Bundles of C*-categories, II: C*-dynamical systems and Dixmier-Douady invariants

Bundles of C* -categories, I I: C* -dynamical systems and Dixmier-Douady in v arian ts Ezio V asselli Dip artimento di Matematic a University of R o me ”L a Sapienza” P.le Aldo Mor o, 2 - 00185 R oma - Italy vasselli@mat.uniroma2.it Octob er 29, 2018 Abstract W e introduce a cohomolog ical inv ariant arising from a class in nonab elian cohomolog y . This inv ariant generalizes th e Dixmier-Douady class and encodes the obstruction to a C* -algebra bundle being t h e ﬁxed-p oint algebra of a gauge action. As an application, the duality breaking for group bun dles vs. tensor C* -categories with n on-simple unit is discussed in the setting of Nistor-T roitsky gauge-equiv ariant K -theory: there is a map assigning a nonab elian gerb e to a tensor category , and “triviality” of the gerb e is equ iva lent to the existence of a dual group bundle. At th e C* -algebraic level, th is correspon d s t o studying C* -algebra bun dles with ﬁb re a ﬁxed-p oint algebra of t h e Cuntz algebra and in this case our inv ariant describ es th e obstruction to ﬁnd ing an emb ed ding into the Cuntz-Pimsner algebra of a vector bund le. AMS Subj. Class.: 18D10, 22D25, 14F05, 55N30. Keywor ds: T ensor C* -category; D u alit y; Cuntz algebra; Group bund le; Gerb e. 1 In tro duction. In a series of works in the last eighties, S. Doplicher and J.E. Rob erts develop e d a n abstract duality for compact gr o ups, motiv ated by q uestions arise d in the context of a lgebraic q uan tum ﬁeld theo ry . In such a scenario, the dual ob ject of a compact gro up is ch a racterized as a tensor C* -categ ory , namely a tenso r categ ory ca rrying an additional C* -algebr aic struc tur e (nor m, conjugation). A t the C* -alge br aic level, one of the main discoveries in that setting ha s b een a machinery per forming a duality theory for compact groups in the context of the Cun tz algebr a ([4]). If d ∈ N and ( O d , σ d ) is the Cun tz C* -dynamical s ystem (her e σ d ∈ end O d denotes the canonica l endomorphism, see [6, § 1]), then every compact subgroup G ⊆ U ( d ) deﬁnes an a utomorphic a ction G → aut O d , G ∋ g 7→ b G ∈ aut O d : b g ( ψ i ) := d X j =1 g ij ψ j , (1.1) where g ij ∈ C , i, j = 1 , . . . , d , are the matrix elements of g and { ψ i } denotes the multiplet of m utually orthogo nal pa rtial iso metries g enerating O d . Let O G denote the ﬁxed- p oint a lgebra of O d w.r.t. the action (1.1). Since σ d commutes w ith the G -action, the restrictio n σ G := 1 σ d | O G ∈ end O G is well-deﬁned. The C* -dynamical system ( O G , σ G ) allows one to rec o nstruct the following ob jects: (1) the gro up G , a s the stabilizer of O G in aut O d ; (2) The categ ory b G of tenso r powers of the deﬁning r epresentation G ֒ → U ( d ) , as the categor y b σ G with o b jects σ r G , r ∈ N , a nd arrows the intertwiner sp ac es o f σ G : ( σ r G , σ s G ) := { t ∈ O G : σ s ( a ) t = tσ r G ( a ) , a ∈ O G } , r , s ∈ N . (1.2) In this way , the ma p G 7→ ( O G , σ G ) (1.3) may b e consider ed as a ”Galois corr espo ndence” for compact subgr oups of U ( d ) . A mor e subtle q ue s tion is when a C* -dyna mica l sy stem ( A , ρ ) , ρ ∈ end A , is isomorphic to ( O G , σ G ) for some G ⊆ U ( d ) . The so lution to this problem (for G contained in the spec ial unitary gr oup SU ( d ) ) has be e n g iv en in [9, § 4]: to get the ab ov e characterizatio n, natura l necessa ry conditions are the tr iv ialit y o f the ce ntre of A and the fa c t that A is generated as a Banach space by the intertwiner spaces ( ρ r , ρ s ) , r , s ∈ N ; a more crucial condition is the existence of an intert winer ε ∈ ( ρ 2 , ρ 2 ) , ε = ε − 1 = ε ∗ (the s ymm et ry ), providing a representation P ∞ → A of the inﬁnite per m utatio n group and implementing suitable ﬂips b etw een e le men ts of ( ρ r , ρ s ) , r, s ∈ N . This structure is an abs tract counterpart of the ﬂip op erator θ ( ψ ⊗ ψ ′ ) := ψ ′ ⊗ ψ , ψ , ψ ′ ∈ H , where H is the Hilb ert s pace of dimension d . In this wa y , a gr oup G ⊆ SU ( d ) is a s so ciated with ( A , ρ, ε ) a nd the int er t winer spaces of ρ are int er preted as G -inv a riant o p era tors betw een tensor p ow ers of H . In this sense G is the gauge gr oup asso ciated w ith ( A , ρ, ε ) , acco rding to the motiv a tion o f Doplic her and Rober ts ([8 ]). The corres p ondence ( A , ρ, ε ) 7→ G is functorial: gro ups G, G ′ ⊆ SU ( d ) are co njugates in U ( d ) if and only if there is an iso morphism α : ( A , ρ, ε ) → ( A ′ , ρ ′ , ε ′ ) of p ointe d C*-dynamic al systems , in the sense that the co nditio ns α ◦ ρ = ρ ′ ◦ α , α ( ε ) = ε ′ , a re fulﬁlled. As we sha ll see in the seq uel, the prev ious co nditions are e q uiv alent to req uire a n is omorphism of s ymmetric tensor C* -ca tegories naturally ass oc ia ted with our C* -dyna mical systems. Our resea rch progr am fo cused o n the study of tenso r C* -categories with non- s imple unit. This means that the space of ar rows of the iden tity ob ject ι is isomorphic to an Abelian C* -algebr a C ( X ) fo r some compact Hausdorﬀ space X . Thu s the mo del categor y , rather tha n the o ne of Hilber t spaces, is the one of Hermitian vector bundles ov er X , that we denote by v ect ( X ) . In a previous work ([25]), we prov ed that every tensor C* -category with symmetry and conjugates can be rega rded in ter ms o f a bundle of C* -catego ries ov er X , with ﬁbres duals of compact groups (see also [28]). By applying a standar d technique, we asso ciate p ointed C* -dynamical systems of the t yp e ( A , ρ, ε ) with ob jects of these categories; as a consequence o f the a bove-men tioned results, each ( A , ρ, ε ) is a contin uous bundle o f C* -alge br as with base X and ﬁbres p ointed C* -dynamica l systems ( O G x , σ G x , θ x ) , x ∈ X . Starting fro m this re s ult, it b ecame natural to sear c h for a class iﬁc a tion of lo c al ly trivial p oin ted C* -dynamical systems ( A , ρ, ε ) with ﬁbr e ( O G , σ G , θ ) , G ⊆ SU ( d ) . In the ﬁrst pap er of the present series, we gav e suc h a classiﬁca tion in terms of the coho mology set H 1 ( X, QG ) , QG := N G \ G , where N G is the normalizer o f G in U ( d ) ([2 5]). In this w ay , QG -co cycles q ∈ H 1 ( X, QG ) are put in one-to-o ne corresp ondence with p ointed C* -dynamical sys tems ( O q , ρ q , ε q ) . F rom a diﬀerent –but equiv alent– p oint of view, H 1 ( X, QG ) desc ribes the set sym ( X , b G ) of isomor phism class es of ”lo cally trivial” symmetric tenso r C* -categories with ﬁbre b G and such that ( ι, ι ) ≃ C ( X ) . In the pre s en t pa p er w e s tudy the Ga lois corresp ondence (1.3) a nd the asso cia ted abstract version in the case where X is non tr ivial. Instead of O d , o ur refer e nce a lgebra is the Cunt z- Pimsner algebra 2 O E asso ciated w ith the mo dule o f sections o f a vector bundle E → X , which y ields a p oint ed C* - dynamical system ( O E , σ E , θ E ) . If G → X is a bundle o f unitary automor phisms of E , then we can construct a p oint ed C* -dynamica l sy stem ( O G , σ G , θ E ) , O G ⊆ O E , from which it is p ossible to recov er G with the same metho d used for compact subgr oups of U ( d ) . This leads to a duality for elements of sym ( X, b G ) vs. G -bundles acting o n vector bundles in the sense o f Nistor and T roitsk y ([20]). An ywa y , what we g et is not a genera lization of the Doplicher- Rob erts constr uction, a s new phenomena arise. Fir s t, in general it is false that a catego ry with ﬁbre b G is the dua l o f a G -bundle; the rea s on is a coho mological obstruc tio n to the embedding into v ect ( X ) : in C* -alg ebraic terms, there a re p ointed C* -dynamical s ystems ( O q , ρ q , ε q ) whic h do not admit a n embedding in to so me ( O E , σ E , θ E ) . Seco ndly , an element of sym ( X , b G ) may b e realized as the dual of non-iso morphic G -bundles: a t the C* -algebr aic level, we ma y ge t iso mo rphisms ( O q , ρ q , ε q ) ≃ ( O G , σ G , θ E ) , ( O q , ρ q , ε q ) ≃ ( O G ′ , σ G ′ , θ E ′ ) , with G not isomor phic to G ′ and E not isomorphic to E ′ . In the present work we give a explanation of these facts in terms of prop erties of H 1 ( X, QG ) , providing a complete geometrical characterization of sym ( X , b G ) for wha t concer ns the duality theory . The ab ove-men tioned coho mo logical ma chinery has its ro o ts in the ge neral framework of princi- pal bundles and can b e applied to generic C* -algebra bundles. Let G be a g roup of automorphisms of a C* -algebra F • and A • denote the ﬁxed p oin t a lg ebra w.r.t. the G -action. It is natural to ask whether an A • -bundle A admits an embedding int o so me F • -bundle. In g eneral, the a nsw er is nega tive and the obstruction is mea sured by a class δ ( A ) ∈ H 2 ( X, G ′ ) , (1.4) where G ′ is an Abelia n quo tien t o f G . When the ab ov e-mentioned em b edding exists, A is the ﬁxed-p oint alg ebra w.r.t. a gauge-action of a gro up bundle G → X with ﬁbre G on an F • -bundle, in the sense of [2 6]. The abov e- men tioned obstr uction for bundles with ﬁbre ( O G , σ G , θ ) a nd the classical Dixmier-Douady inv ariant for bundles with ﬁbre the compact oper ators ([5, Ch.10]), are particular ca ses of this co nstruction. The present work is organiz e d as follows. In § 3 we recall some r e sults re lating p ointed C* -dynamical sy s tems with tensor C* -categ ories. Moreov er , under the h yp othesis that the inclusio n G ⊆ U ( d ) is c ovariant (i.e., the embedding of b G in to the catego r y of tensor p ow ers of H is unique up to unitary natural transfor mations), we give a geometrical character ization of the s pace of embeddings of O G int o O d (Lemma 3.4) and a cohomo logical classiﬁc a tion for s ym ( X, b G ) (Thm.3.5). Note that every inclusion G ⊆ SU ( d ) is cov ariant (in essenc e , this is prov ed in [7, Lemma 6.7]). In § 4 we deﬁne some coho mo logical inv ariants for principa l bundles. Given a n exact sequence o f top ological groups G → N G p → QG and a space X , we c onsider the induced ma p of cohomolo gy sets p ∗ : H 1 ( X, N G ) → H 1 ( X, QG ) and construct a class δ ( q ) ∈ H 2 ( X, G ′ ) v anishing when q is in the image of p ∗ . Moreov er , a nona belian G -gerb e ˘ G is asso cia ted with q , collaps ing to a gro up bundle G when q is in the imag e o f p ∗ . Finally , for each G ⊆ SU ( d ) we deﬁne a Chern class c ( q ) ∈ H 2 ( X, Z ) (Lemma 4 .3). In § 5 we give some pro per ties o f ga uge C* -dynamical systems and apply to them the construc- tion of the previous section. In this wa y w e construct the class (1.4), that w e a pply to pointed C* -dynamical systems (Lemma 5.1, Thm.5.4). The r elation with the cla s sical Dixmier-Douady inv a riant is discussed in P rop.5.5. In § 6 we pr ove a concr ete duality for gro up bundles with ﬁbre G ⊆ U ( d ) . Let E → X be a rank 3 d v ecto r bundle, b E denote the c ategory with ob jects the tensor p ow ers E r , r ∈ N , a nd ar rows the spaces ( E r , E s ) of bundle mor phis ms ; then b E is a sy mmetric tensor C* -category with ( ι, ι ) ≃ C ( X ) . W e consider a g roup bundle G → X with a gaug e action G × X E → E and deﬁne a symmetric tensor C* -subca tegory b G of b E , with arrows G -equiv a riant morphisms ( E r , E s ) G , r, s ∈ N . W e establish a o ne-to-one co r resp ondence b et ween tenso r C* -sub categor ies of b E and ga ug e actions (Prop.6.2). T enso r C* -sub categor ie s of b E with ﬁbr e b G are in one- to -one cor resp ondence with reductions to N G of the structur e group of E (Thm.6.4): this yields a link betw ee n the ca tegorical structure of b E and the geometry of E . In § 7 we discuss the breaking of abstrac t dualit y for categories T with ﬁbre b G . Isomorphism classes [ T ] ∈ s ym ( X, b G ) such that there is an embedding η : T ֒ → v ect ( X ) ar e in one-to-one corres p ondence with elements o f the set p ∗ ( H 1 ( X, N G )) ⊆ H 1 ( X, QG ) (Thm.7.2). F or each η there is a vector bundle E η → X and a G -bundle G η → X acting on E η such that T is isomo r phic to b G η . Applying the results of § 4, we assig n a class δ ( T ) ∈ H 2 ( X, G ′ ) : if there is an embedding T → v ect ( X ) then δ ( T ) v anis hes, and when such a n embedding do es no t exist the r ole of the dual G - bundle is played b y a G -gerb e (Thm.7.6). Finally , we discuss the cases G = SU ( d ) (Ex.7.1, Ex.7.2), G = T (Ex.7.4) a nd G = R d (Ex.7.3, R d denotes the gr oup o f ro ots o f unity). 