T ′ do es not make sense, unless T ′ is the identit y ([4, 1 3.5]). In other ter ms , bmo d X A is a semitensor C* -ca tegory in the sens e of [6]. If we restrict the sets of ar r o ws to ( M , M ′ ) A , M , M ′ ∈ bmo d X A , then w e obtain a t ensor C* -categor y bmo d X, A A , having the same ob jects as bm od X A , and a rrows ( M , M ′ ) A . Note that every ( M , M ′ ) A is a Banach Z -bimo dule in the natural w ay . Let G → X denote a g roup bundle. W e denote the semitensor C* -catego ry with ob jects G - Hilber t A -bimo dules by bmo d G X A , with arrows the spaces ( M , M ′ ) G of equiv ariant op erators . Let M b e a G -Hilber t A -bimodule. The full semitensor C* -subcatego ry of bm od X A with ob jects the internal tensor p o wers M r := M ⊗ A . . . ⊗ A M , r ∈ N , is denoted by M ⊗ ; in pa rticular, we 11 deﬁne M 0 := A . Moreov er, we denote the semitensor C* -ca teg ory with ob jects M r , r ∈ N , by M ⊗ G , with arrows ( M r , M s ) G , r, s ∈ N ; note that ( A , A ) G is the ﬁxed-p oin t a lgebra with res pect to the given gauge action α : G × X b A → b A . Example 4. 6. The fol lowing c onstruction wil l b e use d in the se quel, in the c ont ext of Hilb ert extensions. L et X b e a c omp act Hausdorﬀ sp ac e, E → X a ve ctor bund le, and Z an Ab elian C ( X ) - algebr a with identity 1 . We denote the bund le deﬁne d by the sp e ctru m X ′ of Z by q : X ′ → X . Mor e over, we c onsider a C ( X ) -Hilb ert Z -bimo dule M with left action φ : Z → ( M , M ) , and assume that ther e is an isomorphism M ≃ b E ⊗ C ( X ) Z of right Hilb ert Z -mo dules, wher e E → X is a r ank d ve ctor bund le. Now, M is isomorphic (as a right Hilb ert Z -mo dule) t o the mo dule of se ctions of the pul lb ack bun dle E q := E × X X ′ → X ′ ; note that elements of E q ar e of the typ e ( ξ , x ′ ) , ξ ∈ E x , x ∈ X ′ x , x ∈ X . This al lows one to r e gar d elements of M as inje ctive c ont inuous maps of t he typ e ψ : X ′ → E su ch that ψ ( x ′ ) ∈ E q ( x ′ ) (the asso ciate d se ction of E q is given by the map ψ q ( x ′ ) := ( ψ ( x ′ ) , x ′ ) , x ′ ∈ X ′ ). On the same line of Example 4.4, for every gr oup bun dle p : G → X , G ⊆ U E , we c onsider t he gauge action G × X E q → E q , ( y , ( ξ , x ′ )) 7→ ( y ( ξ ) , x ′ ) , (4.10) obtaine d by extending (3.5 ) to E q . Now, every M x , x ∈ X , is isomorp hic to the mo dule of se ctions of E q | X ′ x → X ′ x (note that E q | X ′ x ≃ E x × X ′ x ≃ C d × X ′ x ); t h us, elements of M x ar e c ontinuous maps fr om X ′ x to E x , and M x is isomo rphic as a right Hilb ert Z x -mo dule to the fr e e m o dule E x ⊗ Z x ≃ C d ⊗ Z x . We obtain the gauge action U : G × X c M → c M , U x y ψ ( x ′ ) := y ( ψ ( x ′ )) , (4.11) y ∈ G , ψ ∈ M p ( y ) , x ′ ∈ X ′ p ( y ) , and c onclude t ha t M is a G -Hilb ert Z -bimo dule. Now, in gener al M ⊗ G is a semitensor C*-c ate gory; anyway, if ( M r , M s ) G ⊆ ( M r , M s ) Z (4.12) (se e (4.9)), then M ⊗ G is a tensor C*-c ate gory. It is cle ar that if the left and right Z - ac tions c oincide, then (4.12) is fulﬁl le d. Anyway, (4.12) hol ds in mor e gener al c ases (se e [16, § 4], and Example 6.3). 5 Dual actions vs. gauge actions. Let ρ b e a unital endo morphism of a C* -algebra A , C ( X ρ ) := { f ∈ A ∩ A ′ : ρ ( f ) = f } , (5.1) E → X ρ a v ector bundle, G ⊆ S U E a co mpact group bundle. A dual G -action on ( A , ρ ) is a C ( X ρ ) -mo nomorphism µ : ( O G , σ G ) ֒ → ( A , ρ ) such that µ ( E r , E s ) G = µ ( σ r G , σ s G ) ⊆ ( ρ r , ρ s ) , r , s ∈ N (in other ter ms, µ is a functor from b G to b ρ ). The cr osse d pr o duct of A by µ is the universal C ( X ρ ) -a lgebra A ⋊ µ b G ge nerated by A and a n is omorphic ima ge of O E by means of a unital C ( X ) -monomor phism j : O E ֒ → A ⋊ µ b G ; moreover, we require that A ⋊ µ b G fulﬁlls the universal prop erties  µ ( t ) = j ( t ) , t ∈ O G ρ ( a ) ψ = ψa , a ∈ A , ψ ∈ j ( b E ) ⊂ j ( O E ) . (5.2) 12 The same argument of [23, Thm 3 .11] shows existence and unicity of A ⋊ µ b G . Let us now denote the C* -bundle a s sociated with A ⋊ µ b G b y b B → X ; then, [23, E q.3.17] implies that there is a gaug e action α : G × X b B → b B , α ( y , v · b j ( ξ )) := v · b j ◦ b y ( ξ ) , (5.3) v ∈ b A , ξ ∈ b O E . If G is endow ed with an inv ariant C ( X ) -v alued functional, then A is the ﬁxed- po in t algebr a with r espect to the a ction (5 .3) (see r emarks after [23, Theorem 3.1 1], [10, Lemma 2.8, Theorem 3.2]). If { ψ l } is a ﬁnite set of g enerators for j ( b E ) ⊂ A ⋊ µ b G , the endomorphism σ ∈ end ( A ⋊ µ b G ) , σ ( b ) := X l ψ l bψ ∗ l , b ∈ A ⋊ µ b G , (5.4) is deﬁned, in such a wa y that σ | A = ρ , σ ◦ j = j ◦ σ E . As an immediate co nsequence of (5.4) and the remarks after (3.7), we ﬁnd j ( E r , E s ) ⊆ ( σ r , σ s ) , r, s ∈ N . (5.5) Lemma 5.1. L et G ⊆ S U E b e a c omp act gr oup bund le with a dual action µ : b G → b ρ . If G ′ ⊆ S U E is a c omp act gr oup bund le with G ⊆ G ′ , then a dual G ′ -action µ ′ is deﬁne d on ( A , ρ ) , and ther e is a C ( X ) -epimorp hism η : A ⋊ µ ′ b G ′ → A ⋊ µ b G . The morphism η is an A -mo dule m ap, intertwines the inn er endomorphisms induc e d by b E , and pr eserves the intertwiners sp ac es ( E r , E s ) , r, s ∈ N . Pr o of. Since ( E r , E s ) G ′ ⊆ ( E r , E s ) G , r, s ∈ N , we ca n deﬁne the dual action µ ′ : ( E r , E s ) G ′ → ( ρ r , ρ s ) , µ ′ ( t ) := µ ( t ) , and consider the crossed pro duct A ⋊ µ ′ b G ′ , endow ed with the C* -morphisms µ ′ : O G ′ → A ⋊ µ ′ b G ′ , j ′ : O E → A ⋊ µ ′ b G ′ . Now, it is clea r that if we consider the g eneric elemen t a · j ( t ) of A ⋊ µ b G , a ∈ A , t ∈ O E , then we ﬁnd  µ ′ ( t ) = µ ( t ) = j ( t ) , t ∈ O G ′ ⊆ O G ρ ( a ) ψ = ψa , ψ ∈ j ( b E ) This means that A ⋊ µ b G fulﬁlles the universal pr operties (5.2) for the dual action µ ′ , th us the desired morphism η is deﬁned b y the universal pro p erty . Note that η is deﬁned as the quotient of A ⋊ µ ′ b G ′ with resp ect to the idea l generated by the rela tions j ′ ( t ) = µ ( t ) , t ∈ O G (recall that µ ( t ) ∈ A ⊂ A ⋊ µ ′ b G ′ , th us the ab o ve equality makes sense). Let A ⊂ B b e an inclusion o f unital C* -alg ebras with common identit y 1 . A Hilb ert A - bimo dule in B is given by a no rm-closed subspa ce M ⊂ B , clo sed with respect to left a nd rig h t multiplication by elemen ts of A , and suc h that the set { ψ ∗ ψ , ψ, ψ ′ ∈ M} coincides with A (see [6]). M is said to hav e supp ort the identity if there is a ﬁnit e set { ψ l } ⊂ M such that P l ψ l ψ ∗ l = 1 . Note that in this case, ev ery ψ ∈ M is of the for m ψ = P l ψ l a l , a l := ψ ∗ l ψ ∈ A , so that M is ﬁnitely generated. In the se q uel, for every r ∈ N we will denote the closed spa n of pro ducts of the type ψ 1 · · · ψ r , ψ i ∈ M , by M r ⊂ B . It is clear tha t M may b e identiﬁed with the r -fold internal tensor p o wer of M . In the sa me wa y , the clo s ed span of elements of the type ψ ′ ψ ∗ , ψ ′ ∈ M s , ψ ∈ M r , may be identiﬁed with the space ( M r , M s ) of rig ht A -module ope rators fr om M r to M s (note that since M is ﬁnitely ge ner ated, every element of ( M r , M s ) is a compact op erator). A version o f the following theorem has been pr o ved in [2 3, Pro p osition 3.12 ]; the pro of that we give here presents the mo diﬁcations due to the fact that w e consider ga uge actions instead of g roup actions. 