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Bundles of C-categories and duality



arXiv:math/0510594v3 [math.CT] 16 Jun 2008

Bundles of C*-categories
Ezio Vasselli Dipartimento di Matematica University of Rome §La Sapienza§ P.le Aldo Moro, 2 - 00185 Roma - Italy vasselli@mat.uniroma2.it June 16, 2008
Abstract We introduce the notions of multiplier C*-category and continuous bundle of C*-categories, as the categorical analogues of the corresponding C*-algebraic notions. Every symmetric tensor C*-category with conjugates is a continuous bundle of C*-categories, with base space the spectrum of the C*-algebra associated with the identity object. We classify tensor C*-categories with ?bre the dual of a compact Lie group in terms of suitable principal bundles. This also provides a classi?cation for certain C*-algebra bundles, with ?bres ?xed-point algebras of Od . Keywords: Tensor C*-category; Continuous ?eld; Principal bundle; Cuntz algebra.

1

Introduction.

C*-categories have been studied in recent years from several points of view, and with several motivations, as duality for (quantum) groups ([12, 35, 10]), quantum ?eld theory ([14, 5]), K -theory ([24]). In the present work, we provide a notion of bundle in the categorical setting, and then show that every C*-category endowed with a tensor structure is a bundle in the above-mentioned sense. This allows us to provide classi?cation results by mean of cohomological methods, and then to approach a duality theory for group bundles and groupoids. A tensor category is described by a collection of objects 老 , 考 , . . . , together with vector spaces denoted by (老, 考 ), . . . , called the spaces of arrows. Arrows t ﹋ (老, 考 ), t∩ ﹋ (考, 而 ) can be composed to obtain an element t∩ ? t ﹋ (老, 而 ), so that in particular every (老, 老) is a ring. Moreover, the objects can be multiplied, via the tensor product 老, 考 ↙ 老 ? 考 , and the existence of an identity, denoted by 牝 , is postulated in such a way that 牝 ? 老 = 老 ? 牝 = 老 for every object 老 . The aim of duality theory is to characterize tensor categories in terms of duals, i.e. categories with objects vector spaces and arrows linear maps equivariant w.r.t. a (possibly unique) dual object. In correspondence with the additional structure carried by the given tensor category, the dual object turns out to have di?erent natures; for example, we may get a compact group ([12]), an algebraic group ([7]), a compact quantum group ([35]), a multiplicative unitary ([10]), just to mention some of the known results. The above-considered cases are characterized by di?erent properties of the tensor structure: the crucial ones are the symmetry for [12, 7] and the braiding for [35, 10], which describe the commutativity of the tensor product. On the other side, a point which is common to all the cases is the following: the identity object is simple, in the sense that the ring (牝, 牝) is isomorphic to the ?eld of scalars. 1

In the present paper, we proceed with the program started in [30] to construct a duality theory for the case in which the given tensor category is symmetric and with non-simple identity. Our setting is the one of tensor C*-categories, considered in [12, 35, 10]; in particular, this implies that (牝, 牝) is a commutative C*-algebra. The condition that 牝 is non-simple is translated as the fact that the spectrum X 牝 of (牝, 牝) does not reduce to a single point. We show that every tensor C*-category has a canonical structure of a bundle, and study the important class of special categories, which turn out to be bundles with *?bre* the dual of a compact Lie group. We provide a complete classi?cation for special categories, in terms of a suitable cohomology set. Such a classi?cation admits a natural translation in terms of continuous bundles of C*-dynamical systems, with ?bre a ?xed-point algebra of the Cuntz algebra Od . Our research program is intended to provide a generalization of the Doplicher-Roberts duality theory ([11, 12, 13]). Anyway, as we will show in a forthcoming paper, dramatic di?erences arise in the case with non-simple identity, concerning non-existence and non-unicity of the dual object, which will turn out to be (when existing) a bundle of compact groups, or more generally a groupoid. These phenomena have a cohomological nature, and their root is the classi?cation provided in the present work. As a future application, we will provide a new kind of twisted topological equivariant K -functor, related to the gauge-equivariant K -theory introduced by V. Nistor and E. Troitsky ([25]). The construction of such a K -functor is outlined in [34]. The motivation of [12] was the search for the gauge group in the setting of superselection sectors in quantum ?eld theory ([13]). Motivated by similar structures arising in low-dimensional quantum ?eld theory and quantum constraints, a duality theory for tensor categories of C*-algebra endomorphisms has been developed in [5]: the set (牝, 牝) is identi?ed as the centre of a given *observable* C*-algebra, and the existence of a subcategory (with simple identity) satisfying the DoplicherRoberts axioms is postulated, with the result that the reconstructed dual object is a compact group. The tensor C*-categories above considered turn out to be *trivial bundles* in the sense of the present paper, as remarked in [32, Ex.5.1]. The present work is organized as follows. In Sec.2, we introduce the notions of multiplier C*category (Prop.2.2) and C0 (X )-category (Def.2.4, X denotes a locally compact, Hausdor? space), which generalize the ones of multiplier C*-algebra and C0 (X )-algebra. As an application, the notion of multiplier C*-bimodule is introduced (Cor.2.3). We prove that every C0 (X )-category has a canonical bundle structure (Prop.2.6), and consider the notion of continuous bundle of C*categories. In Sec.3, we study tensor C*-categories with (牝, 牝) = C . We prove that every tensor C*-category is a C (X 牝 )-category in a canonical way, where C (X 牝 ) is a suitable C*-subalgebra of (牝, 牝) (Prop.3.1). In particular, a symmetric tensor C*-category with (twisted) conjugates is a continuous bundle, having as ?bres duals of compact groups (Thm.3.9, Rem.3.4). The abovequoted theorem has an analogue in the setting of 2 -C*-categories with conjugates ([36, ∫3]); our proof is di?erent, and is based on the notion of (twisted) special object (see (3.16)). Once that a bundle structure is considered for symmetric tensor C*-categories, it is natural to formulate a coherent notion of local triviality (Def.3.10). A special category is a locally trivial, symmetric tensor C*-category such that the ?bre T? is isomorphic (up to direct sums) to the dual of a compact Lie group G (Def.3.13). Special categories with ?bre T? are completely classi?ed by the cohomology set H 1 (X 牝 , QG) (Thm.3.14), where QG is a suitable compact Lie group associated with G (see (3.22)). Finally, we consider a canonical G -action on Od , and give a classi?cation for continuous bundles with ?bre the ?xed-point algebra OG (Sec.4).