2 Preliminaries. 2.1 Keyw ords and Notation. Let X be a lo cally compact Hausdorﬀ space. If { X i } is a cov er of X , then we deﬁne X ij := X i ∩ X j , X ij k := X i ∩ X j ∩ X k . Moreov er , we denote the C* -algebra of contin uous functions on X v anishing at inﬁnity by C 0 ( X ) ; if X is compact, then we denote the C* -algebr a o f contin uous functions o n X by C ( X ) . If U ⊂ X is op en, then we denote the ideal in C 0 ( X ) (or C ( X ) ) of functions v anishing in X − U by C 0 ( U ) . If W ⊂ X is closed, then we deﬁne C W ( X ) := C 0 ( X − W ) ; in par ticular, for every x ∈ X we set C x ( X ) := C 0 ( X − { x } ) . Since in the present pap er we s ha ll deal with ˇ Cech cohomolog y , we assume that every spa ce has go o d cov er s (i.e. each X ij , X ij k , . . . , is e mpt y or contractible). Let A b e a C* -algebra . W e denote the set of automorphisms (resp. endomorphisms) of A , endow ed with p oint wise co n vergence top ology , b y aut A (resp. end A ). A pair ( A , ρ ) , with ρ ∈ end A , is called C* -dynamical system. If ( A , ρ ) , ( A ′ , ρ ′ ) are C* -dynamical systems, then a C* - algebra morphism α : A → A ′ such that α ◦ ρ = ρ ′ ◦ α is denoted by α : ( A , ρ ) → ( A ′ , ρ ′ ) . In particular, if a ∈ A , a ′ ∈ A ′ and α ( a ) = a ′ , then we write α : ( A , ρ, a ) → ( A ′ , ρ ′ , a ′ ) and re fer to α as a morphism of p ointe d C*-dynamic al systems . W e denote the group o f auto morphisms of the po in ted C* -dynamical system ( A , ρ, a ) by aut ( A , ρ, a ) . Let X b e a lo cally compact Haus do rﬀ spa ce. A C 0 ( X ) -algebr a is a C* -algebr a A endowed with a nondegenerate mo rphism from C 0 ( X ) in to the centre of the multiplier alg e bra M ( A ) . It is customary to assume that such a morphism is injective, th us C 0 ( X ) will b e rega rded as a subalgebra of M ( A ) . F or every x ∈ X , we deﬁne the ﬁ br e epimorphi sm as the q uo tien t π x : A → A x := A / ( C x ( X ) A ) and ca ll A x the ﬁbre of A over x . The group of C 0 ( X ) -automor phisms of A is denoted by aut X A . The r estriction of A o n an op en U ⊂ X is given b y the closed idea l obta ined m ultiplying elements of A by elements of C 0 ( U ) , a nd is denoted b y A U := C 0 ( U ) A . W e deno te the (spatial) C 0 ( X ) -tensor pro duct by ⊗ X (see [16, § 1.6 ], where the no tation “ C ( X ) ” is us ed to mean C 0 ( X ) ). Exa mples of C 0 ( X ) -algebr as are co ntin uo us bundles of C* -algebra s in the sense of [17, 5]; we refer to the last r eference for the notion of lo c al ly trivial contin uous bundle. Let A • be a C* -algebra; to b e concis e, we will call A • -bund le a lo ca lly trivial contin uous bundle of C* -algebras 4 with ﬁbre A • ; to a void confusion with bundles in the topolo gical setting, we emphas ize the fac t that an A • -bundle is indeed a C* -algebra . F or standa rd notions ab out ve ct or bund les , we refer to the classics [1, 15, 21]. In the pre s en t work, w e will assume that every vector bundle is endowed with a Hermitian structure. W e shall also consider Banach bund les (see [10],[5, Ch.10]). F or basic prop erties of ﬁbre bundles and principal bundles, w e refer to [12, Ch.4 ,6], [11, I.3]. If p : Y → X is a contin uous map (i.e., a bund le ), then we sa y that p has lo c al se ctions if for every x ∈ X there is a neigh b ourho o d U ∋ x and a con tinuous map s : U → Y such that p ◦ s = id U . If p ′ : Y ′ → X is a contin uous map, then the ﬁbr e d pr o duct is deﬁned as the spa ce Y × X Y ′ := { ( y , y ′ ) ∈ Y × Y ′ : p ( y ) = p ′ ( y ′ ) } . An ex pos ito ry introduction to nonab elian coho mology and ger bes is [2], where a go o d list of references is provided. F or basic pro perties of C* -categorie s and tensor C* -categ ories, we r efer to [7]. In pa rticular, we make use of the ter ms C*-funct or, C*-epifunctor, C*-monofunctor, C*-isofunctor, C*-autofunctor to denote functors pr eserving the C* -structure. F or every r ∈ N we deno te the p ermutation group of o rder r b y P r and the inﬁnite pe r m utatio n group by P ∞ , which is endow ed with natura l inclusions P s ⊂ P ∞ , s ∈ N . F or every r, s ∈ N , we denote the p ermutation exchanging the ﬁr st r ob jects with the remaining s ob jects b y ( r, s ) ∈ P r + s . 2.2 Bundles of C* -categories. A C*-c ate gory C is a categor y having Banach spaces as s ets of ar r ows and endowed with a n inv o- lution ∗ : ( ρ, σ ) → ( σ, ρ ) , ρ, σ ∈ ob j C , suc h that the C* -ident ity k t ∗ ◦ t k = k t k 2 , t ∈ ( ρ, σ ) , is fulﬁlled. In this w ay , e ach ( ρ, ρ ) , ρ ∈ ob j C , is a C* -algebra, whilst ( ρ, σ ) a Hilbe r t ( σ , σ ) - ( ρ, ρ ) - bimo dule (se e [25, 14]). In the pr esent work we will c o nsider C* -catego r ies not necessa rily endow ed with identity arr ows 1 ρ ∈ ( ρ, ρ ) (see [1 9, § 2.1]). In this setting, ( ρ, ρ ) is not necessarily unital a nd we denote the m ultiplier algebra by M ( ρ, ρ ) . Let C b e a C* -category and X a lo cally co mpact Hausdor ﬀ space. C is said to b e a C 0 ( X ) - c ate gory whenever there is a family { i ρ , ρ ∈ o b j C } o f non-degener a te morphisms i ρ : C 0 ( X ) → M ( ρ, ρ ) , called the C 0 ( X ) - struct ur e , such that t ◦ i ρ ( f ) = i σ ( f ) ◦ t , ρ, σ ∈ ob j C , t ∈ ( ρ, σ ) , f ∈ C 0 ( X ) . The previous equality implies that eac h ( ρ, ρ ) , ρ ∈ ob j C , is a C 0 ( X ) -algebr a. W e assume that each i ρ is injectiv e and write f t := i σ ( f ) ◦ t , f ∈ C 0 ( X ) , t ∈ ( ρ, σ ) . F unctors preserv ing the C 0 ( X ) -structure a re called C 0 ( X ) - functors . If U ⊆ X is op en, then we deﬁne the r estriction on U as the C* -catego ry C U having the same ob jects as C and spaces of ar rows ( ρ, σ ) U := C 0 ( U )( ρ, σ ) ; note that C U may la c k s identit y arrows also when C has identit y arrows. If W is closed, then we denote the C* -category having the sa me ob jects as C and space s of arr ows the quotients ( ρ, σ ) W := ( ρ, σ ) / ( C 0 ( X − W )( ρ, σ )) b y C W ; the corres p onding C* -epifunctor π W : C → C W is called the r estriction functor . In par ticular, we deﬁne the ﬁbr e of C over x as C x := C { x } and call π x : C → C x the ﬁbr e fun ctor . F or every ρ, σ ∈ ob j C , t ∈ ( ρ, σ ) , we deﬁne the norm function n t ( x ) := k π x ( t ) k , x ∈ X . It can b e prov ed that n t is upper semicontin uous for each arrow t ; when each n t is contin uous, we say that C is a c ontinuous bund le ov er X . In this case, each ( ρ, σ ) is a cont inuous ﬁeld o f Ba nach spaces o ver X and each ( ρ, ρ ) is a contin uous bundle of C* -algebras. Let C • be a C* -catego ry . The c onst ant bu n d le X C • is the C 0 ( X ) -catego ry ha ving the same ob jects as C • and arrows the spaces ( ρ, σ ) X of co ntin uo us maps v anishing a t inﬁnity from X to ( ρ, σ ) , ρ, σ ∈ ob j C • . A C 0 ( X ) -catego ry C is said to be lo c al ly t rivial whenever for each x ∈ X 5 there is an op en neigh b ourho o d U ∋ x with a C 0 ( U ) -isofunctor α U : C U → U C • , such that the induced ma p α U : ob j C → ob j C • do es not dep end on the choice of U . The functors α U are called lo c al charts . When X is compact, the s a me co nstructions apply with the obvious mo diﬁcations. 3 T e n sor C* -categories and C* -dynamical systems. The present s e ction has t wo purp oses. Firs t, in o r der to mak e the pre sen t pa per eno ugh self- contained, we collect some results from [6, 2 5] in a sligh tly diﬀerent form and recall the notions of sp e cial c ate gory and emb e dding functor . Secondly , we describe the space of certain em b edding functors in terms o f a principa l bundle (Lemma 3.4) a nd pr ovide a classiﬁcation result for bundles with ﬁbre b G , G ⊂ U ( d ) (Thm.3.5); these r esults shall b e applied in § 6 . A t en sor C* -catego ry is a C* -catego ry T with identit y arrows endow ed with a C* -bifunctor ⊗ : T × T → T , c alled the tensor pr o duct . F o r bre vit y , w e denote the tensor pro duct o f ob jects ρ, σ ∈ ob j T b y ρσ , whilst the tensor pro duct o f arrows t ∈ ( ρ, σ ) , t ′ ∈ ( ρ ′ , σ ′ ) , is denoted by t ⊗ t ′ ∈ ( ρρ ′ , σ σ ′ ) . W e assume the existence of an identity obje ct ι ∈ ob j T such that ιρ = ρι = ρ , ρ ∈ ob j T : it ca n b e easily veriﬁed that ( ι, ι ) is an Abelian C* -algebra a nd every spa ce of a r rows ( ρ, σ ) is a Ba nach ( ι, ι ) - bimodule w.r.t. the o p era tion o f tensoring with arr ows in ( ι, ι ) . Let X ι denote the sp e ctrum o f ι ; then T is a C ( X ι ) -category in a natura l wa y . In particular , it can be proved that T is a co n tinuous bundle if certain additional assumptions ar e satisﬁed ([28, 25]). A tenso r C* -catego ry whos e ob jects are r -fold tensor p ow ers of an ob ject ρ , r ∈ N , is denoted by ( b ρ, ⊗ , ι ) ; for r = 0 , we use the conven tion ρ 0 := ι . In the sequel o f the present work, we shall need to keep in evidence an ar row a ∈ ( ρ r , ρ s ) for some r , s ∈ N , so that we intro duce the notation ( b ρ, ⊗ , ι, a ) . More over, we denote ten sor C* -functors α : b ρ → b ρ ′ (the ter m tensor means that ⊗ ′ ◦ ( α × α ) = α ◦ ⊗ , α ( ι ) = ι ′ ) such tha t α ( a ) = a ′ by α : ( b ρ, ⊗ , ι, a ) → ( b ρ ′ , ⊗ ′ , ι ′ , a ′ ) . If ( A , ρ, a ) is a p ointed C* -dynamical s y stem, then the category b ρ with ob jects the pow er s ρ r , r ∈ N , and arrows the intert winer spa ces ( ρ r , ρ s ) , r , s ∈ N , endow ed with the tens or pro duct ρ r ρ s := ρ r + s , t ⊗ t ′ := tρ r ( t ′ ) , t ∈ ( ρ r , ρ s ) , ( ρ r ′ , ρ s ′ ) , is an ex ample of such s ingly gener ated tensor C* -categor ie s with a distinguished arrow. W e denote the C* -algebra generated by the intert winer spa ces ( ρ r , ρ s ) , r , s ∈ N , by O ρ . Actually , every tenso r C* -category ( b ρ, ⊗ , ι ) comes a s so ciated with a C* -dynamical system, in the following way (see [7, § 4] for deta ils ). As a ﬁrst s tep, we consider the maps j r,s ( t ) := t ⊗ 1 ρ ∈ ( ρ r +1 , ρ s +1 ) , t ∈ ( ρ r , ρ s ) and deﬁne the Ba nach s paces O k ρ := lim → r (( ρ r , ρ r + k ) , j r,r + k ) , k ∈ Z . As a seco nd step, we note that co mp osition of a rrows and inv olution induce a well-deﬁned *-alg ebra structure on the direct sum 0 O ρ := ⊕ k O k ρ . It can b e prov ed that there is a unique C* -norm o n 0 O ρ such that the cir cle action b z ( t ) := z k t , z ∈ T , t ∈ O k ρ , extends to an automorphic actio n. In this wa y , the so-obtained C* -co mpletion O ρ comes equipp ed with a contin uous a ction T → aut O ρ with sp ectral subspaces O k ρ , k ∈ N , and also with a c anonic al endomorph ism ρ ∗ ∈ end O ρ , ρ ∗ ( t ) := 1 ρ ⊗ t , t ∈ ( ρ r , ρ s ) , such that ρ ∗ ◦ b z = b z ◦ ρ ∗ , z ∈ T . The pair ( O ρ , ρ ∗ ) is called the DR-dynamic al s yst em asso ciate d with ρ . Sinc e the maps j r,s ar e inje ctive in al l the c ases of inter est in the pr esent work, in the se quel we wil l identify t ∈ ( ρ r , ρ s ) with the c orr esp onding element of O ρ . 