13 Theorem 5.2. L et G ⊆ U E b e a c omp act gr oup bund le with a dual action µ : ( O G , σ G ) → ( A , ρ ) . If A ′ ∩ ( A ⋊ µ b G ) = Z , then t he fol lowing pr op erties hold: 1. Ther e is a ﬁnitely gener ate d C ( X ρ ) - H il b ert Z -bimo dule M ⊂ B with su p p ort 1 , isomorphic to b E ⊗ C ( X ρ ) Z as a right Hilb ert Z -mo dule; 2. The gauge action α (se e (5.3)) r estricts to a gauge action U : G × X c M → c M ; 3. ( M r , M s ) G = ( ρ r , ρ s ) , ∀ r, s ∈ N . Pr o of. Poin t 1 : deﬁne M := n ψ ∈ A ⋊ µ b G : ψ a = ρ ( a ) ψ , a ∈ A o ; since ψ ∗ ψ ′ ∈ A ′ ∩ ( A ⋊ µ b G ) = Z , and f ψ = ψ f , f ∈ C ( X ρ ) , ψ, ψ ′ ∈ M , w e ﬁnd that M is a C ( X ρ ) -Hilb ert Z - bimodule in A ⋊ µ b G . Mor eo ver, (5.2) implies that j ( b E ) ⊂ M . Let now { ψ l } b e a ﬁnite set of genera tors for j ( b E ) ≃ b E . Since j is unital, the relatio ns (3.3) imply that M has supp ort the identit y in A ⋊ µ b G , a nd that M is gener ated by elements o f j ( b E ) as a rig h t Z -mo dule; this implies the isomorphism M ≃ b E ⊗ C ( X ) Z . Point 2 : let us co nsider the restrictions η x : M → M x := η x ( M ) of the ﬁbre epimorphism, and v ∈ M x . By lo cal triviality of E there are sections ϕ 1 , . . . , ϕ d ∈ j ( b E ) ≃ b E fulﬁlling the Cunt z relations η x ( ϕ ∗ i ϕ j ) = δ ij η x (1) , X i η x ( ϕ i ϕ ∗ i ) = η x (1) . In this way , v = P j e j,x η x ( z j ) , where z j ∈ Z and e j,x := η x ( ϕ j ) . By deﬁnition of α , a nd by (2.1), for every y ∈ G x we ﬁnd α ( y , v ) = X k,j y kj e j,x η x ( z j ) where y kj ∈ C are the matrix co eﬃcien ts of y . The a bov e equa lit y implies that α ( y , v ) ∈ M x , th us M is sta ble with resp ect to the g auge action α . W e deno te the restrictio n of α to M b y U . Since α x y ◦ η x ( ψ ∗ ψ ′ ) = η x ( ψ ∗ ψ ′ ) (in fact ψ ∗ ψ ′ ∈ Z ), a nd α x y ◦ η x ( ψ z ) = α x y ◦ η x ( ψ ) · η x ( z ) , z ∈ Z , we conclude that U is actua lly a unitary gauge action in the sens e of Deﬁnition 4 .3, with trivial ass ociated gauge actions o n Z . Poin t 3: if t ∈ ( M r , M s ) G then by deﬁnition of U we ﬁnd t ∈ A . Moreov er, s ince t is sum o f terms of the type ψ ′ ψ ∗ , ψ ∈ M r , ψ ′ ∈ M s , by deﬁnition of M we conclude that t ∈ ( ρ r , ρ s ) (note in fact that ψ ∈ M r implies ρ r ( a ) ψ = ψ a , a ∈ A ). On the conv erse, let t ∈ ( ρ r , ρ s ) ; we ﬁx ﬁnite sets o f g enerators { ψ L } ⊂ M s , { ϕ M } ⊂ M r , in such a way that P L ψ L ψ ∗ L = P M ϕ M ϕ ∗ M = 1 , and note that t LM := ψ ∗ L tϕ M ∈ A ′ ∩ ( A ⋊ µ b G ) = Z (in fact, t LM a = ψ ∗ L tρ r ( a ) ϕ M = ψ ∗ L ρ s ( a ) tϕ M = at LM , a ∈ A ). In this wa y , t = P LM ψ L t LM ψ ∗ M belo ngs to ( M r , M s ) G . 6 Sp ecial Endomorphisms. 6.1 P erm utation symmetry: weak forms. Let A b e a unital C* -algebra with centre Z . A unital endomorphism ρ of A has p ermutation symmetry if there is a unitary representation p 7→ ε ( p ) of the group P ∞ of ﬁnite p erm utations o f N in A , such that:    ε ( S p ) = ρ ◦ ε ( p ) ε := ε (1 , 1) ∈ ( ρ 2 , ρ 2 ) ε ( s, 1) t = ρ ( t ) ε ( r, 1) , t ∈ ( ρ r , ρ s ) , (6.1) 14 where ( r , s ) ∈ P r + s exchanges the ﬁrst r terms with the remaining s , and S is the s hift ( S p )(1) := 1 , ( S p )( n ) := 1 + p ( n − 1 ) , p ∈ P ∞ . The a bov e prop erties imply that ε ( p ) ∈ ( ρ n , ρ n ) , n ∈ N , p ∈ P n ; (6.2) we say that ρ has we ak p ermutation s ymm et ry if just (6.1(1)),(6.2) hold. In such a case, we call symmetry intertwiners the elements of ( ρ r , ρ s ) for which (6.1(3)) holds : ( ρ r , ρ s ) ε := { t ∈ ( ρ r , ρ s ) : ε ( s, 1) t = ρ ( t ) ε ( r , 1) } . W e denote the C* -subalgebra of A g enerated b y the int ertwiner spaces ( ρ r , ρ s ) , r, s ∈ N , by O ρ ; in pa rticular, we deno te the C* -algebra g enerated b y the symmetry intertwiners by O ρ,ε . W e s a y that ρ has p ermutation quasi-symmetry if ( ρ r , ρ s ) = ρ s ( Z ) · ( ρ r , ρ s ) ε = ( ρ r , ρ s ) ε · ρ r ( Z ) . (6.3) Here ρ s ( Z ) · ( ρ r , ρ s ) ε denotes the vector s pa ce spanned by e lemen ts o f the type ρ s ( z ) t , z ∈ Z , t ∈ ( ρ r , ρ s ) ε . F rom (6.3), we see that if ρ has p ermutation q uasi-symmetry , then the obstacle to get per m utation s ymmetry is given by elements of Z that ar e not ρ - in v aria n t. The ab o ve prop erties may b e summar ized by saying that the catego ry b ρ with ob jects ρ r , r ∈ N , and a rrows ( ρ r , ρ s ) is a tensor qua si-symmetric C* -category , whilst the catego r y b ρ ε having the same ob jects as b ρ and arrows ( ρ r , ρ s ) ε is a tenso r sy mm etric C* -sub category of b ρ (see [23, § 4.1 ]). R emark 6.1 . By [7, (2.6)], to get a we ak p ermu tation symmetry for ρ it suﬃc es that ther e is ε ∈ ( ρ 2 , ρ 2 ) su ch that ε = ε ∗ = ε 2 = 1 . In fac t, e ach ε ( p ) , p ∈ P ∞ , may b e r e c over e d in t erms of pr o ducts of the t yp e ερ ( ε ) . . . ρ n ( ε ) , n ∈ N . If E → X is a vector bundle and G ⊆ U E is a compact group bundle, then σ G ∈ end X O G has per m utation symmetry , induced by the ﬂip op erator θ ∈ ( E 2 , E 2 ) G = ( σ 2 G , σ 2 G ) (see [22, Re ma rk 4.5]). W eak forms of p erm utation symmetry are strictly related to dual actions: Lemma 6.1 . L et ( A , ρ ) b e a C*-dynamic al system, E → X ρ a ve ctor bund le, and G ⊆ E a c omp act gr oup bun dle with a dual action µ : ( O G , σ G ) → ( A , ρ ) . Then ρ has a we ak p ermutation symmetry ε . Mor e over, if A ′ ∩ ( A ⋊ µ b G ) = Z , then for every r , s ∈ N the map µ r estricts to an isomorphi sm ( E r , E s ) G → ( ρ r , ρ s ) ε of Banach C ( X ρ ) - bimo dules. Pr o of. Since θ ( r , s ) ∈ ( σ r + s G , σ r + s G ) , r, s ∈ N , deﬁning ε ( r, s ) := µ ◦ θ ( r , s ) we obtain a weak p erm u- tation symmetr y for ρ . No w, if t ∈ ( ρ r , ρ s ) ε then using the deco mposition t = P LM ψ L t LM ψ ∗ M , t LM := ψ ∗ L tψ M (where ψ L ∈ j ( b E s ) , see (3.4)), in the same w ay as in the pro of o f Theorem 5.2 w e ﬁnd t LM ∈ A ′ ∩ ( A ⋊ µ b G ) = Z . Moreov er, we a ls o ﬁnd ρ ( t LM ) = ψ ∗ L ε (1 , s ) ε ( s, 1) tε (1 , r ) ε ( r , 1) ψ M = t LM . This implies t LM ∈ C ( X ρ ) , so that t ∈ j ( E r , E s ) . Since j ( E r , E s ) ∩ A = µ ( E r , E s ) G , we conclude that t ∈ µ ( E r , E s ) G . On the co nverse, the fact that σ G ∈ end X ρ O G has p erm utation symmetry , and the deﬁnition of ε , imply that µ ( σ r G , σ s G ) = µ ( E r , E s ) G ⊆ ( ρ r , ρ s ) ε , a nd this proves that ( ρ r , ρ s ) ε = µ ( E r , E s ) G . 