2

1.1

Notation and Keywords.

Let X be a locally compact Hausdor? space. We denote by: C0 (X ), the C*-algebra of ( C -valued) continuous functions on X vanishing at in?nity; Cb (X ), the C*-algebra of bounded, continuous functions on X ; C (X ), for X compact, the C*-algebra of continuous functions on X ; CU (X ), for U ? X open, the ideal in C0 (X ) of functions vanishing on X ? U ; Cx (X ) := CX ?{x} (X ), x ﹋ X . If {Xi } is a cover of X , then we denote Xij := Xi ﹎ Xj , Xijk := Xi ﹎ Xj ﹎ Xk . If A is a C*-algebra, we denote by autA (resp. endA ) the set of automorphisms (resp. endomorphisms) of A , endowed with the topology of pointwise convegence. A pair (A, 老), where 老 ﹋ endA , is called C*-dynamical system. If (A, 老), (A∩ , 老∩ ) are C*-dynamical systems, a C*algebra morphism 汐 : A ↙ A∩ such that 汐 ? 老 = 老∩ ? 汐 is denoted by 汐 : (A, 老) ↙ (A∩ , 老∩ ). Finally, for every d ﹋ N we denote by U(d) the unitary group, and by SU(d) the special unitary group. 1.1.1 C0 (X ) -algebras.

Let X be a locally compact Hausdor? space. A C0 (X ) -algebra is a C*-algebra A endowed with a non-degenerate morphism from C0 (X ) into the centre of the multiplier algebra M (A) ([21, ∫2]). It is customary to identify elements of C0 (X ) with their images in M (A). We call C0 (X )-morphism a C*-algebra morphism commuting with the above C0 (X )-action; in particular, we denote by autX A (resp. endX A ) the set of C0 (X )-automorphisms (resp. C0 (X )-endomorphisms) of A . It can be proved that C0 (X )-algebras correspond to upper-semicontinuous bundles of C*-algebras with base space X ([26]); in particular, every continuous bundle of C*-algebras ([9, Chp.10],[22]) is a C0 (X )-algebra. The ?bres of A are de?ned as the quotients Ax := A/(Cx (X )A), x ﹋ X . We denote by ?X the minimal C0 (X )-algebra tensor product [21, ∫2]. If A , B are C0 (X )-algebras, a C0 (X )-Hilbert A - B -bimodule is a Hilbert A - B -bimodule M such that af 肉b = a肉f b , f ﹋ C0 (X ), a ﹋ A, b ﹋ B , 肉 ﹋ M. 1.1.2 bi-Hilbertian bimodules.