6 By cons tr uction we have ( ρ r , ρ s ) ⊆ ( ρ r ∗ , ρ s ∗ ) , r, s ∈ N . W e say that ρ is amenable if ( ρ r , ρ s ) = ( ρ r ∗ , ρ s ∗ ) , r, s ∈ N , and in that case b ρ is s aid to b e amenably gener ate d . W e summar ize the ab ov e consideratio ns in the following theorem, which a lso includes a re fo rm ula tion of [25, P r op.19]: Theorem 3.1. The map ( b ρ, ⊗ , ι, a ) → ( O ρ , ρ ∗ , a ) deﬁn es a one-to-one c orr esp ondenc e b etwe en the class of amenably gener ate d tensor C*-c ate gories with a distinguishe d arr ow and the class of p ointe d C*-dynamic al systems ( A , σ , a ) su ch that A is gener ate d by the intertwiner sp ac es of σ . T ensor C*-functors α : ( b ρ, ⊗ , ι, a ) → ( b ρ ′ , ⊗ ′ , ι ′ , a ′ ) ar e in one-t o-one c orr esp ondenc e with morphisms α : ( O ρ , ρ ∗ , a ) → ( O ρ ′ , ρ ′ ∗ , a ′ ) of p ointe d C*-dynamic al systems. The c ate gory b ρ is a c ontinu ous bund le over the sp e ctru m X ι of ( ι, ι ) if and only if O ρ is a c ontinuous bund le over X ι . If b ρ is lo c al ly trivial as a bun d le of C*-c ate gories, then O ρ is lo c al ly trivial as a C*-algebr a bu nd le. A tensor C* -categ ory ( T , ⊗ , ι ) is sa id to b e symmetric if there is a family o f unitary op erator s ε ( ρ, σ ) ∈ ( ρσ, σ ρ ) , ρ, σ ∈ ob j T , implementing the ﬂips ( t ⊗ t ′ ) ◦ ε ( ρ ′ , ρ ) = ε ( σ ′ , σ ) ◦ ( t ′ ⊗ t ) . In particular , if ( b ρ, ⊗ , ι ) is symmetric, then we deﬁne the symmetry op erator ε := ε ( ρ, ρ ) ∈ ( ρ 2 , ρ 2 ) . It is well-known that ε induces a unitary representation o f P ∞ , by considering pro ducts of the type ε ◦ (1 ρ × ε ) ◦ (1 ρ r ⊗ ε ) ◦ . . . , r ∈ N (for example, s e e [6, p.100]). W e denote the unitar ies ar ising from such a repre s en tatio n by ε ( p ) ∈ ( ρ r , ρ r ) , r ∈ N , p ∈ P r ⊆ P ∞ ; in par ticular, we denote the unitary asso cia ted with ( r, s ) ∈ P r + s by ε ρ ( r , s ) ∈ ( ρ r + s , ρ r + s ) . If there is α : ( b ρ, ⊗ , ι, ε ) → ( b ρ ′ , ⊗ ′ , ι ′ , ε ′ ) , then the ab ov e conside r ations imply that α ( ε ρ ( r , s )) = ε ′ ρ ′ ( r , s ) , r , s ∈ N . W e denote the pointed C* -dynamical sy stem asso ciated with ( b ρ, ⊗ , ι, ε ) by ( O ρ , ρ ∗ , ε ) . According to the co nsiderations of the previous section, we ﬁnd that b ρ is a C ( X ι ) -category . Now let ( b ρ • , ⊗ • , ι • , ε • ) b e a s y mmetric tensor C* -categor y such that ( ι • , ι • ) ≃ C ; w e denote the set of isomor phism classes of lo cally trivial symmetric tenso r C* -catego ries ( b ρ, ⊗ , ι, ε ) having ﬁbre b ρ • and such that ( ι, ι ) ≃ C ( X ) by sym ( X , b ρ • ) . With the term isomorphi sm , here we mean a tens o r C* -isofunctor of the type α : ( b ρ, ⊗ , ι, ε ) → ( b ρ ′ , ⊗ ′ , ι ′ , ε ′ ) . A lo cally trivial symmetric tensor C* -categor y ( b ρ, ⊗ , ι, ε ) with ﬁbre b ρ • is called b ρ • -bund le ; the class of b ρ in s ym ( X, b ρ • ) is deno ted by [ b ρ, ⊗ , ι, ε ] or, mor e co ncisely , by [ b ρ ] . R emark 3.1 . The condition α ( ε ) = ε ′ required in the pr evious notion of isomor phism comes from group dua lit y . Let G 1 , G 2 be co mpa ct groups and R G 1 , RG 2 the as so ciated symmetric tensor C* -categories of ﬁnite dimensional, con tinuous, unitar y representations; if α : R G 1 → R G 2 is an isomorphism of tensor categor ies, then a suﬃcient condition to get an iso morphism α ∗ : G 2 → G 1 is that α preserves the symmetry (see [13]). The catego ry hilb of Hilb ert spaces, endow ed with the usual tensor pr oduct, is clearly a sym- metric tensor C* -categor y . Of par ticular interest for the pr e sen t work is the following class of sub c ategories of hilb . Le t H be the standard Hilbert space of dimensio n d ∈ N ; we denote the r - fold tensor p ow er o f H by H r (for r = 0 , w e deﬁne ι := H 0 := C ) and the space of linear op erators from H r to H s by ( H r , H s ) , r , s ∈ N ; mor eov er, we cons ider the ﬂip θ ∈ ( H 2 , H 2 ) . 7 If G ⊆ U ( d ) is a compa ct gr oup, then for every g ∈ G we ﬁnd that the r -fo ld tensor p ow er g r is a unitary o n H r , so that w e co nsider the spaces of G -inv a riant o pera tors ( H r , H s ) G := { t ∈ ( H r , H s ) : t = b g ( t ) := g s ◦ t ◦ g ∗ r , g ∈ G } . (3.1) In particular , we hav e that θ ∈ ( H 2 , H 2 ) G . By deﬁning the c ategory b G with ob jects H r , r ∈ N , and arrows ( H r , H s ) G , we obta in a symmetric tensor C* -category ( b G, ⊗ , ι, θ ) . The pointed C* - dynamical system a sso ciated with ( b G, ⊗ , ι, θ ) in the sense of Thm.3.1 is ( O G , σ G , θ ) , wher e O G , σ G are deﬁned in § 1 . As mentioned in § 1, the ca teg ory b G is amenably gener ated, so that we have equalities ( H r , H s ) G = ( σ r G , σ s G ) , r, s ∈ N . (3.2) If G reduce s to the trivial gr o up, then we obtain the categor y ( b H , ⊗ , ι, θ ) of tensor p ow ers of H and Thm.3.1 yields the Cun tz C* -dynamical system ( O d , σ d , θ ) . If G = U ( d ) , then ( H r , H s ) U ( d ) is non trivia l only fo r r = s ; in s uch a case, ( H r , H r ) U ( d ) is generated as a vector space b y the unitaries θ ( p ) , p ∈ P r . Le t N G denote the normalize r o f G in U ( d ) and Q G := N G/G the quotient gr oup; then, the map (1.1) induces an injective contin uo us a ction QG → aut ( O G , σ G , θ ) , y 7→ b y . (3.3) An o ther symmetr ic tensor C* -category that shall play an impo r tant r o le in the presen t pa per is the ca tegory v ect ( X ) with ob jects vector bundles ov er a compa ct Hausdorﬀ s pace X and arrows vector bundle morphisms. Deﬁnition 3.2. L et ( T , ⊗ , ι, ε ) b e a symmetric tensor C*-c ate gory. A n embe dding functor is a C*-monofunctor E : T → vect ( X ι ) pr eserving t ensor pr o duct and symmetry. W e now descr ibe in geometrica l terms a set of em b edding functors of b G , G ⊆ U ( d ) . T o this end, let us denote the set o f mono morphisms η : ( O G , σ G , θ ) → ( O d , σ d , θ ) by emb O G , and endow it with the p oint wise norm top ology; by Thm.3.1, we can iden tify em b O G with the set of embeddings β : ( b G, ⊗ , ι, θ ) → ( b H , ⊗ , ι, θ ) . In particular, w e denote the group of autofunctors of the type β : ( b G, ⊗ , ι, θ ) → ( b G, ⊗ , ι, θ ) by aut b G . Deﬁnition 3. 3 . The faithful r epr esentation G ⊆ U ( d ) is said to b e cov arian t whenever for e ach η ∈ em b O G ther e is u ∈ U ( d ) such that η = b u | O G . By Thm.3.1 the pro per t y of G ⊆ U ( d ) being cov aria nt is equiv alent to require that the inclus io n functor ( b G, ⊗ , ι, θ ) ⊆ ( b H , ⊗ , ι, θ ) is unique up to tensor unitary natural transfo rmation. B y [7, Lemma 6.7,Thm.4.17 ] (see also the following Thm.3.7) every inclusion G ⊆ SU ( d ) is cov aria n t, th us we conclude that every compact Lie group ha s a faithful cov a r iant repres en tato n (in fact, it is w ell-k nown that every co mpact Lie group G ha s a faithful representation u : G → U ( d ) , so it suﬃces to consider u ⊕ det u ). Anyw ay there a re interesting examples of cov ariant r epresentations whose image is not contained in the sp ecial unitary gro up. Example 3.1. Let G ⊂ U ( d ) deno te the image o f T under the action on H ≃ C d deﬁned b y scalar multiplication. Then b G has spaces of arrows ( H r , H s ) G = δ r s ( H r , H s ) , r, s ∈ N , where δ r s denotes the K roneck er symbol. W e hav e N G = U ( d ) , QG = PU ( d ) . If η ∈ emb O G then η restricts to a C* -isomorphism η : ( H , H ) G = ( H , H ) → ( H , H ) , whic h is the inner automo rphism induced b y a unitary u ∈ U ( d ) . Since ( H r , H r ) ≃ ⊗ r ( H, H ) a nd η ( ⊗ r i t i ) = ⊗ r i η ( t i ) = ⊗ r i b u ( t i ) , t i ∈ ( H, H ) , i = 1 , . . . , r , we c onclude that η = b u | O G and G ⊆ U ( d ) is cov ar iant. 8 Lemma 3.4. L et G ⊆ U ( d ) b e c ovariant. Then em b O G is home omorphic t o the c oset sp ac e U ( d ) \ G . F or e ach lo c al ly c omp act Hausdorﬀ sp ac e Y and c ontinuous map β : Y → emb O G , y 7→ β y , (3.4) ther e is a ﬁnite op en c over { Y l } of Y and c ontinuous maps u l : Y l → U ( d ) such that b u l,y ( t ) = β y ( t ) , y ∈ Y l , t ∈ O G , (3.5) wher e b u l,y ∈ aut O d is deﬁne d by (1.1). Pr o of. W e co nsider the ﬁbration q : U ( d ) → U ( d ) \ G and deﬁne χ : U ( d ) → em b O G , u 7→ b u | O G . The map χ is clearly contin uous and, since G ⊆ U ( d ) is cov ariant, it is also s urjective. Now, (1.1) yields an isomorphism fr om G to the sta biliz e r of O G in aut O d ([6, Cor.3 .3]), thus we ﬁnd that χ ( u 1 ) = χ ( u 2 ) if and only if u ∗ 1 u 2 ∈ G , i.e. q ( u 1 ) = q ( u 2 ) . This pr ov es that emb O G is homeomorphic to U ( d ) \ G . Since U ( d ) is a compact Lie gr o up, the map q deﬁnes a pr incipal G - bundle ov er U ( d ) \ G , thus there is a ﬁnite o pen cover { Ω l } o f U ( d ) \ G a nd lo ca l s e c tions s l : Ω l → U ( d ) , q ◦ s l = i d Ω l . Now, let us iden tify e m b O G with U ( d ) \ G and co nsider the ma p (3.4); deﬁning Y l := β − 1 (Ω l ) w e obtain a ﬁnite op en cover of Y and set u l,y := s l ◦ β y , y ∈ Y l . By deﬁnition of χ , the equa tion (3 .5) is fulﬁlled and the theorem is proved. Let no w p : N G → QG the natural pro jection. The following res ult is a version of [25, Thm.36] for groups no t neces s arily co n tained in SU ( d ) : Theorem 3 .5. If G ⊆ U ( d ) is c ovariant then ther e is an isomorphism QG ≃ aut b G , and for e ach c omp act Hausdorﬀ s p ac e X ther e is a bije ctive map Q : s ym ( X, b G ) → H 1 ( X, QG ) . Pr o of. Using Thm.3 .1 w e identify aut b G with aut ( O G , σ G , θ ) . By [25, Lemma 32 ], to prov e the theorem it suﬃces to verify that (3.3) is a n isomor phism. Now, the same ar gument of the previo us Lemma shows that if η ∈ aut b G then there is u ∈ U ( d ) suc h that b u ∈ aut O d restricts to η on O G ; since O G is b u -sta ble , for each g ∈ G we ﬁnd tha t b u ◦ b g ◦ b u − 1 is the identit y on O G , th us by [6, Cor.3.3 ] there is g ′ ∈ G such that g ′ = u g u ∗ . W e conclude that u ∈ N G and since b u ◦ b g | O G = b u | O G = η for all g ∈ G we ﬁnd η = b y , where y = p ( u ) and b y is the ima ge of y ∈ QG under (3.3). R emark 3 .2 . Let aut ( O d ; O G ) denote the g roup of automor phisms of ( O d , σ , θ ) that restrict to ele- men ts of aut O G . The ar gumen t o f the previous theo r em shows tha t (1.1) induces the iso mo rphism N G → aut ( O d ; O G ) , in such a wa y that for each u ∈ N G we hav e b u | O G = b y , y := p ( u ) ∈ QG . This yields a s lig h t generaliza tion o f [25, Thm.34]. Given a b G -bundle ( b ρ, ⊗ , ι, ε ) , with G ⊆ U ( d ) cov a riant, we denote the asso ciated class in H 1 ( X, QG ) by Q [ b ρ ] . Now, let us consider a s ymmetric tensor C* -category ( b ρ, ⊗ , ι, ε ) . F or ev er y n ∈ N , w e deﬁne the antisymmetric pr oje ction P ρ,ε,n := 1 n ! X p ∈ P ( n ) sign( p ) ε ρ ( p ) . (3.6) 9 The ob ject ρ is said to b e sp e cial if ther e is d ∈ N and a partial isometr y S ∈ ( ι, ρ d ) with supp ort P ρ,ε,d , such that ( S ∗ ⊗ 1 ρ ) ◦ (1 ρ ⊗ S ) = ( − 1 ) d − 1 d − 1 1 ρ ⇔ S ∗ ρ ∗ ( S ) = ( − 1 ) d − 1 d − 1 1 . (3.7) In such a ca se, d is ca lled the dimension of ρ . When ρ is an endomorphism and is sp ecial in the ab ov e sens e, we say that ρ satisﬁes the sp e cial c onjugate pr op erty (s e e [9 , § 4]). Spec ia l ob jects play a pivotal ro le in the Doplicher-Rob erts theory . F r om the vie wp oint of gro up duality they are an abstra ct characteriz ation of the notion of repres en tation with deter minan t 1 (see [7 , § 3 ]). F rom the C* -alg ebraic p oint of view, they are a n essential to ol for the cr ossed pro duct deﬁned in [9, § 4 ]. Let G ⊆ SU ( d ) . Then the o b ject H o f b G is sp ecial and has dimensio n d . In fact, we consider the isometry S g e ne r ating the to tally antisymmetric tenso r p ow er ∧ d H , and no te that u d ◦ S = det u · S = S , u ∈ SU ( d ) , s o that S ∈ ( ι, H d ) SU ( d ) ⊆ ( ι, H d ) G and (3.7) follows from [6, Lemma 2 .2]. In pa rticular when G = SU ( d ) the spaces ( H r , H s ) SU ( d ) , r, s ∈ N , are generated by the op erator s θ ( p ) , p ∈ P ∞ , and S ∈ ( ι, H d ) SU ( d ) , by closing w.r.t. comp o sition and tensor pro duct. Deﬁnition 3. 