6.2 Sp ecial endomorphisms and Crossed Pro ducts. The notion of sp e cial c onjugate may b e re garded as an alge br aic counterpart of the pr operty of a ﬁnite-dimens ional group repr esen tation of b eing with determinant one, and plays a crucial role 15 in the setting of the Do plicher-Rob erts theory ([10, § 4 ],[11, p.58 ]). T o take into acc o un t the class of examples in § 3, we generalize the deﬁnition of sp ecial conjugate in the following sense: we say that ρ is (we akly/quasi) sp e cial if ρ has (weak/quasi) permutation symmetry , and for some d ∈ N , d > 1 , ther e is a Hilber t C ( X ρ ) -bimo dule R ⊂ ( ι, ρ d ) ε such that  R ∗ ρ ( R ′ ) = ( − 1) d − 1 d − 1 R ∗ R ′ , R, R ′ ∈ R RR ∗ := span { R ′ R ∗ : R , R ′ ∈ R} = C ( X ρ ) P ε,d (6.4) When R exis ts, it is unique ([2 3, Lemma 4.8]). Let K ( R ) denote the C* -alg ebra of r igh t C ( X ρ ) - mo dule op erator o f R ; then, (6.4(2)) implies that there is an isomor phism K ( R ) ≃ C ( X ρ ) . The previous isomo rphism, and the Serr e-Sw an theor em, imply that R is the module of s ections of a line bundle ov er X ρ ; th us, the ﬁr st Chern class c 1 ( ρ ) ∈ H 2 ( X ρ , Z ) of such a line bundle is a complete inv ar ian t of R . The Chern class c 1 ( ρ ) v anishes if and only if ρ fulﬁlles the s pecial conjugate prop ert y in the sense of Doplicher and Roberts ([10, Lemma 4 .2 ]); this is equiv a le nt to require that there is a partial isometry S ∈ R , S ∗ S = 1 , S S ∗ = P ε,d . The class of ρ is de ﬁned by d ⊕ c 1 ( ρ ) ∈ N ⊕ H 2 ( X ρ , Z ) . (6.5) A w eakly sp ecial endo morphism is denoted by ( ρ, ε, R ) . A class of examples fo llo w. With the nota tion of § 3, if G ⊆ S U E then ( σ G , θ, ( ι, λE )) ∈ end X O G is sp ecial, w her e λE ⊂ E d is the d -fold exterior p o wer of E ; moreover, c ( σ G ) = d ⊕ c 1 ( E ) , whe r e d is the rank of E , and c 1 ( E ) is the ﬁrst Cher n class (see [23, § 2.2,Exa mple 4.2]). The following basic result has b een prov ed in [23, Theor em 5.1]. Theorem 6.2. L et ( ρ, ε, R ) ∈ end A b e a we akly sp e cial endomorphi sm with class d ⊕ c 1 ( ρ ) , d ∈ N , c 1 ( ρ ) ∈ H 2 ( X ρ , Z ) . Then, for every r ank d ve ctor bun d le E → X ρ with ﬁrst Chern class c 1 ( ρ ) , ther e is a dual S U E -action µ : O S U E → A . Mor e over, O ρ,ε is a c ontinuous bu nd le of C*-alg ebr as over X ρ , with ﬁbr es isomorphic t o O G x , x ∈ X ρ , wher e G x ⊆ SU ( d ) is a c omp act Lie gr oup unique up to c onjugacy in SU ( d ) . Final ly, ther e is an inclusion O S U E ֒ → O ρ,ε of C*-algebr a bund les. The image of µ is the C* -algebra gener ated by ε , R b y clo s ing with r espect to the action of ρ . W e hav e µ ( θ ) = ε , µ ( ι, λE ) S U E = R (see [23, Lemma 4.6 ]). F or every rank d v ector bundle E → X ρ with ﬁrst Chern class c 1 ( ρ ) , we in tro duce the nota tio n B E := A ⋊ µ [ S U E . (6.6) By (5.4), B E comes equipped with the endomorphism σ ∈ e n d B E . B y deﬁnition of crossed pr oduct by a dual action, there are C ( X ρ ) -mo rphisms  j : ( O E , σ E ) ֒ → ( B E , σ ) µ : ( O S U E , σ S U E ) ֒ → ( A , ρ ) , j | O S U E = µ . (6.7) Note that σ has w eak p erm utation symmetry induced by ε . Since σ E ∈ end X ρ O E has symmetry θ , we ﬁnd j ( E r , E s ) = j ( σ r E , σ s E ) θ ⊆ ( σ r , σ s ) ε . (6.8) Moreov er, there is a gaug e action α : S U E × X b B E → b B E , (6.9) so that it is of interest to in vestigate some technical prop erties of the bundle p : S U E → X ρ : 16 Lemma 6. 3 . L et X b e a c omp act Hausdorﬀ sp ac e, d ∈ N , and E → X a r ank d ve ctor bund le. Then t he r e is an invariant C ( X ) -functional ϕ : C ( S U E ) → C ( X ) . Mor e over, for every gr ou p bund le G ⊆ S U E t h e natur al pr oje ction p G : S U E → Ω := G \S U E has lo c al se ctions: in other terms, for every ω ∈ Ω ther e is an op en neighb ourho o d W ∋ ω with a c ontinuous map s W : W → S U E such that p G ◦ s W = i d W . Pr o of. F or every u ∈ U ( d ) , we consider the gr o up automorphism b u ∈ aut SU ( d ) deﬁned by adjoin t action b u ( g ) := ug u ∗ , g ∈ SU ( d ) ; we main tain the sa me no ta tion fo r the C* -algebra automorphism induced on C ( SU ( d )) , so that b u ∈ aut C ( SU ( d )) . If ( { X i } , { u ij } ) is an U ( d ) -co cycle a ssociated with E (see [1 4 ]), then S U E is deﬁned as the bundle with co cycle ( { X i } , { b u ij } ) . Let now λ ij ( x ) := det u ij ( x ) , x ∈ X ij , and v ij ( x ) := λ ij ( x ) u ij ( x ) , x ∈ X ij ; in this wa y , for ev ery pair i, j a contin uous map v ij : X ij → SU ( d ) is deﬁned. Since [ b u ij ( x )]( g ) = [ b v ij ( x )]( g ) , g ∈ SU ( d ) , w e co nc lude that the bundle S U E ha s a n asso ciated SU ( d ) -co cycle ( { X i } , { b v ij } ) . Let now ϕ 0 : C ( SU ( d )) → C denote the Haa r meas ure of SU ( d ) ; s ince SU ( d ) is unimo dular, for every v ∈ SU ( d ) we ﬁnd ϕ 0 ◦ b v = ϕ 0 . Thus, applying [24, Pr o position 4.3], we conclude that the desired functiona l ϕ : C ( S U E ) → C ( X ) exists. Ab out existence of lo cal sections, we note that S U E is a loc ally trivial bundle with ﬁbr e the compact Lie group SU ( d ) , a nd apply [24, Lemma 2.5]. R emark 6.2 . Existenc e of the invariant C ( X ) -functional ϕ : C ( S U E ) → C ( X ) al lows one to deﬁne an invariant me an m ϕ : B E → A ; using m ϕ we c an c onclude that A is the ﬁxe d-p oint algebr a of B E with r esp e ct to t he action (6.9). W e ca n now inv estigate the algebr aic structure of the cross ed pro duct B E . Let us denote the centre of B E by C E and the spe c tr um o f C E by Ω E . If f ∈ C ( X ρ ) , then ψ f = ρ ( f ) ψ = f ψ , ψ ∈ j ( b E ) , thus f comm utes with A and j ( O E ) ; this implies f ∈ C E , s o that there is a unital inclusion C ( X ρ ) ֒ → C E (i.e., C E is a C ( X ρ ) -a lgebra). W e consider the s urjectiv e map deﬁned by the Gel’fand transform q : Ω E → X ρ . (6.10) The gauge action (6.9) restricts to a gauge action on C E ; b y Gel’fand duality , there is a right action α ∗ : Ω E × X S U E → Ω E , α ∗ ( ω , y ) := ω ◦ α x y , x = p ( y ) = q ( ω ) , (6.11) in the sense of [19, § 3]. R emark 6.3 . C E is a c ontinuous bund le of C*-algebr as on X ; for e ach x ∈ X , ther e is a close d gr ou p G x ⊆ SU ( d ) such that the ﬁbr e C E ,x is isomorphic to C ( G x \ SU ( d )) , i.e. Ω x := q − 1 ( x ) is home omorphic t o G x \ SU ( d ) (se e [24, L emma 5.1]). Lemma 6 . 4. If ρ has p ermutation quasi-symmet r y, t h en the fol lowi ng pr op erties hold: 1. A ′ ∩ B E = C E ∨ Z , wher e C E ∨ Z denotes the C*-algebr a gener ate d by C E , Z . 