6. A sp ecial category is a lo c al ly trivial, symmetric tensor C*-c ate gory ( b ρ, ⊗ , ι, ε ) with ﬁbr e ( b ρ • , ⊗ • , ι • , ε • ) , su ch that ρ • is a sp e cial obje ct . The dimension of the ob ject ρ generating the sp ecial ca tegory b ρ is by deﬁnition the dimension of the sp ecial ob ject ρ • and is denoted by d . The main motiv atio n of the present work is the search o f embedding functors for sp ecial categ ories. The ﬁrs t step in this direction is given by the following clas s iﬁcation res ult, proved in [25, Thm.36]: Theorem 3. 7 . L et ( b ρ, ⊗ , ι, ε ) b e a sp e cial c ate gory with ﬁbr e ( b ρ • , ⊗ • , ι • , ε • ) . Then: (1) ρ is amenable; (2) L et d ∈ N denote the dimension of ρ ; then ther e is a c omp act Lie gr oup G ⊆ SU ( d ) such that ( b ρ • , ⊗ • , ι • , ε • ) ≃ ( b G, ⊗ , ι, θ ) ; (3) Ther e is a bije ction sym ( X ι , b ρ • ) ≃ H 1 ( X ι , QG ) ; (4) O ρ is an O G -bund le. In g eneral, the o b ject g e ne r ating a sp ecial category is not s pecia l. The obstr uction to ρ b eing sp ecial is enco ded by the Chern class intro duced in [25, § 3.0.3 ], c ( ρ ) ∈ H 2 ( X ι , Z ) , (3.8 ) constructed by observing that the ( ι, ι ) - mo dule R ρ := { ψ ∈ ( ι, ρ d ) : P ρ,ε,d ψ = ψ } is the set of sections of a line bundle L ρ → X ι . The inv ar ia n t c ( ρ ) is deﬁned as the ﬁrst Chern class o f L ρ . 4 Cohomology classes and principal bundles. In the pr esen t sectio n we give an exact sequence and a coho mological inv ar iant f o r a cla s s of principal bundles. T his ele mentary construction has imp ortant cons e quences in the setting o f abstr act duality for tensor C* -categ o ries a nd can b e r egarded as a gener alization of the Dixmier-Douady inv ariant. Let G a top o logical gr oup with unit 1 and X a lo cally compact, pa racompact Hausdorﬀ spa ce endow ed with a (go o d) op en cover { X i } . A G -c o cycle is given by a family g := { g ij } of contin uous maps g ij : X ij → G satisfying o n X ij k the c o cycle r elations g ij g j k = g ik (whic h imply g ij g j i = g ii = 1 ). In the sequel we will denote the ev a luation o f g ij on x ∈ X ij by g ij,x . W e say that g is c ohomolo gous to g ′ := { g ′ ij } whenever there ar e maps v i : X i → G such that 10 g ij v j = v i g ij on X i . This deﬁnes an equiv alence relation over the set of G -co c ycles, a nd passing to the inv erse limit over op en go o d cov ers pr ovides the ˇ Cech c ohomolo gy set H 1 ( X, G ) (see [15, I.3 .5]), which is a p ointed se t with distinguished element the class of the trivial c o cycle 1 , 1 ij,x ≡ 1 . T o b e concise, sometimes in the s e quel co cycles will b e denoted simply by g or { g ij } , and their cla sses in H 1 ( X, G ) by [ g ] o r [ g ij ] . It is well-kno wn that H 1 ( X, G ) classiﬁes the principal G -bundles over X . When G is Abelian, H 1 ( X, G ) co inc ide s with the ﬁrs t cohomolo gy gr oup with co eﬃcients in the sheaf S X ( G ) of germs of c o n tinuous maps from X in to G ([11, I.3.1]). W e now pa ss to give a deﬁnition of nonab elian ˇ Ce ch 2 -c ohomolo gy . T he ba sic o b ject providing the co eﬃcients of the theory is no w given b y a cr osse d m o dule (a lso called 2 –gr oup , see [2, § 3]), which is de ﬁned b y a mo rphism i : G → N of to polo gical gro ups and a n action α : N → aut G , such that i is e quiv ar ian t for α and the adjoint actions G → aut G , G ∋ g 7→ b g , N → aut N , N ∋ u 7→ b u : b u ◦ i = i ◦ α ( u ) , b g = α ◦ i ( g ) . The cross ed mo dule ( G, N , i, α ) is denoted for short by G → N . T o b e concise we write g := i ( g ) ∈ N , g ∈ G , and α ( u ) := b u , u ∈ N . The equiv aria nce relatio ns ensur e that no co nfusio n will a rise from this notatio n. Example 4.1. Let N b e a top o logical g roup a nd G a no rmal subgro up of N : then considering the inclusion i : G → N and the adjoint a ction α : N → aut G , u 7→ b u , y ields a c rossed mo dule G → N . A c o cycle p air b := ( u , g ) with co eﬃcients in the cro ssed mo dule G → N is g iv en b y families of maps u ij : X ij → N , g ij k : X ij k → G , satisfying the co cycle r elations  u ij u j k = g ij k u ik g ij k g ikl = b u ij ( g j kl ) g ij l , where b u ij : X ij → aut G is deﬁned by means of α . Co cycle pair s b := ( u , g ) , b ′ := ( u ′ , g ′ ) a re said to b e c ohomolo gous whenever ther e is a pair ( v , h ) of families of maps v i : X i → N , h ij : X ij → G , such that  v i u ′ ij = h ij u ij v j h ik g ij k = b v i ( g ′ ij k ) h ij b u ij ( h j k ) . It ca n be prov ed that cohomology of co cycle pa irs deﬁnes a n equiv alence relatio n ([2 , § 4]). The set of coho mo logy cla sses of co cycle pairs is by deﬁnition the c ohomolo gy set r elative to the cover { X i } with c o eﬃcients in t he cr osse d mo dule G → N ; passing to the limit w.r.t. cov er s yields the ˇ Ce ch c ohomolo gy set ˘ H 2 ( X, G → N ) with distinguis hed element the class of the trivial co cycle pair 1 := (1 , 1) , 1 ij,x := 1 ∈ N , 1 ij k ,x := 1 ∈ G . The s ym b ol ˘ H is used to emphasize that we dea l with nonab elian cohomolo gy sets. Note that our nota tio n is not universally us e d in litera ture: sometimes the sy m b ol ˘ H 1 ( X, G → N ) is used instead of ˘ H 2 ( X, G → N ) (see for example [2]). The cohomolog y class of the co cycle pa ir b = ( u , g ) is denoted by [ b ] ≡ [ u , g ] . R emark 4.1 . (1) Each N -co cycle u := { u ij } deﬁnes the co cycle pa ir d u := ( u , 1 ) ; (2) If G is Abelia n and α is the trivial a ction, then each co cycle pair ( u , g ) deﬁnes the co cycle g = { g ij k } in the second (Ab elian) cohomolog y of G . 11 An impo rtant class of examples is the following: let G be a top ologica l gro up and i : G → aut G , i ( g ) := b g , deno te the adjoint action; then taking α : aut G → aut G as the identit y map yields a cro ssed module G → aut G . Th us we can deﬁne the cohomolo g y set ˘ H 2 ( X, G → aut G ) with elements clas ses of co cycle pairs ( λ, g ) o f the type λ ij : X ij → aut G , g ij k : X ij k → G :  λ ij λ j k = b g ij k λ ik g ij k g ikl = λ ij ( g j kl ) g ij l , where each b g ij k : X ij k → aut G is deﬁned b y a djoin t action. R emark 4.2 . Acco rding to the cons ide r ations in [2, § 2], ˘ H 2 ( X, G → aut G ) clas siﬁes the G -g erb es on X up to isomo rphism. In the present pap er we use the term G -g erb e to mean a principal 2 –bundle ov er X with ﬁbre the crossed mo dule G → aut G . In this w ay , co cycle pairs with coeﬃcients in G → aut G are interpreted as tra nsition maps for G - gerb es, and G -bundles deﬁne G -gerb es s uc h that the as so ciated co cycle pair s a re of the type d λ = ( λ, 1 ) , λ ∈ H 1 ( X, aut G ) (see Rem.4.1). W e deﬁne the maps γ ∗ : H 1 ( X, N ) → H 1 ( X, aut G ) , [ u ij ] 7→ [ b u ij ] , (4.1) ˘ γ ∗ : ˘ H 2 ( X, G → N ) → ˘ H 2 ( X, G → aut G ) , [ { u ij } , g ] 7→ [ { b u ij } , g ] . (4.2) Let now N de no te a top olog ical group and G ⊆ N a normal subgroup. Deﬁning QG := N \ G yields the exact sequence 1 → G i ֒ → N p → QG → 1 . (4.3) Let N G denote the smaller no rmal subgroup of N co n taining the set [ G, N ] := { g ug − 1 u − 1 , g ∈ G, u ∈ N } . W e cons ider the quotient map π N : N → N ′ := N \ N G and deﬁne G ′ := π N ( G ) . Note that by construction G ′ is Abelia n; when G is co n tained in the c e n tre of N we hav e that [ G, N ] is trivial and N G = { 1 } , N = N ′ , G = G ′ . Lemma 4.1. L et G b e a normal sub gr oup of the top olo gic al gr oup N and supp ose that the ﬁbr ation p : N → Q G := N \ G has lo c al se ctions. Then for every lo c al ly c omp act, p ar ac omp act Hausdorﬀ sp ac e X t her e is an isomorphism of p ointe d sets ν : H 1 ( X, QG ) → ˘ H 2 ( X, G → N ) . (4.4) Mor e over, ther e is a c ommut ative diagr am G i / / π G   N π N   G ′ i ′ / / N ′ (4.5) which yields the map π N , ∗ : ˘ H 2 ( X, G → N ) → H 2 ( X, G ′ ) . (4.6) Pr o of. The fact that there is an isomor phism as in (4.4 ) is proved in [2, Lemma 2], anyw ay for the reader’s con venience we give a sketc h of the pro o f. Let q := { y ij } b e a QG -co cycle; since p ha s lo cal sectio ns , up to p erforming a reﬁnement of { X i } there a re maps u ij : X ij → N suc h that 12 y ij = p ◦ u ij (it s uﬃce to deﬁne u ij := s ◦ y ij , where s : U → N , U ⊆ QG , y ij ( X ij ) ⊆ U , is a lo cal section). Since p ◦ ( u ij u j k u − 1 ik ) = y ij y j k y ki = 1 , we conclude that there is g ij k : X ij k → G suc h that u ij u j k = g ij k u ik . It is trivial to chec k that ( { u ij } , { g ij k } ) is a co cycle pair , a nd we deﬁne ν [ q ] := [ { u ij } , { g ij k } ] , [ q ] := [ y ij ] ∈ H 1 ( X, QG ) . (4.7) On the other side, if b := ( { u ij } , { g ij k } ) is a co cyc le pair then deﬁning p ∗ [ b ] := [ p ◦ u ij ] yields an inv erse of ν . W e now pr ove (4.6). Deﬁning π G := π N | G yields the commutativ e diagra m (4.5); if b := ( u , g ) , g := { g ij k } , is a co cycle pair, then we deﬁne π N , ∗ [ b ] := [ π N ◦ g ij k ] and this yields the desired map (note in fact that π N ◦ b u ij ( g j kl ) = π N ◦ g j kl , so that { π N ◦ g ij k } is a 2– G ′ -co cycle). Now, by functoriality o f H 1 ( X, · ) there is a se quence o f maps of p oin ted sets H 1 ( X, G ) i ∗ − → H 1 ( X, N ) p ∗ − → H 1 ( X, QG ) . (4.8) In fact, p ∗ ◦ i ∗ [ g ] = [ 1 ] for each g ∈ H 1 ( X, G ) . In the following result we give an o bstruction to p ∗ being surjective. Lemma 4.2. L et G b e a normal sub gr oup of the top olo gic al gr oup N such that the ﬁbr ation p : N → Q G := N \ G has lo c al se ctions. Then we have t he fol lowing se quenc e of maps of p ointe d sets: H 1 ( X, G ) i ∗ / / H 1 ( X, N ) p ∗ / / γ ∗   H 1 ( X, QG ) δ / / ˘ γ ∗ ◦ ν   H 2 ( X, G ′ ) H 1 ( X, aut G ) d ∗ / / ˘ H 2 ( X, G → aut G ) (4.9) Her