2. C E is gener ate d by elements of the typ e t ∗ ϕ , wher e t ∈ ( ι, ρ r ) ε , ϕ ∈ j ( ι, E r ) , r ∈ N . Pr o of. Poin t 1: As a ﬁrs t step, we note that if E is trivial then the re sult follows by [24, P ropos itio n 7.1]. Let us now consider the gener al case in which E is nontrivial; it is co n v enient to rega r d B E , A a s C ( X ρ ) -a lgebras. Let x ∈ X ρ , U an op en neighbourho od of x triv ializing E . W e denote the restrictions of B E , A over U b y B E ,U , A U (see § 2); note that ther e ar e ob vious inclusions A ′ U ∩ A U ⊆ Z , B ′ E ,U ∩ B E ,U ⊆ C E . (6.12) 17 Now, S U E | U ≃ U × SU ( d ) , and [2 4 , Pro position 3 .3] implies that there is a strong ly contin uous action α U : SU ( d ) → B E ,U (in fact, SU ( d ) ma y be regar ded as a gro up o f co nstan t sections s panning U × SU ( d ) ). It is clea r that SU ( d ) a cts on B E ,U in such a wa y that the ﬁxed point a lgebra B α U E ,U coincides with A U . Thu s, the arg umen t us e d for E ≃ X ρ × C d implies that A ′ U ∩ B E ,U is gener ated as a C* -alg ebra by A ′ U ∩ A U and B ′ E ,U ∩ B E ,U ; by (6.12), we conclude A ′ U ∩ B E ,U ⊆ C E ∨ Z . W e now pick a ﬁnite op en cover { U k } trivializing E with a subo rdinate partition of unity { λ k } ⊂ C ( X ρ ) . If b ∈ A ′ ∩ B E , then b = X l λ k b , λ k b ∈ A ′ U k ∩ B E ,U k ⊆ C E ∨ Z . Poin t 2: let t ∈ ( ι, ρ r ) ε , ϕ ∈ j ( ι, E r ) . Then (5.5) implies t ∗ ϕa = t ∗ ρ r ( a ) ϕ = at ∗ ϕ , a ∈ A ; moreov er, (6.8) implies that, for every ψ ∈ j ( b E ) , ψ t ∗ ϕ = σ ( t ∗ ϕ ) ψ = ρ r ( t ) ∗ σ r ( ϕ ) ψ = t ∗ ε (1 , r ) ε ( r , 1 ) ϕ ψ = t ∗ ϕψ . W e co nclude that t ∗ ϕ commu tes with A and j ( b E ) , so that t ∗ ϕ ∈ C E . T o prov e that the set { t ∗ ϕ } is total in C E , we pro ceed as in [8, Lemma 5(1)], [24, P ropos ition 7.1]. As a ﬁrst step, we assume that E is trivial, in such a wa y tha t we hav e a strongly contin uous a c tio n α : SU ( d ) → aut X ρ C E such that C α E = C ( X ρ ) . By F ourier analysis, the set o f elemen ts o f C E that tra nsform like v ectors in irre ducible repre sen tations of SU ( d ) is dense in C E . Thus, we consider n -ples { T i } ⊂ C E such that α y ( T i ) = X j T j u j i ( y ) , y ∈ SU ( d ) , where u is some irreducible representation o f SU ( d ) . Since u is a subrepresentation of some tensor power of the deﬁning representation of SU ( d ) , we ﬁnd that ther e is r ∈ N and constant orthonor mal sections { ϕ i } of E r ≃ X ρ × C d r transforming like the T i ’s; mo reo ver, we may regar d the ϕ i ’s as elements of j ( ι, E r ) fulﬁlling the relations ϕ ∗ i ϕ j = δ ij 1 . In this way , we ﬁnd W ∗ := X i T i ϕ ∗ i ∈ A . Multiplying on the r igh t by ϕ j , we conclude T j = W ∗ ϕ i ; moreov er, it is clear that W ∈ ( ι, ρ r ) , and ρ ( W ) = X i σ ( ϕ i ) T ∗ i = X i ε ( r , 1) ϕ i T ∗ i = ε ( r, 1) W , i.e., W ∈ ( ι, ρ r ) ε . This proves Poin t 2 for tr iv ial vector bundles. In the general cas e, the sa me argument used for the pro of of Poin t 1 s ho ws that { t ∗ ϕ } is total in C E . The following Theorem generaliz e s several results, namely [10, Theorem 4.1] for Z = C 1 , [24, Theorem 7.2] for ρ symmetric and c 1 ( ρ ) = 0 , and [3 , Theorem 4.1 3] (for a single endomorphism) in the case in which each ( ρ r , ρ s ) ε , r, s ∈ N , is fr ee as a Banach C ( X ρ ) -bimo dule. 18 Theorem 6.5 . L et ( ρ, ε, R ) ∈ end A b e a quasi-sp e cial endomorphism with class d ⊕ c 1 ( ρ ) , d ∈ N , c 1 ( ρ ) ∈ H 2 ( X ρ , Z ) . F or every r ank d ve ct or bund le E → X ρ with ﬁrst Chern class c 1 ( ρ ) , the fol lowi ng ar e e quivalent: 1. ther e is a s e ction s : X ρ → Ω E , q ◦ s = id X ρ (r e c al l (6.10)); 2. ther e is a c omp act gr oup bund le G ⊆ S U E , and a dual action ν : O G → A such that ν ( E r , E s ) G = ( ρ r , ρ s ) ε , r , s ∈ N , and A ′ ∩ ( A ⋊ ν b G ) = Z . Mor e over, A is the ﬁxe d-p oint algebr a with r esp e ct t o t he gauge G -action on A ⋊ ν b G . Pr o of. (1) ⇒ (2) : B y Le mma 6 .3 , Lemma 6.4, and from the fa c t that ther e is s ∈ S X ρ (Ω E ) , we ﬁnd that the triple ( B E , S U E , α ) fulﬁlles the pr operties requir ed for the pro of of [24, Theorem 6.1]. Th us, there is a gr oup bundle G ⊆ S U E and a dynamical system ( F , G , β ) , endow ed with a C ( X ρ ) -e pimorphism η : B E → F which is inj ective on A , and suc h that η ( A ) = F β , η ( A ) ′ ∩ F = η ( Z ) . W e reca ll that η is deﬁned as the quotient B E → B E / ( C 0 (Ω E − s ( X ρ )) B E ) (6.13) (note that C 0 (Ω E − s ( X ρ )) is an ideal of C E ), whilst G := { y ∈ S U E : α ∗ ( s ( x ) , y ) = s ( x ) , x := p ( y ) } . (6.14) T o econo mize in notatio ns , we deﬁne A η := η ( A ) , Z η := η ( Z ) , ρ η := η ◦ ρ ◦ η − 1 | η ( A ) ∈ end A η , in such a wa y that w e hav e a C ( X ρ ) -is omorphism η | A : ( A , ρ ) → ( A η , ρ η ) . By deﬁnitio n of η , G , we obtain b η ◦ α ( y , w ) = β ( y, b η ( w )) , y ∈ G , w ∈ b B E (6.15) (see [24, § 6.1]). Now, since η is a C ( X ρ ) -mo rphism, we ﬁnd that η restricts to a unitary is omor- phism from j ( b E ) on to η ◦ j ( b E ) (in fa ct if ψ , ψ ′ ∈ b E a nd f := h ψ , ψ ′ i , then f = η ( j ( ψ ) ∗ j ( ψ ′ )) = j ( ψ ) ∗ j ( ψ ′ ) ). This fact has t wo consequences: ﬁrs t, b y univ ersa lit y of the Cuntz-Pimsner a lgebra there is a C ( X ) -monomorphism j η : O E ֒ → F , j η := η ◦ j ; moreov er, since C ( X ρ ) is contained in the centre of F , if { ψ l } is a ﬁnite set of gener ators for j η ( b E ) then we can deﬁne σ η ∈ end X ρ F , σ η ( t ) := P l ψ l tψ ∗ l , in such a w ay that σ η ◦ η = η ◦ σ , σ η ◦ j η = j η ◦ σ E . This also implies σ η | A η = ρ η . Applying (5.3) to B E , and us ing (6.1 5 ), we ﬁnd that j η is G -equiv aria nt, i.e. β ( y , b j η ( ξ )) = b j η ◦ b y ( ξ ) , y ∈ G , ξ ∈ b O E . Thu s, j η restricts to a C ( X ρ ) -mo nomorphism ν : ( O G , σ G ) → ( A η , ρ η ) . If t ∈ ( σ r G , σ s G ) , then ν ( t ) ρ r η ( a ) = η ( j ( t ) ρ r ( a ′ )) = η ( j ( t ) σ r ( a ′ )) = η ( σ s ( a ′ ) j ( t )) = ρ s η ( a ) ν ( t ) , where a ′ ∈ A , a := η ( a ′ ) ∈ A η ; this implies ν ( E r , E s ) G ⊆ ( ρ r η , ρ s η ) , r , s ∈ N . Thus, ν is a dual action, and F fulﬁlles the universal prop erties  ν ( t ) = j s ( t ) , t ∈ O G ψ a = ρ η ( a ) ψ , a ∈ A η , ψ ∈ j s ( b E ) . By identif ying A and A η , w e conclude that F is isomo r phic to A ⋊ ν b G . By co nstruction o f F , we also ﬁnd A ′ ∩ ( A ⋊ ν b G ) = Z , and that A is the ﬁxed-p oin t alg ebra w ith res p ect to the G -a ction 19 on A ⋊ ν b G . Fina lly , the fact that ( ρ r , ρ s ) ε = ν ( E r , E s ) G , r, s ∈ N , follows fro m Lemma 6 .1. (2) ⇒ (1) : applying Lemma 5.1, w e ﬁnd that ν restr icts to the dual a c tion µ : O S U E → A , µ ( t ) := ν ( t ) , t ∈ O S U E ⊂ O G , int ro duced in Theorem 6.2, in fact ν ( θ ) = ε , ν ( ι, λE ) = R . Again by Lemma 5.1, we ﬁnd that there is a C ( X ρ ) -e pimorphism η : B E → A ⋊ ν b G . No w, since A ′ ∩ ( A ⋊ ν b G ) = Z , we ﬁnd that the cen tre of A ⋊ ν b G coincides with C ( X ρ ) , in fact z ∈ Z belo ngs to the centre of A ⋊ ν b G if and only if z ψ = ψ z = ρ ( z ) ψ , ψ ∈ j ( b E ) . Th us, η ( C E ) = C ( X ρ ) , and η | C E : C E → C ( X ρ ) is a C ( X ρ ) -e pimorphism. By Gel’fand equiv alence, we conclude that there is a section s : X ρ → Ω E . Deﬁnition 6.6. The triple ( A ⋊ ν b G , G , β ) of Point 2 of The or em 6.5 is c al le d a Hilb ert extension of ( A , ρ ) . In such a c ase, we say that t he p air ( E , G ) is a gauge-e qui v arian t pair asso ciated with ( A , ρ ) ; in p articular, the gr oup bund le G → X ρ is c al le d a gauge group of ( A , ρ ) . T o emphasize the dependence of G on s ∈ S X (Ω E ) w e use the notation G ≡ G s . W e no w inv estigate the dependence o f the sys tem ( A ⋊ ν b G s , G s , β ) o n the se c tion s . As a preliminary remark, w e note that (6.11) induces a gro up action α ∗ : S X ρ (Ω E ) × S U E → S X ρ (Ω E ) , α ∗ ( s, u ) ( x ) := α ∗ ( s ( x ) , u ( x )) , x ∈ X . (6.16) Theorem 6. 