e d ∗ is induc e d by the map d λ := ( λ, 1 ) , and the squ ar e is c ommutative. When G is c ontaine d in the c entr e of N , the u pp er horizontal r ow is exact and G ′ = G . Pr o of. The pr o o f of the Le mma is based on the maps introduced in Lemma 4.1. Deﬁne δ := π N , ∗ ◦ ν . If u := { u ij } is an N -co cycle and q := { y ij := p ◦ u ij } then by deﬁnition of ν we ﬁnd ν [ q ] = [ u , 1 ] = d ∗ [ u ] (se e Rem.4.1 ); mo reov er, δ [ q ] = π N , ∗ [ u , 1 ] = [ 1 ] and this prov es that p ∗ ( H 1 ( X, N )) ⊆ ker δ . W e now prove that the sq uare is comm utative. T o this end, note tha t for each N -co cy c le u := { u ij } we ﬁnd d ∗ ◦ γ ∗ [ u ] = [d { b u ij } ] = [ { b u ij } , 1 ] ; on the other side, if q := { y ij } := { p ◦ u ij } then ˘ γ ∗ ◦ ν ◦ p ∗ [ u ] = ˘ γ ∗ ◦ ν [ q ] = ˘ γ ∗ [ u , 1 ] = [ { b u ij } , 1 ] , and we conclude that the squar e is commutativ e. Finally , we prov e that the upp er ho rizontal row is exa ct when G is contained in the centre of N ; to this end, it suﬃces to v er ify that ker δ ⊆ p ∗ ( H 1 ( X, N )) . Now, we hav e G = G ′ and the ma p π N , ∗ takes the form π N , ∗ [ u , g ] := [ g ] . Since ν is bijective we have that δ [ q ] = [ 1 ] if and only if π N , ∗ [ u , g ] = [ 1 ] , where [ u , g ] = ν [ q ] . This means that g = { g ij k } is a trivia l G - 2 -c o cy c le , so that there are maps h ij : X ij → G such that h ij h j k = g ij k h ik ; the pair (1 , { h ij } ) deﬁnes a 2 -co cycle equiv a lence b et ween ( u , g ) a nd ( u ′ , 1) , where u ′ := { u ij h j i } is, by constructio n, an N -co cyc le . By deﬁnition of ν we have p ∗ [ u ′ ] = [ q ] , a nd this pr ov es p ∗ ( H 1 ( X, N )) = ker δ . Th us the upp er hor iz o n tal row is exact as desired. Note that by cla ssical results when N is a compact L ie gro up a nd G is closed, G , QG , G ′ are compact Lie g roups a nd the ﬁbration N → QG has lo cal sections. An in teres ting class of examples is the following. Let U b e the unitary group of an inﬁnite dimensional Hilb ert s pace; then, the c e ntre of U is the torus T and PU := U / T is the pro jective unitary gro up. In this case, δ takes the form  δ : H 1 ( X, PU ) → H 2 ( X, T ) ≃ H 3 ( X, Z ) δ [ q ] := [ g ] , q := { y ij } , g := { g ij k } (4.10) 13 (where { g ij k } is deﬁned by (4.7)) and it is well-known that it is an isomorphism (see [5, § 10.7 .12] and following sectio ns). In the fo llowing Lemma, we deﬁne a Chern class for a Q G -c ocy cle when G ⊆ SU ( d ) and N G is the no rmalizer of G in U ( d ) . Lemma 4. 3 . L et G ⊆ SU ( d ) . Then ther e is a map c : H 1 ( X, QG ) → H 2 ( X, Z ) ; if q is a trivial QG -c o cycle t hen c [ q ] = 0 . Pr o of. It suﬃce to note that the determinant deﬁnes a g roup mor phism det : N G → T . Since G ⊆ SU ( d ) , we ﬁnd that det fa c torizes through a morphism det Q : QG → T . The functor ialit y o f H 1 ( X, · ) , and the well-known isomorphis m H 1 ( X, T ) ≃ H 2 ( X, Z ) , c omplete the pro of. 5 Bundles of C* -algebras and cohomology c lasses. In the present section w e giv e an application of the cohomolog y class δ deﬁned in Lemma 4 .2 to bundles of C* -algebras. T o this end, in the following lines we present some constructions inv olving principal bundles a nd C* -dynamical systems. Let F • be a C* -algebra and X a loca lly compact, paraco mpact Hausdorﬀ space. T he n the cohomolog y set H 1 ( X, aut F • ) can b e interpreted as the set of is o morphism classes of F • -bundles, in the following wa y: for eac h aut F • -co cycle u := { u ij } , denote the ﬁbre bundle with ﬁbre F • and trans ition maps { u ij } b y π : b F → X (see [12, 5 .3.2]); by construction, b F is endow ed with lo cal charts π i : b F | X i := π − 1 ( X i ) → X i × F • , where { X i } i is an op en cov er of X , in such a way that π i ◦ π − 1 j ( x, v • ) = ( x, u ij,x ( v • )) , x ∈ X ij , v • ∈ F • . (5.1) The set of sections t : X → b F , p ◦ t = id X , such that the norm function { X ∋ x 7→ k t ( x ) k} v a nishes a t inﬁnity has a natura l str ucture of F • -bundle, that w e deno te by F u . On the conv ers e, given an F • -bundle F , using the metho d exp osed in [2 6, § 3.1 ] (s e e also the r elated refer ences), we can co ns truct a ﬁbre bundle π : b F → X with ﬁbre F • , in suc h a wa y that F is is omorphic to the C 0 ( X ) -algebr a o f sections of b F . The corr e spo ndence F 7→ b F is functorial: C 0 ( X ) -morphisms τ : F 1 → F 2 corres p ond to bundle morphisms b τ : b F 1 → b F 2 such that τ ( t ) = b τ ◦ t , t ∈ F 1 . Let F 1 , F 2 be F • -bundles and K a subgro up of aut F • ; a C 0 ( X ) -isomor phism β : F 1 → F 2 is sa id to b e K -e quivariant if ther e is an o pen cover { X i } i ∈ I trivializing b F 1 , b F 2 by means of lo cal charts π i,k : b F k | X i → X i × F • , k = 1 , 2 , i ∈ I , with automorphisms β i,x ∈ K , i ∈ I , x ∈ X i , satisfying b β ◦ π − 1 i, 1 ( x, v • ) = π − 1 i, 2 ( x, β i,x ( v • )) , v • ∈ F • (roughly sp eaking , at the loca l level β is desc ribed b y automorphisms in K ). In such a ca se, we say that F 1 is K - C 0 ( X ) - isomorphic to F 2 . Moreov er , we say tha t an F • -bundle F has struct u r e gr oup K if F = F u for some K -co cycle u . It is easy to verify that K -cocycles u , v a re equiv alent in H 1 ( X, K ) if and only if the asso ciated F • -bundles are K - C 0 ( X ) -isomor phic. 14 R emark 5.1 . Let ( A • , ρ • , a • ) be a p ointed C* -dynamical s y stem and K := aut ( A • , ρ • , a • ) ⊆ aut A • . An A • -bundle A ha s structure group K if and only if there is ρ ∈ end X A and a ∈ A with lo cal charts π i : b A| X i → X i × A • , such that π i ◦ b ρ ( v ) = ρ • ( v • ) , π i ( a ) = ( x, a • ) , v ∈ b A , ( x, v • ) := π i ( v ) . In this case, we say that ( A , ρ, a ) is a lo cally trivial p ointed C* -dynamical system . Now, β : A → A ′ is a K - C 0 ( X ) -isomor phism if a nd only if β is an isomorphism of p ointed C* -dynamical systems. So that, H 1 ( X, K ) des crib e s the set of isomo r phism cla sses o f lo cally trivial p ointed C* - dynamical sy stems ( A , ρ, a ) with ﬁbre ( A • , ρ • , a • ) . In the sequel, we shall make use o f the following fact: if N is a subgroup of K a nd n is an N -co cycle, then we may reg ard n a s a K - co cycle; thus, if A is an A • -bundle with structure gro up N , then A deﬁnes a lo cally tr ivial p oint ed C* -dynamical system ( A , ρ, a ) . The next lemma is an applicatio n of the previous ideas. Lemma 5.1. L et d ∈ N , G ⊆ U ( d ) b e c ovariant and QG := N G \ G . Then for e ach c omp act Hausdorﬀ sp ac e X ther e ar e one-to-one c orr esp ondenc es b etwe en: (1) QG -c o cycles; (2) lo c al ly trivial p ointe d C*-dynamic al systems with ﬁ br e ( O G , σ G , θ ) ; (3) b G -bund les. Pr o of. Cons ider the p ointed C* -dynamical system ( O G , σ G , θ ) with the action (3.3), then apply Rem.5.1, Thm.3.5 and Thm.3.1. The fo llowin g construction may b e regarded as an analog ue of the notion of gro up action in the setting of C* -bundles and app eared in [26, § 3.2 ]. Let G be a subgroup of aut F • . A gauge C*-dynamic al system with ﬁbr e ( F • , G ) is given b y a triple ( F , G , α ) , where F is an F • -bundle, η : G → X is a bundle with ﬁbre G and α : G × X b F → b F is a contin uo us map such that for each x ∈ X there is a neighbouro o d U of x with lo cal charts η U : G | U → U × G , π U : b F | U → U × F • , (5.2) satisfying π U ◦ α ( y , v ) = ( x, y • ( v • )) , (5.3) where x := π ( v ) = η ( y ) , ( x, y • ) := η U ( y ) , ( x, v • ) := π U ( v ) (so that y • ∈ G ⊆ aut F • and v • ∈ F • ). W e sa y that ( F , G , α ) ha s structure gro up K if F has structure gro up K . Usual contin uous actions are r elated with gauge C* -dynamical sys tems in the following wa y: if S is a set of sectio ns of G which is a ls o a g r oup w.r.t. the op erations deﬁned p oint wise, then there is an a ction S → aut X F ; in pa rticular, every contin uous action G → aut X F can be regarded as a gauge a c tion o n F of the bundle G := X × G (see [26, § 3.2 ] for details). The ﬁxe d-p oint algebr a of ( F , G , α ) is given by the C 0 ( X ) -algebr a F α := { t ∈ F : α ( y , t ( x )) = t ( x ) , x ∈ X , y ∈ η − 1 ( x ) } . Let A • ⊆ F • denote the ﬁxed-p oint algebr a w.r .t. the G -action. Then (5.3) implies that F α is an A • -bundle. 15 W e now exp ose the main construction of the present s ection. Again, w e cons ider a C* -algebra F • and a subgroup K o f aut F • ; moreov er , we pick a subgro up G of K a nd denote the ﬁxed-p oint algebra w.r .t. the G -action by A • . W e consider the normalizer o f G in K and the asso ciated quotient gr oup, as follows:  N G := { u ∈ K : u ◦ g ◦ u − 1 ∈ G } p : N G → QG := N G \ G . (5.4) By co nstruction, for every u ∈ N G , g ∈ G , a ∈ A • there is g ′ ∈ G such that g ◦ u ( a ) = u ◦ g ′ ( a ) = u ( a ) . The ab ov e equalities imply that the N G -action on F • factorizes through a QG -action QG → aut A • , p ( u ) 7→ u | A • , u ∈ N G ; (5.5) th us, applying the a bove pr oce dure, fo r e very QG -co cycle q we can construct an A • -bundle A q . Lemma 5.2 . L et q := ( { X i } , { y ij } ) ∈ H 1 ( X, QG ) and A q denote the asso ciate d A • -bund le. Then the fol lowing ar e e quivalent: 1. Ther e is a gauge C*-dynamic al system ( F , G , α ) with ﬁbr e ( F • , G ) and struct u r e gr oup N G , such that A q is QG - C 0 ( X ) -isomorphic t o F α ; 2. ther e is an N G -c o cycle n such that [ q ] = p ∗ [ n ] , wher e p ∗ : H 1 ( X, N G ) → H 1 ( X, QG ) is the map induc e d by (5.4(2)). Pr o of. (1) ⇒ (2): Let us denote the N G - co cycle as so ciated with F by n := ( { X i } , { u ij } ) . W e assume that { X i } tr ivializes G a nd b F (otherwise, we p erform a r e ﬁnement of { X i } ), so that w e hav e lo cal charts η i : G | X i → X i × G , π i : b F | X i → X i × F • fulﬁlling (5.3), with { π i } related with { u ij } by mea ns of (5.1). Let us consider the ﬁbre bundle b F α → X asso ciated with F α ; then w e hav e an inclusion b F α ⊆ b F and (5.3) implies π i ( b F α | X i ) = X i × A • ⇒ X ij × A • = π i ◦ π − 1 j ( X ij × A • ) . W e conclude by (5.1) that v ij,x := u ij,x | A • ∈ aut A • for every x ∈ X ij and pair i, j . Mo reov er, by (5.5) we ﬁnd that v ij = p ◦ u ij , as i, j v ary , yield a set of tra ns ition maps for b F α . Finally , s ince A is QG - C 0 ( X ) -isomor phic to F α , w e conclude that [ q ] = p ∗ [ n ] . (2) ⇒ (1): Let n := { u ij } . W e deﬁne F as the F • -bundle with co cycle n and G → X a s the ﬁbre bundle with ﬁbre G and tr a nsition maps γ ij,x ( g ) := u ij,x ◦ g ◦ u − 1 ij,x , x ∈ X ij . Suc h tra nsition maps deﬁne a co cycle with class γ ∗ [ n ] ∈ H 1 ( X, aut G ) . Now, we no te that u ij,x ◦ g ( v • ) = γ ij,x ( g ) ◦ u ij,x ( v • ) , g ∈ G , v • ∈ F • , x ∈ X ij . This implies that if we consider the ma ps α i : ( X i × G ) × X i ( X i × F • ) → X i × F • , α i (( x, g ) , ( x, v • )) := ( x, g ( v • )) , then there is a unique gauge action α : G × X b F → b F with lo cal c ha rts { η i } of G asso ciated with { γ ij } and { π i } of b F asso ciated with { u ij } , fulﬁlling α i = π i ◦ α ◦  η − 1 i × π − 1 i  for every index i . Since A has QG -co cycle { p ◦ u ij } , rea s oning a s in the ﬁrs t part of the pro of we conclude that A is QG - C 0 ( X ) -isomor phic to F α . 