7. L et G s , G s ′ ⊆ S U E b e c omp act gr oup bun d les with dual actions ν : ( O G s , σ G s ) → ( A , ρ ) , ν ′ : ( O G s ′ , σ G s ′ ) → ( A , ρ ) , such t h at A ′ ∩ ( A ⋊ ν b G s ) = Z , A ′ ∩ ( A ⋊ ν ′ b G s ′ ) = Z . Then the fol lowi ng ar e e quivalent: 1. Ther e is u ∈ SU E with s ′ = α ∗ ( s, u ) ; 2. Ther e is an isomorphi sm δ : A ⋊ ν b G s → A ⋊ ν ′ b G s ′ such t ha t δ | A = i d A (in the sense t ha t δ is an A -mo dule map), δ ◦ σ η = σ η ′ ◦ δ . 3. G s and G s ′ ar e c onjugates in S U E , i.e. ther e is u ∈ SU E such that G s ′ = u G s u ∗ . Pr o of. (1) ⇒ (2) : W e re tain the notation introduced in the pro of of Theorem 6.5 . Since s ′ = α ∗ ( s, u ) , w e ﬁnd α u ( C 0 (Ω E − s ( X ρ ))) = C 0 (Ω E − s ′ ( X ρ )) . This implies that if we co nsider the C ( X ρ ) -e pimorphisms η : B E → A ⋊ ν b G s , η ′ : B E → A ⋊ ν ′ b G s ′ (see (6.1 3 )), then we obtain α u (ker η ) = ker η ′ . This last equa lit y allows one to deﬁne δ : A ⋊ ν b G s → A ⋊ ν ′ b G s ′ , δ ◦ η ( b ) := η ′ ◦ α u ( b ) , b ∈ B E . Since η , η ′ are faithful on A , we conclude that δ is faithful on A . F o r the same reason, δ is faithful on j s ( O E ) ⊂ A ⋊ ν b G s . Finally , since σ η ◦ η = η ◦ σ , σ η ′ ◦ η ′ = η ′ ◦ σ , we c o nclude that δ ◦ σ η = δ ◦ σ η ′ . (2) ⇒ (3) : As a preliminary remark, we note that the minimality condition A ′ ∩ ( A ⋊ ν b G s ) = Z implies that the cen tre of A ⋊ ν b G s is C ( X ρ ) . By construction of σ , it is clear that j s ( b E ) ⊆ ( ι, σ ) ; on the other s ide, if ψ ∈ ( ι, σ ) , then c l := ψ ∗ l ψ ∈ ( ι, ι ) = C ( X ρ ) , and this implies that ψ = P l ψ l c l ∈ j s ( b E ) . Th us, ( ι, σ ) = j s ( b E ) ; since δ ( ι, σ ) = ( ι, σ ′ ) , we conclude that δ deﬁnes a unitary C ( X ρ ) -mo dule oper ator u : b E → b E , which e x tends to a C ( X ρ ) -a utomorphism b u ∈ aut X ρ O E fulﬁlling δ ◦ j s = j s ′ ◦ b u . (6.17) Since δ ( R ) = R , R ∈ R ⊂ A , and since R = j s ( ι, λE ) = j s ′ ( ι, λE ) , we co nclude that b u r estricts to the identit y on e le men ts of ( ι, λE ) . In other terms, u ∈ SU E . Now, j s | O G = ν , j s ′ | O G s ′ = ν ′ ; th us, applying (6.17), and by using the fact that δ | A = i d A ⇒ δ ◦ ν = ν , 20 we ﬁnd ν = ν ′ ◦ b u . By Point 2 of Theor em 6.5 ν : O G s → O ρ,ε and ν ′ : O G s ′ → O ρ,ε are isomo rphisms, th us w e hav e that b u ∈ aut X ρ O E restricts to a C ( X ρ ) -is omorphism from O G s to O G s ′ . F rom Lemma 3.2 we conclude that G s and G s ′ are conjugates. (3) ⇒ (1 ) : By L e mm a 3.2, w e hav e that b u ∈ aut X ρ O E restricts to an iso morphism from O G s ′ onto O G s . Mo reo ver, by (6.9) there is a n automor phism α u ∈ aut X ρ B E such that α u ◦ j = j ◦ b u , wher e j is deﬁned b y (6.7). Let us now deﬁne ( j ′ := j s ◦ b u : O E → A ⋊ ν s b G s ν ′ := ν ◦ b u : O G s ′ → A ⋊ ν s b G s . Since j ′ ( t ) = ν ′ ( t ) , t ∈ O G s ′ , and since ψ a = ρ ( a ) ψ , a ∈ A , ψ ∈ j ′ ( b E ) , we conclude that A ⋊ ν s b G s fulﬁlles the universal prop erties (5.2) for the cr ossed pr oduct A ⋊ ν s ′ b G s ′ . Thus, by universalit y there is an isomorphism β u : A ⋊ ν s ′ b G s ′ → A ⋊ ν s b G s , β u ◦ j s ′ ( t ) = j ′ ( t ) , t ∈ O E . In other terms, w e hav e β u ◦ j s ′ ( t ) = j s ◦ b u ( t ) . In this wa y , we obtain a co mmutative diagram B E η   α u / / B E η ′   A ⋊ ν s ′ G s ′ β u / / A ⋊ ν G s where η , η ′ are deﬁned as in (6 .13). F r om the abov e diagr a m, it is evident that ker( η ◦ α u ) = ker( β u ◦ η ′ ) = k er η ′ . In particular, c ∈ C E ∩ ker( η ◦ α u ) if and only if c ∈ C E ∩ ker η ′ . But by construction of η , η ′ , we hav e  C E ∩ ker( η ◦ α u ) = C 0 (Ω E − α ∗ ( s, u ∗ )( X ρ )) C E ∩ ker η ′ = C 0 (Ω E − s ′ ( X ρ )) F ro m the ab ov e equalities, we conclude that s ′ ( X ρ ) = α ∗ ( s, u ∗ )( X ρ ) ; so that s ′ = α ∗ ( s, u ∗ ) , and the Theorem is pr o ved (up to a res caling u ∗ 7→ u ). Deﬁnition 6.8. L et ( ρ, d, R ) b e a quasi-sp e cial endomo rphism. Hilb ert extensions ( A ⋊ ν b G s , β , G s ) , ( A ⋊ ν b G s , β , G s ′ ) of ( A , ρ ) ar e said to b e equiv alent when Po int 2 of the pr evious the or em is f ulﬁl le d. Example 6.1. L et E → X b e a ve ctor bund le, G ⊆ S U E a gr ou p bund le. Then the c anonic al endomorphi sm σ G is sp e cial, and O E ≃ O G ⋊ µ b G , wher e µ is the identity of O G . Example 6. 2. The sup ersele ction stru ctur es c onsider e d in [2, 3] deﬁne endomorphisms with p er- mutation quasi-symmetry. In p articular, the endomorphisms ρ ∈ end A c onsider e d in [24, § 7] ar e quasi-sp e cial, and have trivial Chern class c 1 ( ρ ) . Thus, we may pick a trivial ve ctor bun dle E := X ρ × C d , and c onstruct t h e cr osse d pr o duct B E by the dual action of X ρ × SU ( d ) . It is pr ove d in [24, L emma 7.3] that the c entr e of B E is isomorphic to C ( X ρ ) ⊗ C ( G \ SU ( d )) , wher e G ⊆ SU ( d ) is a c omp act gr oup un iq ue up t o c onjugation in SU ( d ) ; thus, it is cle ar that the sp e ctrum Ω E = X ρ × G \ SU ( d ) admits se ctions, and we c an c onstruct the cr osse d pr o duct A ⋊ ν b G , wher e G = X ρ × G . Now, we may c onsider as wel l a r ank d ve ctor bund le E ′ → X ρ with trivial ﬁrst Chern class, and c onstruct the cr osse d pr o duct B E ′ by the dual S U E ′ -action: as shown in [24, § 6.2], it is not ensur e d that the r esulting gauge gr oup G ′ → X ρ is isomorphic to G . We c onclude that unicity of the gauge gr ou p of ( A , ρ ) is ensur e d when we r estrict ourselves t o c onsider t ri vial gr oup bun d les acting on trivial ve ctor bund les. In this way, we obtain the ”algebr aic Hilb ert sp ac es” c onsider e d in the ab ove-cite d r efer enc es. 21 6.3 The mo duli space of Hilb ert extensions. Lacking of existence and unicity of the Hilbe r t extensio n has b een alr eady discussed in the case in which the int ertwiners spaces are lo cally trivia l as co n tin uous ﬁelds of Banach spa ces by means of cohomolog ical metho ds ([25]). In the pr e s en t section we pr esen t some immediate co ns equences of Theorem 6.7: this will allows us to g iv e a classiﬁcation of the Hilb ert extensions of a q uasi-specia l C* -dynamical system by means of the spac e of sectio ns of a bundle, without any ass umpt ion on the structure of the int ertwiner spa ces. Let ( ρ, d, R ) ∈ end A b e a weakly sp ecial endomo rphism. T o simplify the expo s ition we as sume that X ρ is connected (otherwise, it is p ossible to decomp ose ( A , ρ ) into a direct sum indexed by the clop ens of X ρ ). W e deno te the set of isomorphis m cla sses of vector bundles E → X ρ with rank d and ﬁrs t Chern class c 1 ( ρ ) by E ( ρ ) , and deﬁne the Ab elian C* -alge bra C ρ := M [ E ] ∈E ( ρ ) C E . In the prev io us deﬁnition we considered the C* -algebr as C E , thus fo r each [ E ] ∈ E ( ρ ) w e made a choice of E in its class [ E ] ∈ E ( ρ ) . If E and E ′ are isomorphic (i.e., [ E ] = [ E ′ ] ∈ E ( ρ ) ), then C E is isomorphic to C E ′ , th us the isomo r phism class of C ρ do es not dep end on the choice. Let us now in tro duce the compact gro up bundle: S U E ( ρ ) := Y E ∈E ( ρ ) S U E → X ρ ; since each comp onen t S U E , E ∈ E ( ρ ) , of S U E ( ρ ) is full, we conclude that the set of s ections SU E ( ρ ) is a s ection group for S U E ( ρ ) . Lemma 6.9. C ρ is a c ontinuous bund le of A b elia n C*-algebr as with b ase sp ac e X ρ . Mor e over, ther e is a gauge action α : S U E ( ρ ) × X ρ b C ρ → b C ρ . Pr o of. Since each C E , E ∈ E ( ρ ) , is a contin uous bundle of Ab elian C* -algebras , we ﬁnd that C ρ is a contin uous bundle o f Ab elian C* -alg ebras, with as s ociated C* -bundle b C ρ → X ρ . Note tha t ea c h ﬁbre C ρ,x , x ∈ X ρ , is isomor phic to the direc t sum ⊕ E C E ,x of the ﬁbres of the C* -alg e br as C E , E ∈ E ( ρ ) . Let us denote the generic element of S U E ( ρ ) by e u := { u E ∈ S U E } E ∈E ( ρ ) ; fo r every c := ⊕ E c E ∈ b C ρ (with c E ∈ b C E ), w e deﬁne α ( e u , c ) := ⊕ E α E ( u E , c E ) , where α E : S U E × X ρ b C E → b C E is the g auge action deﬁned as in (6.9), restricted to C E . It follows from the pr e vious Lemma tha t the sp ectrum Ω ρ of C deﬁnes a bundle q : Ω ρ → X ρ . By deﬁnition of C ρ , we may reg a rd Ω ρ as the disjoint union ˙ ∪ E Ω E . F or every x ∈ X ρ , the ﬁbre Ω ρ,x := q − 1 ( x ) is ho meo morphic to the disjoint union ˙ ∪ |E ( ρ ) | G x \ SU ( d ) , wher e G x ⊆ SU ( d ) is the group deﬁned b y means of Theorem 6 .2 (see also Remark 6.3). By Gel’fand duality , there is a gauge action α ∗ : S U E ( ρ ) × X ρ Ω ρ → Ω ρ . (6.18) Let us co nsider the space of sections S X ρ (Ω ρ ) of Ω ρ . Since X ρ is connected, every s ∈ S X ρ (Ω ρ ) has image cont ained in some connected comp onen t of Ω ρ . Since each Ω E app ears as a clop en in 22 Ω ρ , w e conclude that s ( X ρ ) ⊆ Ω E for some E ∈ E ( ρ ) , so that s is actually a sec tio n o f Ω E . Now, the action (6.18) induces an action α ∗ : SU E ( ρ ) × S X ρ (Ω ρ ) → S X ρ (Ω ρ ) , α ∗ ( u, s )( x ) := α ∗ ( u ( x ) , s ( x )) , x ∈ X . The previous element ary rema r ks, and Theo rem 6.7, imply the following Theorem 6.10. L et ( ρ, d, R ) b e a quasi-sp e cial en d omorphism of a C*-algebr a A . (If X ρ is c onn e cte d,) then ther e is a one-to-one c orr esp ondenc e b etwe en the set of H i lb ert extensions of ( A , ρ ) and the sp ac e of se ctions S X ρ (Ω ρ ) , assigning to e ach s ∈ S X ρ (Ω ρ ) the t rip le ( A ⋊ ν b G s , G s , β ) . Two Hilb ert extensions of ( A , ρ ) asso ciate d with se ctions s, s ′ ar e e qu iv alent if and only if s ′ = α ∗ ( s, u ) for some u ∈ SU E ( ρ ) . Some r emarks follow. First, the ﬁbration Ω ρ → X ρ may lack sectio ns, and in this case there are no Hilb ert extensions of ( A , ρ ) . Moreover, the SU E ( ρ ) -action on S X ρ (Ω ρ ) may be no t transitive, and this mea ns that there could b e non-equiv a le n t Hilber t extensio ns of ( A , ρ ) . It could b e interesting to study the behaviour of Hilber t extensions A ⋊ ν b G s n for sequences { s n } conv erging to a given s ∈ S X ρ (Ω ρ ) . The case with Z = C 1 studied in [1 0 , § 4] yields a s pecial endomorphism ρ with C ( X ρ ) ≃ C , so that E ( ρ ) has a s unique element the Hilber t space C d , d ∈ N , and Ω ρ reduces to a homogeneo us space; this means that there is a unique (up to equiv alence) Hilber t extensio n. More g e nerally , if Z 6 = C 1 , a canonica l endo mo rphism ρ in the sense of [3, § 4] whic h is also quasi- special ha s Chern class c 1 ( ρ ) = 0 , and a Hilb ert extensio n of ( A , ρ ) in the sense of the a bov e-cited reference corres p onds to a constan t se c tion of Ω E ≃ X ρ × G \ SU ( d ) , Ω E ⊂ Ω ρ , where E ∈ E ( ρ ) is the trivia l rank d vector bundle ov er X ρ . 6.3.1 A class of examples. Non-uniquenes s of the Hi lbe rt extension. Let A := O S U E , ρ := σ S U E be deﬁned as in § 3 fo r a ﬁxed rank d v ector bundle E → X , d ∈ N . There is a natural ide ntiﬁcation X ≡ X ρ , and ρ is a sp ecial endomorphism w ith class d ⊕ c 1 ( E ) (see § 6.2). Now, if E ′ → X is a n arbitrar y vector bundle having the sa me rank and ﬁrst Chern class as E , then the dua l a ction µ ′ : O S U E ′ → A (6.19) deﬁned as in Theorem 6.2 is an isomorphism; in fact, A is the ρ -sta ble alg ebra generated by θ , ( ι, λE ) (see [22, P ropos ition 4 .17]), and this is exa ctly the image of µ ′ (see remar ks a fter Theorem 6.2). In this wa y , by [23, Example 3.2] we ﬁnd O E ′ ≃ A ⋊ µ ′ \ S U E ′ (in particular, O E ≃ A ⋊ µ [ S U E , see a lso Lemma 6 .13 b e low). Since C E ′ := O ′ E ′ ∩ O E ′ = C ( X ) , we conclude that Ω E ′ = X , so that Ω ρ is a disjoint union of copies of X : Ω ρ ≃ ˙ ∪ E ′ ∈E ( ρ ) X . This means that for every E ′ ∈ E ( ρ ) there is a unique sec tio n s E ′ : X → Ω ρ , with image coin- ciding with the copy of X lab elled b y E ′ . F or ea c h of such sectio ns, there is a Hilb ert extension ( O E ′ , S U E ′ ) of ( A , ρ ) , not nece s sarily equiv alen t to ( O E , S U E ) . F or example, take X co inciding with the sphere S 2 n , n > 2 , and E suc h that S U E is nont rivia l (it is well-known that such v ector bundles exist o n S 2 n , see [1 4 , I.3.13]). Since c 1 ( E ) = c 1 ( ρ ) = 0 (in fact, H 2 ( S 2 n , Z ) = 0 ), we may pick E ′ := S 2 n × C d , the as sociated dual action (6.1 9 ), and obtain the ”trivial” Hilber t ex tension ( C ( S 2 n ) ⊗ O d , S 2 n × SU ( d ) ) , with S U E not isomorphic to S 2 n × SU ( d ) . 23 6.3.2 Another class of examples. Non-exis tence of the Hil bert extens ion. Let X denote the 2 -sphere. W e consider a lo cally trivial pr incipal SO (3) -bundle Ω → X with no se ctions , and endowed with the trans la tion actio n λ : SO (3) → aut X C (Ω) such that C (Ω) λ = C ( X ) . Such a bundle exis ts beca us e H 1 ( X, SO (3)) ≃ π 1 ( SO (3)) ≃ Z 2 is non trivial. Now, SO (3) is a q uotien t of SU (2) (in fact, SO (3) ≃ SU (2 ) / Z 2 ), thus w e may lift λ to an action λ : SU (2) → aut X C (Ω) . By Gel’fa nd duality , we wr ite λ u c ( ω ) = c ( ω u − 1 ) , u ∈ SU (2) , c ∈ C (Ω) , ω ∈ Ω . W e now consider the C* -algebra B := C (Ω , O 2 ) , endow ed with the action α : SU (2) → aut X B , α u b ( ω ) := b u ( b ( ω u − 1 )) , b ∈ B , u ∈ SU (2) , ω ∈ Ω , deﬁned as in (2.1). As in [24, Lemma 6.8], we ﬁnd A ′ ∩ B = B ′ ∩ B = C (Ω) , where A := B α the ﬁxed-p oin t C ( X ) -alg e bra with resp ect to the SU (2) -action. W e now equip B with the e ndo morphism σ ( b ) := P 2 i =1 ψ i bψ ∗ i , b ∈ N , where ψ 1 , ψ 2 are the is ometries genera ting O 2 ⊂ B . By a standard argument, w e ﬁnd that σ restricts to an endomorphism ρ ∈ end X A (see (6.26) b elo w). Let us denote the trivial rank 2 vector bundle by E → X ρ , and its sp ecial unitary bundle by S U E = X × SU (2) . Then O E = C ( X , O 2 ) and O S U E = C ( X , O SU (2) ) . If we r egard C ( X ) as the C* -subalgebra of C (Ω) of functions whic h are inv ar ian t with r espect to the right transla tion action induced by λ , then we ﬁnd an obvious inclusion O S U E ⊆ A ; moreov er, it is clear that O E ⊆ B . If we denote the ab ov e-ment ioned inclus io ns by µ : O S U E → A and j : O E → B , then it is clear that j ( t ) = µ ( t ) , t ∈ O S U E . Thus, b y universalit y of the crossed pro duct by a dual action, we conclude the following: Prop osition 6. 