16 Corollary 5.3. With the notation of t he pr evious L emma, if γ ∗ [ n ] = [ 1 ] then ther e is a c ontinuous action α • : G → aut F with ﬁxe d-p oint algebr a Q G - C 0 ( X ) -isomorphic t o A . Pr o of. Since γ ∗ [ n ] = [ 1 ] there is an isomorphism G ≃ X × G . Thus, the gauge action α : G × X b F induces the co n tinuous action α • : G → aut X F (see [26, Cor.3.4]). Theorem 5.4. L et G ⊆ K ⊆ aut F • , A • denote the ﬁxe d-p oint algebr a of F • w.r.t. t he G -action and Q G deﬁne d as in (5.4(2)). F or e ach A • -bund le A with structur e gr oup QG t her e is a class δ ( A ) ∈ H 2 ( X, G ′ ) (5.6) fulﬁl ling t he fol lowing pr op erty: if A is Q G - C 0 ( X ) -isomorphic to t he ﬁx e d-p oint algebr a of a gauge C*-dynamic al system ( F , G , α ) with ﬁ br e ( F • , G ) and stru ctur e gr oup N G , then δ ( A ) = [ 1 ] . The c onverse is also true when G lies in the c entr e of N G . Pr o of. Applying Lemma 4 .2 we deﬁne δ ( A ) := δ [ q ] , where q is the QG - c ocy cle asso cia ted with A (of cour s e, there is a n a buse of the notation δ in the previo us deﬁnition, but this should not create confusion). The theorem now follows applying Le mma 5.2. The class δ may be also interpreted as an obs truction to co nstructing cov aria n t r epresentations of a gauge C* -dynamical system over a contin uous ﬁeld of Hilbert spaces. Since this po in t goes beyond the purp ose of the pres en t work, we p ostp one a complete discussio n to a forthcoming pa per. In the following lines we disc us s the relation b etw een the class δ and the Dixmier-Douady inv a riant. Let H denote the standard s eparable Hilb ert s pace, U the unita r y group of H endow ed with the norm top olo g y , T the torus a cting on H by scalar multiplication, PU := U / T the pro jective unitary gro up, K r the C* -algebra of compact op erators acting on the tensor p ower H r , r ∈ N , and ( H r , H r ) ⊃ K r the C* -algebra of bo unded op erator s . Moreov er , let O ∞ denote the Cun tz algebra; it is well-kno wn that there is a contin uous action U → aut O ∞ , (5.7) deﬁned as in (1.1), which restr icts to the cir cle action T → aut O ∞ . The construction (5.4) with K = U , G = T yields QG = PU and the ac tion PU → aut O 0 ∞ , γ 7→ b γ . Now, O 0 ∞ can b e constructed using a universal co nstruction on K , as follows (see [3]). Co nsider the inductive s tr ucture . . . j r − 1 → ( H r , H r ) j r → ( H r +1 , H r +1 ) j r +1 → . . . , j r ( t ) := t ⊗ 1 , (5.8) where 1 ∈ ( H , H ) is the identit y , and denote the as so ciated C* -a lg ebra by B ∞ . Then, O 0 ∞ is the C* -subalgebra of B ∞ generated by the imag es o f the K r ⊂ ( H r , H r ) , r ∈ N . The P U -a ction on O 0 ∞ preserves the inductive structure: if i r : K r → O 0 ∞ , r ∈ N , a r e the na tur al inc lus ions, then b γ ◦ i r ( t ) ∈ i r ( K r ) , γ ∈ P U , t ∈ i r ( K r ) , (5.9) and in pa rticular P U acts o n i 1 ( K ) as the usual adjoint a c tio n: b γ ◦ i 1 ( t ) = i 1 ◦ γ ( t ) , t ∈ K , γ ∈ P U . (5.10) 17 Let us deno te the catego ry of O 0 ∞ -bundles ov er X with a rrows P U - C 0 ( X ) -isomor phisms by bun PU ( X, O 0 ∞ ) . By the a b ov e results, each O 0 ∞ -bundle A ∞ with structure group PU is determined by a PU -co cycle q , and the class δ ( A ∞ ) = δ [ q ] ∈ H 2 ( X, T ) ≃ H 3 ( X, Z ) (5.11) measures the obstr uction to ﬁnding a gauge dynamical sys tem with ﬁbre (5 .7) and ﬁxed-p oint algebra A ∞ . Now, we denote the category o f K - bundles ov er X with ar r ows C 0 ( X ) -isomor phisms by bun ( X, K ) ; each K -bundle A is determined b y a PU -co cycle q , and its Dixmier-Douady inv a riant ([5, Ch.10]) is computed by (4.10): δ DD ( A ) = δ ( q ) . (5.12) Prop osition 5.5. F or e ach lo c al ly c omp act, p ar ac omp act Hausdorﬀ sp ac e X , ther e is an e quiva- lenc e of c ate gories bun ( X , K ) → bun PU ( X, O 0 ∞ ) , A 7→ A ∞ , and δ DD ( A ) = δ ( A ∞ ) , A ∈ bun ( X , K ) . (5.13) Pr o of. Let A b e a K -bundle with asso cia ted PU -co cycle q . The multiplier algebra M A of A can be constructed as the C 0 ( X ) -algebr a of bounded sections of the bundle b B → X with ﬁbre ( H, H ) and transition ma ps deﬁned b y q . F or each r ∈ N , we consider the C 0 ( X ) -tensor pro ducts M A r := M A ⊗ X . . . ⊗ X M A , A r := A ⊗ X . . . ⊗ X A and the o b vious inclusions A r ⊂ M A r . W e hav e the inductive limit structure . . . j r − 1 → M A r j r → M A r +1 j r +1 → . . . , j r ( t ) := t ⊗ 1 , where 1 ∈ M A is the iden tity . The sys tem ( M A r , j r ) yields the inductive limit algebra M A ∞ and we deﬁne A ∞ as the C* -subalgebr a of M A ∞ generated by the images of the C* -algebras A r , r ∈ N . If β : A → A ′ is a C 0 ( X ) -isomor phism, then it natura lly e x tends to C 0 ( X ) -isomor phisms β r : A r → A ′ r , r ∈ N , and ﬁnally to a C 0 ( X ) -isomor phism β ∞ : A ∞ → A ′ ∞ . On the conv er se, let A ∞ be a O 0 ∞ -bundle with structur e gro up PU and asso ciated P U - c ocy cle q . Since the PU - action o n O 0 ∞ preserves the inductive str ucture (5.8), and since the PU -action on O 0 ∞ restricts to the natural PU -action on K ⊂ O 0 ∞ (see (5.9) and (5.10)), for each r ∈ N there is a K r -bundle A r ⊂ A ∞ with asso cia ted PU -co cyc le q , with A 1 generating A ∞ as ab ove; th us o ur functor is surjective on the sets of ob jects. If β ′ : A ∞ → A ′ ∞ is an is omorphism in bun PU ( X, O 0 ∞ ) then by PU -equiv a riance we ﬁnd β ′ | A r = A ′ r for ea c h r ∈ N . Deﬁning β := β ′ | A 1 we easily ﬁnd β ′ = β ∞ ; th us our functor is sur jectiv e o n the sets o f arrows. Fina lly , (5.13) follows b y (5.11) a nd (5 .12). 6 Gauge-equiv arian t bundles, and a conc rete dualit y . Let X b e a co mpact Hausdor ﬀ space. In the pres en t section we give a duality theor y in the setting of the c a tegory v ect ( X ) o f vector bundles ov er X , r elating suitable sub categor ies of v e ct ( X ) with gauge equiv ariant vector bundles in the sens e o f [20]. Let d ∈ N and π : E → X a vector bundle of rank d . W e denote the Hilb ert C ( X ) -bimo dule of sections of E by S E , endow ed with coinciding left and rig h t C ( X ) -a ctions. F or each r ∈ N , we denote the r - fold tensor p ow er of E in the se nse of [15, § I.4 ], [1, 1 .2] by E r (for r = 0 , we deﬁne E 0 := ι := X × C ) a nd by ( E r , E s ) the set o f vector bundle morphisms fr om E r int o E s . The Serr e- Swan equiv alence implies that every ( E r , E s ) is the C ( X ) -bimo dule of sections o f a vector bundle π r s : E r s → X , having ﬁbr e ( H r , H s ) ≡ M d r ,d s ([15, Thm.5.9 ]). In explicit terms, E r s ≃ E s ⊗ E r ∗ , 18 where E r ∗ is the r -fold tensor p ow er of the conjugate bundle and every t ∈ ( E r , E s ) can b e regar ded as a contin uous map t : X → E r s , π r s ◦ t = id X . W e denote the tensor catego ry with ob jects E r , r ∈ N , and arr ows ( E r , E s ) by b E . It is clear that ( ι, ι ) = C ( X ) . Moreover, the ﬂip op erator θ E ∈ ( E 2 , E 2 ) : θ E ( x ) ◦ ( v ⊗ v ′ ) := v ′ ⊗ v , v , v ′ ∈ E x , x ∈ X , (6.1) deﬁnes a symmetry on b E . Th us, ( b E , ⊗ , ι, θ E ) is a symmetric tensor C* -categ o ry; we denote the asso ciated p ointed C* -dynamical s ystem by ( O E , σ E , θ E ) . Prop osition 6. 1. L et d ∈ N and H denote the standar d r ank d Hilb ert sp ac e. (1) F or e ach c omp act Hausdorﬀ s p ac e X ther e is an isomorphism Q : sym ( X , b H ) → H 1 ( X, U ( d )) . (2) If E → X is a r ank d ve ctor bund le, then the c ate gory ( b E , ⊗ , ι, θ E ) is a b H -bun d le and al l the elements of sym ( X , b H ) ar e of t his typ e; (3) If u is an U ( d ) -c o cycle asso ciate d with E as a set of tr ansition maps, t hen Q [ b E ] = [ u ] ; ( 4) O E is the Cunt z -Pimsner algebr a asso ciate d with S E and is an O d -bund le with structur e gr oup U ( d ) . Pr o of. (1) W e apply Thm.3.5 to the case G = { 1 } , so that N G = QG = U ( d ) . (2) Let E → X be a vector bundle; we consider a loca l c ha rt π U : E | U → U × H and note that, by functoriality , for each r, s ∈ N there ar e lo cal charts π r s U : E r s | U → X × ( H r , H s ) . This yields the desired lo cal chart b π U : b E → U b H . Let no w ( b ρ, ⊗ , ι, ε ) b e a b H -bundle; to prov e that b ρ ≃ b E for some vector bundle E we note that the Hilber t C ( X ) -bimodule ( ι, ρ ) deﬁnes a lo ca lly trivial co n tinuous ﬁeld o f Hilber t spa ces with ﬁbr e H ; we denote the vector bundle a s so ciated with ( ι, ρ ) b y E , and a pplying the Serre-Swan equiv alence we obtain an isomorphis m β : S E ≃ ( ι, E ) → ( ι, ρ ) , whic h extends to the desired isomorphisms β r s : ( E r , E s ) → ( ρ r , ρ s ) , r , s ∈ N . (3) W e pick an U ( d ) -co cycle u ′ with cla ss Q [ b E ] . By deﬁnition of Q w e hav e that u ′ yields tra nsition maps fo r the vector bundles E r s , r, s ∈ N , by means of the a ction b u ( t ) := u s ◦ t ◦ u ∗ r , u ∈ U ( d ) , t ∈ ( H r , H s ) (compare with (3.1)). In particular, for r = 0 , s = 1 , we conclude tha t u ′ deﬁnes, up to co cycle equiv alence, a set of transition maps for E and th us [ u ′ ] = Q [ b E ] = [ u ] . (4) It suﬃces to reca ll [2 2, Prop.4 .1 , Prop.4.2 ]. R emark 6.1 . T o be co ncise, we deno te the tota lly an tisymmetr ic pro jections deﬁned as in (3.6) by P n := P E ,θ E ,n ∈ ( E n , E n ) , n ∈ N . By deﬁnition of the tota lly antisymmetric line bundle ∧ d E := P d E d we have that E is a t wisted spe c ial ob ject, with ’categorical Chern cla ss’ (3.8) coinciding with the ﬁrs t Chern class c 1 ( E ) . If c 1 ( E ) = 0 then E is a specia l ob ject a nd the conjugate bundle E ∗ app ears as the o b ject asso ciated with the pro jection P d − 1 ∈ ( E d − 1 , E d − 1 ) (see [7, Lemma 3.6]). Clearly , the existence of the c onjugate bundle do es no t dep end on the v anishing of c 1 ( E ) , anyw ay in general it is false that E ∗ ≃ P d − 1 E d − 1 . Let b ρ be a tensor C ( X ) -subca tegory of ( b E , ⊗ , ι ) ; we denote the spaces of arrows of b ρ by ( E r , E s ) ρ , r, s ∈ N . F or every r , s ∈ N , we deﬁne the set E r s ρ := { t x ∈ E r s : x ∈ X , t ∈ ( E r , E s ) ρ } and denote the r estriction of π r s on E r s ρ by π ρ r s . In this w ay , we obtain Banach bundles π ρ r s : E r s ρ → X . (6.2) Let t ∈ ( E r , E s ) . If t ∈ ( E r , E s ) ρ , then by deﬁnition t x ∈ E r s ρ for every x ∈ X . On the converse, suppo se that t x ∈ E r s ρ , x ∈ X ; then for every x ∈ X there is t ′ ∈ ( E r , E s ) ρ such that t x = t ′ x . By 19 contin uity , fo r every ε > 0 there is a neighbour ho o d U ε ∋ x with s up y ∈ U ε   t y − t ′ y   < ε . Thus, [5, 10.1.2 (iv)] implies that t ∈ ( E r , E s ) ρ . W e conclude that t ∈ ( E r , E s ) ρ ⇔ t x ∈ E r s ρ , ∀ x ∈ X . (6.3) Let p : G → X be a g roup bundle with ﬁbres co mpact gro ups G x := p − 1 ( x ) , x ∈ X . According to [20], a gauge action on E is given by a contin uous map α : G × X E → E , such that each restriction α x : G x × E x → E x , x ∈ X , is a unitary repr esentation on the Hilb ert space E x ; to ec onomize on no tation, we deﬁne G α,x := α x ( G x ) , u α,x := α x ( u ) , u ∈ G x . In this w ay , E is a G -e quivariant ve ctor bund le in the sense of [20, § 1], with trivial a c tion on X . Moreov er , every π r s : E r s → X is a G -vector bundle, with action α r s : G × X E r s → E r s , ( u, v ) 7→ α r s ( u, v ) := b u α,x ( v ) , x := p ( u ) = π r s ( v ) ∈ X , where b u α,x ( v ) is deﬁned as in (1.1). W e denote the ca tegory with ob jects E r , r ∈ N , a nd a rrows ( E r , E s ) α := { t ∈ ( E r , E s ) : α r s ( u, t ( x )) = t ( x ) , u ∈ G , x := p ( u ) } (6.4) by b α . Clearly , ( b α, ⊗ , ι ) is a tensor C* -categor y with ( ι, ι ) = C ( X ) and ﬁbr es b G α,x , x ∈ X , deﬁned as in (3.1). Since θ E ( x ) = θ , x ∈ X , we co nclude that θ E ∈ ( E 2 , E 2 ) α , thus there is an inclusion functor E : ( b α, ⊗ , ι, θ E ) → ( b E , ⊗ , ι, θ E ) . Let us consider the bundle U E → X of unitary a utomorphisms of E (see [1 5, I.4.8 ]). It is well known that U E has ﬁbre the unitary group U ( d ) ; if { u ij } is the U ( d ) -co cycle asso ciated with E , then U E has as so ciated aut U ( d ) -co cycle γ ij,x ( u ) := u ij,x · u · u ∗ ij,x , x ∈ X ij , u ∈ U ( d ) . Note that U E is compact as a top ological s pace. In the same way the bundle S U E → X of sp ecial unitary automorphisms of E is deﬁned: it has ﬁbre SU ( d ) and the same tra nsition maps as U E . Of cour s e, there is an inclusion S U E ⊂ U E . Now let b e G → X b e a closed subbundle of U E , no t necess arily lo cally trivial. Then there is an obvious gauge action α : G × X E → E . In or der to emphasize the picture of G as a subbundle of U E , we use the notatio ns b G := b α , ( E r , E s ) G := ( E r , E s ) α , and call b G the dual of G . Clearly , each ( E r , E s ) G is the mo dule of sections o f a Banach bundle π G r s : E r s G → X , r , s ∈ N . W e deﬁne ( O G , σ G , θ E ) as the po inted C* -dynamica l sys tem ass ocia ted with ( b G , ⊗ , ι, θ E ) . Clearly , there is a ca nonical monomorphis m E ∗ : ( O G , σ G , θ E ) → ( O E , σ E , θ E ) . 20 Actions on the vector bundle E → X by (gener a lly noncompact) groups G of unitary automor- phisms have b een consider ed in [2 3, § 4]. This a pproach has the disadv antage to asso ciate the same dual to very diﬀerent groups (see [23, Ex.4.2]). According to [23, Def.4.7], we ca n asso cia te a g roup bundle G ⊆ U E to G , in such a way tha t the map {G 7→ b G } is one-to-o ne ([23, P rop.4.8]). F or this reason in the present pap er we passed to consider the notion of gauge action. The following result is a diﬀerent version of [23, P rop.4.8]; since the pro of is es sen tia lly the sa me, it is o mitted. Prop osition 6. 2. L et E → X b e a ve ctor bund le. The map {G 7→ b G } deﬁnes a one-to-one c orr esp ondenc e b etwe en the set of close d subbund les of S U E and the s et of symmetric t ensor C*- sub c ate gories b ρ of b E such that ( ι, ∧ d E ) ⊆ ( ι, E d ) ρ . Let G ⊆ U ( d ) . A b G -bund le in b E is a b G - bundle b ρ endow ed with an inclusion ( b ρ, ⊗ , ι, θ E ) ⊆ ( b E , ⊗ , ι, θ E ) . Let us deno te the inclus io n map by i : N G → U ( d ) , and the quotient pro jection by p : N G → QG ; by functoria lit y of H 1 ( X, · ) , there are maps i ∗ : H 1 ( X, N G ) → H 1 ( X, U ( d )) , p ∗ : H 1 ( X, N G ) → H 1 ( X, QG ) . (6.5) Moreov er , b y (4 .1) each N G -co cycle n = { u ij } deﬁnes an aut G - c ocy cle { b u ij } with class γ ∗ [ n ] . Theorem 6. 3. L et G ⊆ U ( d ) b e a c omp act gr oup. F or e ach c omp act Hausdorﬀ sp ac e X and N G -c o cycle n = { u ij } , ther e ar e a ve ctor bund le E → X with U ( d ) -c o cycle { i ◦ u ij } and a ﬁbr e G - bund le G ⊆ U E with tr ansition maps { b u ij } . The c ate gory ( b G , ⊗ , ι, θ E ) is a b G - bund le with asso ciate d c ohomolo gy class p ∗ [ n ] ∈ H 1 ( X, QG ) . Mor e over, ther e is a gauge action α : G × X b O E → b O E with ﬁbr e ( O d , G ) and ﬁxe d-p oint algebr a O G . Pr o of. Clea r ly , there a re E and G deﬁned as ab o ve. Since b y construction G ⊆ U E , the action α is deﬁned, together with the tensor C* -categ ory ( b G , ⊗ , ι, θ E ) a nd the po inted C* -dynamical system ( O G , σ G , θ E ) . By (3.3) we can regar d QG as a subgro up of aut b G , so the QG -co cycle q := { p ◦ u ij } deﬁnes a symmetric tensor C* -category ( b ρ q , ⊗ , ι, ε q ) and a p ointed C* -dynamical system ( O q , ρ q , ε q ) . T o pr ov e that b G has as so ciated co cycle q , it suﬃces to give a QG - C ( X ) - isomorphism O q ≃ O G . T o this end, we note that Lemma 5.2 implies that O q is QG - C ( X ) - isomorphic to the ﬁxed- p oint algebra O α E ; th us, in order to get the desired isomorphism, it suﬃces to pro ve that O G = O α E . Now, it is clea r that O G ⊆ O α E . T o prov e the oppos ite inclusio n, we consider the Haa r functional ϕ : C ( G ) → C ( X ) and the induced inv ariant mean m : O E → O α E in the s ense of [26, § 4]. By deﬁnition of α we hav e m (( E r , E s )) = ( E r , E s ) G ⊂ O G , r, s ∈ N , so that if t ∈ O α E is a no r m limit of the type t = lim n t n , t n ∈ span ∪ r s ( E r , E s ) , then t = m ( t ) = P n m ( t n ) , with t n ∈ ( E r , E s ) G . Thus, O α E = O G and this completes the pro of. In the following theor em we characterize the b G -bundles in b E that arise as a bove. Theorem 6.4 . L et G ⊆ U ( d ) b e c ovariant, E → X a ve ctor bund le with U ( d ) -c o cycle u and b ρ a b G -bund le in b E with Q G -c o cycle q (in the sense of L emma 5.1). Then the stru ctur e gr oup of E c an b e r e du c e d to N G , i.e. ther e is an N G - c o cycle n such that [ u ] = i ∗ [ n ] . Mor e over [ q ] = p ∗ [ n ] and b ρ = b G , wher e G ⊆ U E is a ﬁbr e G -bun d le with class γ ∗ [ n ] ∈ H 1 ( X, aut G ) . 21 Pr o of. W e a sso ciate to b ρ the lo cally trivia l pointed C* -dynamical system ( O ρ , ρ ∗ , θ E ) with ﬁbre ( O G , σ G , θ ) , equipp ed with the inclusion ( O ρ , ρ ∗ , θ E ) ⊆ ( O E , σ E , θ E ) . There is a ﬁnite open cov er { X i } and lo cal charts η i : O ρ | X i → C 0 ( X i ) ⊗ O G , deﬁning the QG -co cycle q := { y ij } such that b y ij,x = η i,x ◦ η − 1 j,x , x ∈ X ij . Now, up to p erfor ming a reﬁnemen t, we may assume that { X i } trivia lizes E , so that there a re lo cal charts π i : E | X i → X i × H with asso ciated U ( d ) - c ocy cle u :=  u ij := π i ◦ π − 1 j  . Mo reov er, each π i induces a lo cal chart b π i : O E → C 0 ( X i ) ⊗ O d . Let us deﬁne O ρ,i := b π i ( O ρ ) . W e introduce the C 0 ( X i ) -isomorphisms β i := b π i ◦ η − 1 i , β i : C 0 ( X i ) ⊗ O G ≃ − → O ρ,i ⊆ C 0 ( X i ) ⊗ O d , so that for e a c h pair i, j we ﬁnd b y ij,x = β − 1 i,x ◦ b u ij,x ◦ β j,x , x ∈ X ij . (6.6) Now, each β i may b e rega rded as a contin uous map β i : X i → emb O G , thus by Lemma 3.4 there is an op en cover { Y il } l of X i and contin uous maps w il : Y il → U ( d ) such that β i,x ( t ) = b w il,x ( t ) , t ∈ O G , x ∈ Y il . (6.7) W e extra c t from { Y il } il a ﬁnite op en cover { Y h } of X ; to econo mize on nota tion, we intro duced the index h instead o f i, l , so that we hav e maps w h satisfying (6.7) for each h a nd x ∈ Y h . Since E | Y h is trivial, we hav e that u is equiv a len t to a co cycle deﬁned by tra nsition maps u hk : Y hk → U ( d ) ; with a s ligh t abuse of notation, we denote this co cy c le again by u . Of course, the same pro cedure applies to q = { y hk } . By (6.6), w e ﬁnd b y hk,x = b w h,x ◦ b u hk,x ◦ b w − 1 k,x , x ∈ Y hk . (6.8) Now, u is equiv alent to the U ( d ) - coc ycle n := { z hk := w h u hk w ∗ k } and (6.8) b ecomes b y hk,x ( t ) = b z hk,x ( t ) , x ∈ Y hk , t ∈ O G . In other terms, each z hk,x ∈ aut ( O d , σ d , θ ) restr ic ts to the automor phis m b y hk,x ∈ aut O G , so by Rem.3.2 w e c o nclude that n takes v alues in N G and yields a reduction to N G of the structure group o f E . More over, using ag ain Rem.3.2 we hav e [ q ] = p ∗ [ n ] . The fac t that G has cla s s γ ∗ [ n ] follows by applying the previous theo r em to the N G -co cycle n . Corollary 6.5. L et b ρ b e a sp e cial c ate gory with an inclusion ( b ρ , ⊗ , ι, θ E ) → ( b E , ⊗ , ι, θ E ) . Then: (1) Ther e is a c omp act gr oup G ⊆ SU ( d ) such that E has an asso ciate d N G -c o cycle n ; (2) b ρ has class Q [ b ρ ] = p ∗ [ n ] ∈ H 1 ( X, QG ) ; (3) Ther e is a gr oup bund le G ⊆ S U E such that b ρ = b G . Pr o of. (1) By Thm.3.7 there is a compact group G ⊆ SU ( d ) suc h that b ρ has ﬁbr e b G and an asso ciated QG -co cycle q ; th us, by the pr evious theor em we co nclude that E has an asso ciated N G -co cycle n . (2) The previous theorem implies [ q ] = p ∗ [ n ] . (3) W e apply ag a in the previous theorem. 22 7 Cohomological in v arian ts and dualit y breaking. In the presen t section w e approa c h the following question: given a cov ar ian t inc lus ion G ⊆ U ( d ) and a b G -bundle ( b ρ, ⊗ , ι, ε ) , is there any G -equiv ariant vector bundle E → X ι with an isomo rphism ( b G , ⊗ , ι, θ E ) ≃ ( b ρ, ⊗ , ι, ε ) ? This is w ha t we ca ll the problem o f abstr act duality , as – diﬀerently from the previous section – our catego ry b ρ is no t pr e sen ted as a sub category of v ect ( X ι ) . W e will g iv e a complete answer to the previo us ques tion in terms o f the coho mology set H 1 ( X ι , QG ) , reducing the pro blem of a bs tract duality to (rela tiv ely) simple computations in volving co cycle s and principal bundles. As a prelimina ry step we a nalyze the setting o f C* -bundles. Let X b e a compact Hausdorﬀ space. By Le mma 5 .1 we hav e that H 1 ( X, QG ) descr ibes the set of is omorphism cla sses o f lo cally trivial, p ointed C* -dynamical sy s tems with ﬁbre ( O G , σ G , θ ) . F or every QG -c ocy cle q , we denote the asso ciated p ointed C* -dynamical system by ( O q , ρ q , ε q ) . Theorem 7.1. With the ab ove not ation, for e ach Q G -c o cycle q and ( O q , ρ q , ε q ) , t he fol lowing ar e e quivalent: 1. Ther e is a r ank d ve ctor bu n d le E → X with a C 0 ( X ) -monomorphism η : ( O q , ρ q , ε q ) → ( O E , σ E , θ E ) ; 2. Ther e is a gauge C*-dynamic al system ( O , G , α ) with ﬁbr e ( O d , G ) and stru ctur e gr oup U ( d ) , such that O q is QG - C 0 ( X ) -isomorphic t o the ﬁxe d-p oint algebr a O α ; 3. Ther e is an N G -c o cycle n such t hat p ∗ [ n ] = [ q ] . Pr o of. (3) ⇒ (2): w e consider the Cun tz algebra O d endow ed with the N G -action (1.1), which factorizes through the action QG → aut O G . Then we apply Lemma 5.2 with F • = O d and A • = O G . (2) ⇒ (1): Let n := { u ij } denote the U ( d ) -co cycle asso ciated with O and E → X be the ra nk d v ector bundle with tra nsition maps { u ij } . Acco r ding to Pr op.6.1 there is a U ( d ) - C 0 ( X ) -isomor phism O ≃ O E , so that, to b e concise, we identify O with O E . Now, by co ns truction of O E there is an inclusion S E ⊂ O E , to which c orresp onds an inclusion E ⊂ b O E ; since ( O E , G , α ) has ﬁbre ( O d , G ) , with G acting on O d as in (1.1), we conclude that E is G - stable and the map α : G × X b O E → b O E restricts to an a ction G × X E → E , (7.1) i.e., E is G -equiv aria n t. By Thm.6 .3, we conclude that O α = O G . Mor eov er, by Thm.3.5 we have QG = aut ( O G , σ G , θ ) , th us , fro m Rem.5.1 we conclude tha t the given QG - C 0 ( X ) -isomor phism β : O q → O G yields monomorphis ms ( O q , ρ q , ε q ) β − → ( O G , σ G , θ E ) ֒ → ( O E , σ E , θ E ) . (7.2) (1) ⇒ (3): Apply again Lemma 5 .2 with F • = O d and A • = O G . Now, by Thm.3.5 there are maps ( sym ( X , b G ) → H 1 ( X, QG ) , [ b ρ, ⊗ , ι, ε ] 7→ Q [ b ρ ] H 1 ( X, QG ) → sym ( X , b G ) , [ q ] 7→ [ b ρ q , ⊗ , ι, ε q ] (7.3) which a re the inv erses one of each other. The fo llo wing r esult is the transla tion of Thm.7.1 in categoric al terms; the pro of is an immedia te application of Thm.3 .1, Thm.3.5 and Thm.6.4, th us it will b e omitted. 