11. ρ ∈ end X A is a sp e cial endomorphism with class 2 ⊕ 0 ∈ N ⊕ H 2 ( X, Z ) , and B is c anonic al ly isomorphic to the cr osse d pr o duct A ⋊ µ [ S U E . Thu s, Ω coincides with the bundle Ω E deﬁned in the previous sections, a nd Ω E lacks of sections. Moreov er, X b eing the tw o sphere, every vector bundle E ′ → X with rank 2 and trivial ﬁrst Chern class is is o morphic to E (see [14, V.3.25]). Thus, we conclude that Ω ρ = Ω E lacks of sections, and ( A , ρ ) do es not a dmit Hilb ert extensio ns. 6.4 The mo del, and a dualit y t heorem . Let X be a compact Hausdo rﬀ spa ce X , and Z an Abelia n C ( X ) -algebra with identit y 1 . W e consider a C ( X ) -Hilber t Z -bimo dule M ≃ b E ⊗ C ( X ) Z with left action φ : Z → ( M , M ) , deﬁned as in Exa mple 4.6 for s ome ra nk d vector bundle E → X . The Cun tz-Pimsner algebra O M asso ciated with M ma y be presented a s the o ne genera ted by Z a nd a se t of generator s { ψ l } n l =1 of b E , with rela tions ψ ∗ l ψ m = h ψ l , ψ m i , X l ψ l ψ ∗ l = 1 , z ψ l = φ ( z ) ψ l , (6.20) z ∈ Z , h ψ l , ψ m i ∈ C ( X ) . It is clea r that O M is a C ( X ) -algebra; on the other side, note that in gener al Z is not contained in the centre of O M . It is easy to verify that the C* -bundle b O M → X has ﬁbres iso morphic to the Cunt z-Pimsne r algebra s O M x asso ciated with the ﬁbres M x , x ∈ X ; we denote the ﬁbre epimorphisms b y η x : O M → O M x . Since each M x is is o morphic to C d ⊗ Z x as a rig h t Hilb ert Z x -mo dule (see Example 4.6), w e ﬁnd that there is a set { ψ x,i } d i =1 of o r thonormal gener ators for M x . At the level of Cuntz-Pimsner alg ebra, { ψ x,i } d i =1 app ears a s a set of or thonormal partial isometr ies with total suppo rt the identit y 1 x ∈ O M x ; by universality 24 of the Cuntz relatio ns , we obtain a unital monomor phism j x : O d → O M x . Now, the fo llowing endomorphism is deﬁned: τ M ( t ) := X l ψ l tψ ∗ l , t ∈ O M . (6.21) It turns out that τ M is weakly-symmetr ic: the representation θ : P ∞ → O M is deﬁned by θ ( p ) := X ψ l p (1) · · · ψ l p ( r ) ψ ∗ l r · · · ψ ∗ l 1 , p ∈ P r ⊂ P ∞ . (6.22) Moreov er, for every x ∈ X w e hav e a unital endomor phism τ x ∈ end O M x : τ x ( t ) := X i ψ x,i t ψ ∗ x,i , in such a w ay tha t τ x ◦ η x = η x ◦ τ M , x ∈ X . By adapting a standard argument used in the s ettin g of the Cuntz alg ebra, we ﬁnd that if u ∈ U ( d ) and uψ x,i := P k ψ x,k u ki (where u ki ∈ C denotes the matrix co eﬃcien t of u ), then τ x ( t ) = X i ( uψ x,i ) t ( uψ x,i ) ∗ . (6.23) In the next Lemma, w e prov e that τ M is sp ecial. T o this end, for every r ∈ N we cons ide r the totally an tisymmetric bundle λ ( E r ) ⊆ E r d r , and in tro duce the C* -subalg ebra S of O M generated by elements of the spaces ( ι, λ ( E r )) ⊆ ( ι, τ r d r M ) θ , r ∈ N . Lemma 6.12. With t h e ab ove n otatio n, it t u rns out S ′ ∩ O M = Z ; mor e over, τ M is a sp e cial endomorphi sm, with ( τ r M , τ s M ) = ( τ r M , τ s M ) θ = ( E r , E s ) , r, s ∈ N . Pr o of. As a ﬁrst step, note that we may ident ify C ( X ) with the C* -algebra { z ∈ Z : ψz = φ ( z ) ψ , ψ ∈ M} (6.24) (otherwise, we ma y consider the sp e ctrum X ′ of (6.24), and pass to the pullbac k E ′ := E × X X ′ ). Let t ∈ S ′ ∩ O M ; then, for ev ery x ∈ X we ﬁnd η x ( t ) ∈ η x ( S ) ′ ∩ O M x . Now, since each λ ( E r ) → X , r ∈ N , is a lo cally trivial line bundle, we ﬁnd that η x ( S ) is g enerated by a s e quence of partial isometries { S x,r } r . Since S x,r is a gener ator of the totally antisymmetric space of E r x , we hav e the following analo gue of (6.4): S ∗ x,r τ r d r x ( S x,r ) = ( − 1) r d r − 1 d − r 1 x . Thu s, applying [6, Prop osition 3.5(b)], we conclude η x ( t ) ∈ Z x , and t ∈ Z . Fina lly , by [22, Lemma 5.5] we ﬁnd ( τ r M , τ s M ) = ( τ r M , τ s M ) θ = ( E r , E s ) , so that τ M has p erm utation sy mmet ry; moreover, τ M is a lso sp ecial, in fact ( ι, λ E ) ⊆ ( ι, τ d M ) fulﬁlles the req uired prop erties. Let now p : G → X , G ⊆ U E , be a compact group bundle; then, for every x ∈ X , the inclusion G x ⊆ U ( d ) induces a n actio n G x → U M x ≃ by r igh t Z x -mo dule o perator s . Let b φ b e the morphism deﬁned in (4.1); we make the following assumptions:  [ y , b φ ( v )] = 0 , y ∈ G , v ∈ Z p ( y ) ( M r , M s ) G ⊆ ( M r , M s ) Z , (6.25) where ( M r , M s ) Z is deﬁned as in (4.9). The equa lit y (6.2 5 .1) ensures that M is a G -Hilbert Z -bimo dule fulﬁlling (4 .8), thus b y Lemma 4 .5 there is a gaug e action α : G × X b O M → b O M . The 25 equality (6.25.2) ensures that M ⊗ G is a tensor C* -category , in accord with the rema rks in Example 4.6. Let us no w consider the C ( X ) -algebra O G M ⊆ O M generated by the bimo dules ( M r , M s ) G , r , s ∈ N . In the same wa y as in Lemma 3 .1 , we ﬁnd that if there is a n inv a rian t C ( X ) -functional deﬁned on G , then O G M is the ﬁx ed-point algebr a of O M with res pect to α . If y ∈ G x , x ∈ X , then α ( y , ψ x,i ) = X k ψ x,k y ki ; th us, by using (6.23), α ( y , τ x ( t )) = X i ( y ψ x,i ) α ( y , t ) ( y ψ x,i ) ∗ = τ x ◦ α ( y , t ) . (6.26) The pr evious equality implies that (6.21) res tr icts to an endomorphism τ G ∈ end X O G M . Since θ ( r, s ) ∈ ( τ r M , τ s M ) ∩ O G M , we conclude that τ G has w eak p erm utation symmetry . Lemma 6.13. L et G ⊆ S U E b e a c omp act gr oup bund le fulﬁl ling (6.25 ) and endow e d with an invariant C ( X ) -functional ϕ : C ( G ) → C ( X ) . Then the c anonic al endomorph ism τ G ∈ end X O G M is quasi-sp e cial, and O M ≃ O G M ⋊ ν b G is a Hilb ert extension of ( O G M , τ G ) . Pr o of. It is clear that τ G is weakly special, and that τ s G ( Z )( τ r G , τ s G ) θ ⊆ ( τ r G , τ s G ) , r, s ∈ N . T o pr o ve the opp osite inclusion w e consider a ﬁnite set { ψ l } of gener ators for b E ; by co nstruction, ψ l ∈ ( ι, τ M ) θ for every index l , so that ψ M ∈ ( ι, τ r M ) θ , | M | = r . F or every t ∈ ( τ r G , τ s G ) , w e deﬁne z LM := ψ ∗ L tψ M , and no te that for every b ∈ O G M it turns out bz LM = b ψ ∗ L tψ M = ψ ∗ L τ s G ( b ) tψ M = ψ ∗ L tτ r G ( b ) ψ M = z LM b . In par ticular, z LM commutes with elements of ( ι, ( λE ) r ) ⊆ ( ι, M r d r ) G , r ∈ N ; by Lemma 6.1 2, we conclude tha t z LM ∈ ( O G M ) ′ ∩ O M = Z . In this way , we ﬁnd t = P LM ψ L z LM ψ ∗ M = P LM ρ s ( z LM ) ψ L ψ ∗ M . No te that ψ L ψ ∗ M ∈ ( τ r M , τ s M ) θ , thus t ∈ ρ s ( Z )( τ r M , τ s M ) θ . Now, for every r , s ∈ N , the inv aria nt functional ϕ induces an inv aria n t mean m : ( τ r M , τ s M ) θ → ( τ r G , τ s G ) θ . Since m is a n O G M -mo dule map, we obtain t = m ( t ) = X LM m ( ρ s ( z LM ) ψ L ψ ∗ M ) = X LM ρ s ( z LM ) m ( ψ L ψ ∗ M ) , with m ( ψ L ψ ∗ M ) ∈ ( τ r G , τ s G ) θ . This pr o ves that τ G is quasi-sp ecial. W e now pro ceed with the pro of that O M is a Hilber t extension o f ( O G M , τ G ) . T o this end, w e note that by considering the inclusion O G M ⊆ O M , it turns out ( τ r G , τ s G ) θ ⊆ ( τ r M , τ s M ) θ = ( E r , E s ) (see Lemma 6.12); in particula r, it is clear by deﬁnition o f α that ( τ r G , τ s G ) θ = ( E r , E s ) G , r, s ∈ N . In this wa y , the inclusio ns O G ⊆ O G M ⊆ O M and (6.21) imply that O M = O G M ⋊ ν b G , where ν is the inclusion O G ⊆ O G M . Finally , since the C* -a lg ebra S is contained in O G M (in fact, G ⊆ S U E , and every elemen t of S is S U E -in v ariant), we conclude from Lemma 6.12 that ( O G M ) ′ ∩ O M = Z . Example 6 .3. L et Z b e a unital C ( X ) - al gebr a, and K a c omp act gr oup. Then ther e is a discr ete Ab elian gr oup C ( K ) whose elements ar e classes [ v ] of irr e ducible r epr esentations of K with r esp e ct to a su itab le e quivalenc e r elation (se e [3] and r elate d r efer enc es: C ( K ) is c al le d the chain gr oup of K , and is isomorphic to the Pontryagi n dual of the c ent r e of K ). Le t us now assume that 26 ther e is an action α : C ( K ) → aut X Z . If w is an irr e ducible r epr esentation of K with r ank d , then we c onsider the C ( X ) -Hilb ert Z -bimo du le M w := C d ⊗ Z with left action z ψ := ψ α [ w ] ( z ) , ψ ∈ M w , z ∈ Z , and deﬁne M := M w ⊕ M w ⊕ Z (wher e Z is r e gar de d as the fr e e, r ank one Hilb ert Z - bi mo dule). In this way, M b e c omes a K -Hilb ert Z -bimo dule, with action U : K × M → M , U y := w y ⊕ w y ⊕ 1 . It c an b e pr ove d that ( M r , M s ) K ⊆ ( M r , M s ) Z , r, s ∈ N , so that M ⊗ K is a tensor C*-c ate gory, and τ K ∈ end X O K M is an endomorphism with p ermutation quasi-symmetry fulﬁl ling ( τ r K , τ s K ) = ( M r , M s ) K , r, s ∈ N ; mor e over, ( O K M ) ′ ∩ O M = Z (se e [16, § 4] for details). In p articular, if w takes values in SU ( d ) , then τ K is quasi-sp e cial. W e can now characterize quasi- special endomorphisms in ter ms of C* -dynamical systems of the type ( O G M , τ G ) . The pro of is a direct consequence of Theorem 5.2, E q.(5.4), L emma 6.1, and Eq.(6.21), so that it is omitted. Theorem 6. 14. L et ( ρ, ε, R ) ∈ end A b e a quasi-sp e cial endomorph ism with class d ⊕ c 1 ( ρ ) ∈ N ⊕ H 2 ( X ρ , Z ) . Then for every gauge-e quivariant p air ( E , G ) asso ciate d with ( A , ρ ) (se e D eﬁni- tion 6.6), ther e is a G - H il b ert Z -bimo dule M ≃ b E ⊗ C ( X ρ ) Z such that the fol lowing diagr am is c ommu tative: ( O ρ,ε , ρ ) ⊆ / / ≃   ( O ρ , ρ ) ≃   ( O G , σ G ) ⊆ / / ( O G M , τ G ) (6.27) (the symb ol ” ≃ ” stands for an isomorphism) Mor e over, ther e ar e isomorphisms of strict tens o r C*-c ate gories b G ≃ b ρ ε , M ⊗ G ≃ b ρ such that, for e ach r, s ∈ N , the fol lowing diagr am is c ommutative: ( ρ r , ρ s ) ε ≃   ⊆ / / ( ρ r , ρ s ) ≃   ( E r , E s ) G ⊆ / / ( M r , M s ) G (6.28) The pr e v ious theorem can b e use d as a star ting po in t for the cons truction of an abstr act dua lit y theory for c o mpact g roup bundles vs. tensor C* -categor ies with ob jects non-sy mmetric Hilb ert bimo dules. This shall b e done in a forthco ming work. References [1] M.F. Atiy ah: K - Theory , Benjamin, N ew Y ork, 1967 [2] H. Baumg¨ artel, F. Lled´ o: Sup erselection Structures for C* - al gebras with Nontrivial Center, Rev. Math. Phys. 9 , 785-81 9 (1997) [3] H. Baumg¨ artel, F. Lled´ o: Dualit y of compact groups and Hilb ert C* -sy stems F or C* -algebras with a nontrivial cen ter, Int. J. Ma th. 15 (8), 759-812 (2004) [4] B. Blac k adar: K -Theory of Op erator Algebras, 1995 [5] J. Cuntz: Simple C* -algebras Generated by Isometries , Comm. Math. Phys. 57 , 173-185 (1977) [6] S. D opli cher, C. Pinzari, R. Zuccante: The C* -algebra of a Hilb ert bimodu le, Bollettino UMI Serie VI II 1B , 263-282; arXiv:funct-an/9707006v1 (1998) 27 [7] S. Doplicher, J.E. R oberts: Duals of Compact Lie Groups Realized in the Cuntz A lge bras and Their Actions on C* -A lg ebras, J. F u nc. A nal. 74 , 96-12 0 (1987) [8] S. Doplicher, J.E. Rob erts: Compact Group Actions on C* -Algebras, J. Op er. Theory 91 , 227-284 (1988) [9] S. Doplicher, J.E. R oberts: A New Duality Theory for Compact Group s, I n v. Math. 98 , 157-218 (1989) [10] S. Doplic her, J.E. Rob erts: Endomorphisms of C* -algebras, Cross Prod ucts and Dualit y for Compact Groups, Ann. Math. 130 , 75-119 (1989) [11] S. Doplicher, J.E. Rob erts: Why there is a ﬁ eld algebra with a compact gauge group describing the sup ersel ection structure in particle p h ysics, Comm. Math. Phys. 131 , 51 -107 (1990) [12] M.J. Dupr´ e: Classifying Hilb ert bundles I, J. F un ct. A n al . 15 , 244-278 (1974) [13] K.H Hofman, K. Keimel: Sheaf theoretic concepts in analysis: b undles and sheav es of Banac h spaces, Banac h C ( X ) -bimodu le s, in: Applications of Sheav es, Lecture Notes in Mathematics 753, Springer- V erlag, New Y ork, Heidelb erg, Berlin, 1979 [14] M. Karoubi: K -Theory , Springer V erlag, Berlin - Heidelberg - New Y ork, 1978 [15] G.G. Kasparo v, Equiv arian t K K -Theory and the No viko v Conjecture, Inv. Math. 91 , 147-201 (1988) [16] F. Lled´ o, E. V asselli: Realization of minimal C* -dyn ami cal systems in terms of Cuntz-Pimsner algebras, preprint arXiv:math/07 02775 v1 (2007) [17] G. Mack, V. Schomerus: Conformal ﬁeld algebras with quantum symmetry from th e th eo ry of sup er- selection sectors, Comm. Math. Ph ys. 134 , 139-196 (1990) [18] A.S. Mishc henko, A.T. F omenko: The Index of Elliptic Op erators ov er C* -algebras, Math. USSR Izvestij a 15 (1) 87-112 (1980) [19] V. Nistor, E. T roitsky: An index for gauge-inv ariant operators and the Dixmier-D o uady inv ariant, T rans. AMS. 356 , 185-218 (2004) [20] M. Pimsner: A Class of C* -algebras Generalizing b oth Cuntz-Krieger algebras and Cross Prod uct by Z , in: F ree Probabilit y Theory , Ed. D. V oiculescu, Fields In sti tute Communications 12 , 189-212 (1997) [21] G. Segal, Equiv arian t K -theory , Inst. Hautes ´ Etudes Sci. Publ. Math. 34 , 129-151 (1968) [22] E. V asselli: Crossed p roducts by en d omo rphisms, vector bundles, and group dualit y , Int. J. Math. 16 (2), 137-171 (2005) [23] E. V asselli: Crossed prod ucts b y endomorphisms, vector bundles, and group duality , I I , In t. J. Math. 17 (1) 65-96 (2006) [24] E. V asselli: Some R ema rks on Group Bundles and C* -dyn a mical Systems, p reprin t arXiv math.OA/03 01124 , to appear on Comm. Math. Phys. (2006 ) [25] E. V asselli: Bundles of C* -categories and du al ity , preprint arXiv:math.CT /0510594 v2 (2005 ); E. V as- selli: Bund le s of C* -categories, J. F unc. Anal. 247 351-377 (2007) 28

Gauge-equivariant Hilbert bimodules and crossed products by endomorphisms

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