23 Theorem 7 .2. L et d ∈ N and G ⊆ U ( d ) b e c ovariant. F or e ach b G -bund le ( b ρ, ⊗ , ι, ε ) , the fol lowing ar e e quivalent: 1. Ther e is an emb e dding functor E : b ρ ֒ → vect ( X ι ) ; 2. Ther e is a ve ctor bund le E → X ι and a c omp act G -bund le G ⊆ U E with an isomorphism ( b ρ, ⊗ , ι, ε ) ≃ ( b G , ⊗ , ι, θ E ) ; 3. Ther e is an N G -c o cycle n such t hat p ∗ [ n ] = Q [ b ρ ] . W e ca ll a gauge gr oup asso ciate d with b ρ the bundle G → X app earing in Thm.7.2, whos e isomorphism c la ss is lab eled by γ ∗ [ n ] ∈ H 1 ( X, aut G ) . It follows fro m the previous theor em that the set of em b edding functors E : b ρ ֒ → v e ct ( X ) is in o ne-to-one co rresp ondence with the set of N G - co cycles n such that p ∗ [ n ] = Q [ b ρ ] , that we denote by Z 1 ( X, N G ; b ρ ) . As we sha ll see in the se q uel, Z 1 ( X, N G ; b ρ ) may contain mor e than a co homology clas s, o r b e empty . Let n , n ′ ∈ Z 1 ( X, N G ; b ρ ) and G , G ′ denote the ass ocia ted gauge groups. In genera l, G may b e not isomor phic to G ′ ; an example of this phenomenon with G = SU (2 ) is provided in Ex.7 .2. Corollary 7.3. L et G ⊆ U ( d ) b e c ovariant. If t her e is a c ont inu ous monomorphism s : QG → N G , p ◦ s = id QG , t hen for e ach b G - bund le ( b ρ, ⊗ , ι, ε ) ther e is at le ast one emb e dding functor b ρ ֒ → vect ( X ι ) . Pr o of. By functoria lit y there is s ∗ : H 1 ( X ι , QG ) → H 1 ( X ι , N G ) such that p ∗ ◦ s ∗ is the iden tity on H 1 ( X ι , QG ) . Thus Q [ b ρ ] = p ∗ [ n ] , n := s ∗ ◦ Q [ b ρ ] and this means that the desired em b edding functor exists. Example 7.1. Let G = SU ( d ) , so that N G = U ( d ) and QG = T . By (7.3), we have the map Q : sym ( X , \ SU ( d )) → H 1 ( X, T ) . Elementary co mputations show that the quotient map p : U ( d ) → T is the determinant; we deﬁne the co n tinuous s ection s : T ֒ → U ( d ) , s ( z ) = z ⊕ 1 d − 1 , where 1 d − 1 is the identit y o f M d − 1 . Since s is multiplicativ e, w e co nclude by Cor.7.3 that fo r each \ SU ( d ) -bundle b σ there is at least one embedding functor E : b σ → v ect ( X ) . F or future r eference, we c o nsider the w ell-known isomorphis m B : H 1 ( X, T ) → H 2 ( X, Z ) . Moreov er , we recall the r eader to (3.8). Corollary 7.4. L et ( b ρ, ⊗ , ι, ε ) b e a sp e cial c ate gory such that ρ has dimension d ∈ N and Chern class c ∈ H 2 ( X ι , Z ) . Then t her e is an \ SU ( d ) -bund le ( b σ , ⊗ , ι , ε ) with an inclusion functor ( b σ , ⊗ , ι, ε ) → ( b ρ, ⊗ , ι, ε ) . (7.4) If E : b ρ ֒ → v ect ( X ι ) is an emb e dding funct or and E := E ( ρ ) , t hen ther e is a fac t orization b σ / / E ≃   b ρ E   d S U E ⊆ / / b E (7.5) and E has ﬁrst Chern class c 1 ( E ) = c . 24 Pr o of. W e deﬁne b σ as the tenso r C* -sub category of b ρ g enerated by the symmetr y op erator s ε ρ ( r , s ) , r , s ∈ N , and the elemen ts o f R ρ (see (3.8) a nd follo wing rema rks). The o b vio us inclusio n b σ ⊆ b ρ yields the functor (7.4). If E is an em b edding functor then E ( P ρ,ε,d ) = P d (see Rem.6.1); this implies E ( R ρ ) = ( ι, ∧ d E ) and we conc lude that c ( ρ ) = c 1 ( E ) . Finally , since the spaces of arrows of [ S U E are gener ated by the ﬂips θ E ( r , s ) = E ( ε ρ ( r , s )) , r , s ∈ N , and element s of ( ι, ∧ d E ) , we obtain the desired factor ization (7.5 ). Corollary 7 .5. L et ( b σ , ⊗ , ε , ι ) b e an \ SU ( d ) -bund le with B ◦ Q [ b σ ] ∈ H 2 ( X ι , Z ) . Then the set of emb e dding functors E : b σ → v ect ( X ι ) c oincides with the set of ve ctor bund les over X ι of r ank d and ﬁrst Chern class B ◦ Q [ b σ ] . F or the notion of c onjugate in the setting of tensor C* -categor ie s, we refer the reader to [18, § 2]. Theorem 7.6. L et d ∈ N and G ⊆ U ( d ) b e c ovariant. Then for every b G -bund le ( b ρ, ⊗ , ι, ε ) the fol lowing invariants ar e assigne d:  δ ( b ρ ) := δ ◦ Q [ b ρ ] ∈ H 2 ( X ι , G ′ ) , ˘ γ ∗ ( b ρ ) := ˘ γ ∗ ◦ Q [ b ρ ] ∈ ˘ H 2 ( X ι , G → aut G ) . The class ˘ γ ∗ ( b ρ ) deﬁnes a G -gerb e ˘ G over X ι , unique up to isomorphism, which c ol lapses to a G - bund le G if and only if ther e is an emb e dding functor E : b ρ → v ect ( X ι ) , and in such a c ase δ ( b ρ ) = [ 1 ] . When G ⊆ SU ( d ) , the Chern class c ( ρ ) ∈ H 2 ( X ι , Z ) , deﬁne d in (3.8), fulﬁl les the fol lowing pr op erties: if c ( ρ ) = 0 then ρ is a sp e cial obje ct and the closur e for sub obje cts of b ρ has c onjugates; if E : b ρ → v ect ( X ι ) is an emb e dding funct or t hen c ( ρ ) is the ﬁrst Chern class of E ( ρ ) . Pr o of. W e pick a co cycle pair b in the co homology cla s s ˘ γ ∗ ( b ρ ) and deﬁne ˘ G as the G -gerb e with transition ma ps deﬁned by b according to Rem.4.2 a nd Lemma 4.2. Embeddings E : b ρ → v ect ( X ι ) are in one-to-one corr espo ndence with N G -co cycles n such that p ∗ [ n ] = Q [ b ρ ] and the asso cia ted G - bundles G deﬁne cohomology classes γ ∗ [ n ] ∈ H 1 ( X ι , aut G ) . Commutativit y of the square in (4.9) implies that d ∗ ◦ γ ∗ [ n ] = ˘ γ ∗ ◦ Q [ b ρ ] ∈ ˘ H 2 ( X ι , G → aut G ) , and this proves that ˘ G is isomorphic to the ger be deﬁned by G according to Rem.4 .2. The relation betw e e n the existence of E and the v a nishing of δ ( b ρ ) is pr oved applying Thm.7.2 a nd Lemma 4.2. Let now G ⊆ SU ( d ) . If c ( ρ ) = 0 then R ρ is a free Hilb ert ( ι, ι ) -mo dule a nd there is a n iso metry S ∈ R ρ . So that ρ is s pecia l, and [7, Lemma.3.6] implies tha t the conjugate ρ is a sub ob ject in b ρ . Using [1 8, Thm.2 .4 ] we conclude that the tenso r p ow ers ρ r , r ∈ N , and their sub o b jects, hav e conjugates. The previous theorem sugg ests that in genera l the dual ob ject of a symmetric tenso r C* -category is a nonab elian gerb e rather tha n a gro up bundle. Clearly , we should s ay in pr ecise terms in which sense a tensor C* -categor y is the r epr esentation c ate gory of a gerb e. This co uld b e done conside r ing the notion of a ction of ger bes on bundles of 2 -Hilb ert spaces. An alternative p oin t o f view is to consider Hilb ert C* -bimo dules ra ther than bundles: this situation is analogo us to what ha ppens in t wis ted K -theory , where we can use e quiv alently (Abelian) gerb es or bimodules with co eﬃcients in a co n tinuous trace C* -algebr a to deﬁne the same K -g r oup. These as pects will b e clariﬁed in a forthcoming pap er ([27]). 25 Example 7. 2. Let n ∈ N and S n denote the n -sphere. W e dis c uss the map p ∗ in the case G = SU ( d ) , d > 1 : p ∗ : H 1 ( S n , U ( d )) − → sym ( S n , \ SU ( d )) ≃ H 1 ( S n , T ) ≃ H 2 ( S n , Z ) . A well-known arg umen t implies H 1 ( S n , U ( d )) ≃ π n − 1 ( U ( d )) ([12, Ch.7.8]); thus, by classical r e sults ([12, Ch.7.12]) we have H 1 ( S 2 , U ( d )) ≃ H 1 ( S 4 , U ( d )) ≃ Z , H 1 ( S 1 , U ( d )) ≃ H 1 ( S 3 , U ( d )) ≃ 0 ; moreov er , H 2 ( S 2 , Z ) ≃ Z , H 2 ( S n , Z ) ≃ 0 , n 6 = 2 . Thu s the cas es S 1 , S 3 are trivial. In the other cases, we have the following: • n > 2 . The map p ∗ is trivial and the unique element of s ym ( S n , \ SU ( d )) is the cla ss of the trivial bundle. Now, it is a gener a l fact that if E → X is a vector bundle, then the contin uous bundle ( E , E ) is trivial if and only if E is the tensor pro duct of a triv ia l bundle b y a line bundle. In the cas e X = S n , n 6 = 2 , every line bundle is trivial, thus we co nclude tha t ( E , E ) is tr iv ial if and only if E is trivial. Since ( E , E ) is gener ated as a C ( X ) -mo dule by the sp ecial unitary group of E , we co nclude that E → S n is trivial if and o nly if S U E → S n is trivial. Thu s , [ S U E is tr iv ial for every E → S n , in spite of the fact that S U E is tr ivial if and o nly if E = S n × C d . In particular , this holds for S 2 m , m = 2 , . . . , where nont r ivial vector bundles exist. • n = 2 . W e recall that the Chern character C h : K 0 ( S 2 ) → H 0 ( S 2 , Z ) ⊕ H 2 ( S 2 , Z ) is a r ing iso morphism. The ter m H 0 ( S 2 , Z ) ≃ Z corresp onds to the rank, whilst H 2 ( S 2 , Z ) ≃ Z is the ﬁrs t Chern cla ss. By well-known stability pro perties of vector bundles (see [12, Ch.8, Thm.1.5] or [15, I I.6.10]), we ﬁnd that ra nk d vector bundles E , E ′ → S 2 are iso morphic if and only if [ E ] = [ E ′ ] ∈ K 0 ( S 2 ) , i.e . C h [ E ] = C h [ E ′ ] . This implies that p ∗ is one-to-one for n = 2 . Example 7.3. W e deﬁne R d ⊂ SU ( d ) as the gr o up of diag onal matric e s of the type g := diag( z , . . . , z ) , wher e z ∈ T is a ro ot of unity of order d . Then N R d = U ( d ) acts tr ivially on R d and R ′ d = R d . W e hav e the ex a ct se quence o f p ointed s ets H 1 ( S 2 , R d ) i ∗ → H 1 ( S 2 , U ( d )) p ∗ → H 1 ( S 2 , QR d ) δ → H 2 ( S 2 , R d ) . Now, every principal R d -bundle over S 2 is trivial, and the universal coeﬃcient theorem yie lds H 2 ( S 2 , R d ) ≃ Hom( Z , R d ) ≃ Z d . Thus we hav e 0 → Z p ∗ → H 1 ( S 2 , QR d ) δ → Z d , and p ∗ is injectiv e. W e now prove that there is a left in verse s : Z d → H 1 ( S 2 , QR d ) for δ with trivial in tersectio n with p ∗ ( Z ) . This suﬃces to prove that H 1 ( S 2 , QR d ) ≃ Z ⊕ Z d . T o this end, we embed R d in T and regar d each R d - 2 -co cycle g := { g ij k } a s a T - 2 -co cycle. In this w ay , the argument of the pro of of [5, Thm.10 .8.4(2)] implies that there is an 1 - U ( d ) -co chain u := { u ij } such that u ij u j k u − 1 ik = g ij k . Thus, we deﬁne the map s : H 2 ( S 2 , T ) → ˘ H 2 ( S 2 , R d → U ( d )) ≃ H 1 ( S 2 , QR d ) , s [ g ] := [ u , g ] , 26 which clear ly yields the desired left inverse (recall the deﬁnition of δ ). W e conclude that sym ( S 2 , b R d ) ≃ Z ⊕ Z d . The ﬁrst dir ect summand corr esp o nds to the term H 1 ( S 2 , U ( d )) whos e iso morphism with Z is realized b y mea ns of the determinant (see [12, § 7.8]); this implies that the pro jection of sym ( S 2 , b R d ) on Z is the Chern class. On the other side, by construction the pro jection o n Z d corres p onds to the class δ . Example 7.4. Let G ⊂ U ( d ) b e a s in Ex.3.1 and b ρ b e a b G - bundle; then it is eas y to chec k that the set of embeddings b ρ → vec t ( X ) is in one-to- one cor resp ondence with vector bundles E → X such that ( E , E ) ≃ ( ρ, ρ ) . In particular when X is the 3 –sphere S 3 then every vector bundle E → S 3 is trivial and b ρ admits an embedding if and only if ( ρ, ρ ) is trivia l. O n the other side, the (cla ssical) Dixmier-Douady inv ariant is a complete in v aria n t for bundles with ﬁbre M d and base space S 3 , th us we conclude that δ : sym ( S 3 , b G ) → H 2 ( S 3 , G ) = H 3 ( S 3 , Z ) = Z is an iso mo rphism. Ac kno wl e dgemen ts. The author would like to thank Ma uro Sper a for drawing his atten tion to gerb es, and an anonymous r eviewer for sug gesting se veral improv ements on the ﬁrs t version o f the present pap er. References [1] M.F. 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Bundles of C-categories, II: C-dynamical systems and Dixmier-Douady invariants

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