The following notion appeared in [18]. Let A , B be C*-algebras, M a Hilbert A - B -bimodule such that the C*-algebra K (M) of compact, right B -module operators is contained in A . Then, besides the usual B -valued scalar product ﹞, ﹞ B , there exists an A -valued scalar product 肉, 肉 ∩ A := 牟肉,肉∩ , where 肉, 肉 ∩ ﹋ M , and 牟肉,肉∩ ﹋ K (M) ? A , 牟肉,肉∩ ? := 肉 肉 ∩ , ? B , ? ﹋ M . We say that M is a bi-Hilbertian bimodule if 肉 2 = 牟肉,肉 , 肉 ﹋ M , and use the notation A MB . A morphism of bi-Hilbertian bimodules is a bounded linear map 耳 : A MB ↙ A∩ M∩ B∩ , with C*-algebra morphisms 耳l : A ↙ A∩ , 耳r : B ↙ B ∩ such that 耳(a肉b) = 耳l (a)耳(肉 )耳r (b), 耳r ( 肉, 肉 ∩ B ) = 耳(肉 ), 耳(肉 ∩ ) B∩ , 耳l ( 肉, 肉 ∩ A ) = 耳(肉 ), 耳(肉 ∩ ) A∩ , 肉, 肉 ∩ ﹋ M , a ﹋ A , b ﹋ B . 1.1.3 Bundles.

For standard notions about vector bundles, we refer to the classics [2, 20, 29]. In the present paper, we will assume that every vector bundle is endowed with a Hermitian structure (see [20, I.8]). If X is a locally compact Hausdor? space, we denote by vect(X ) the category having as objects vector bundles with base space X , and arrows vector bundle morphisms. In the present work, we will also deal with Banach bundles in the sense of [15] (i.e., bundles of Banach spaces that are not necessarily locally trivial). About the related notion of continuous ?eld of Banach spaces, we refer to [9, Chp.10].

3

2

Multipliers, C0(X ) -categories.

A category C not necessarily endowed with identity arrows is said to be a C*-category if every set of arrows (老, 考 ), 老, 考 ﹋ obj C , is a Banach space such that the composition of arrows de?nes linear maps satisfying t∩ ? t ≒ t∩ t , t ﹋ (老, 考 ), t∩ ﹋ (考, 而 ); moreover, an antilinear isometric 2 cofunctor ? : C ↙ C is assigned, in such a way that t? ? t = t . Our C*-category C is said to be unital if every 老 ﹋ obj C admits an identity arrow 1老 ﹋ (老, 老) such that t ? 1老 = t , t ﹋ (老, 考 ). Functors between C*-categories preserving the above structure are said C*-functors; in particular, we will use the term C*-monofunctor, resp. C*-autofunctor, resp. C*-epifunctor in the case in which the given C*-functor induces injective, resp. one-to-one, resp. surjective Banach space maps on the spaces of arrows. If D is a C*-subcategory of C , then we use the notation D ? C ; in such a case, every space (老, 考 )D of arrows in D is a Banach subspace of (老, 考 ). Basic references for C*-categories are [12, 18]; non-unital C*-categories have been considered in [24]. For the notions of subobjects and direct sums, we refer to [12, ∫1]. The closure for subojects of a C*-category C is the C*-category Cs with objects E = E ? = E 2 ﹋ (老, 老), 老 ﹋ obj C , and arrows (E, F ) := {t ﹋ (老, 考 ) : t = F ? t = t ? E } ; by construction, Cs has subobjects. The additive completition of C is the C*-category C+ with objects n -ples 老 := (老1 , . . . , 老n ), n ﹋ N , and arrows spaces of matrices (老, 考 ) := {(tij ) : tij ﹋ (老j , 考i )} ; by construction, C+ has direct sums (see [24, Def.2.12]). A (unital) C*-category with a single object is a (unital) C*-algebra. A C*-category with two objects is a bi-Hilbertian C*-bimodule, together with the conjugate bimodule.

2.1

Multipliers.

Let C be a C*-category. Then, every (老, 老), 老 ﹋ obj C , is a C*-algebra, and every (老, 考 ) can be endowed with the following structure of a bi-Hilbertian (考, 考 )- (老, 老)-bimodule: b, t ↙ b ? t , t, a ↙ t ? a t, t∩ 老 := t? ? t∩ ﹋ (老, 老) , t∩ , t := t∩ ? t? ﹋ (考, 考 ) ,



a ﹋ (老, 老), b ﹋ (考, 考 ), t, t∩ ﹋ (老, 考 ); in fact, t 2 = t, t 老 = t, t 考 . Let 老, 考, 缶 ﹋ obj C ; according to the above-de?ned structure, we say that a bounded linear map T : (缶, 老) ↙ (缶, 考 ) is a right (缶, 缶 ) -module operator if T (t ? a) = T (t) ? a , t ﹋ (缶, 老), a ﹋ (缶, 缶 ). In the same way, a bounded linear map T ∩ : (老, 缶 ) ↙ (考, 缶 ) is a left (缶, 缶 ) -module operator if T ∩ (a ? t) = a ? T ∩ (t), t ﹋ (老, 缶 ), a ﹋ (缶, 缶 ). A multiplier from 老 to 考 is a pair (T l , T r ), where T l : (老, 老) ↙ (老, 考 ) is a right (老, 老)-module operator and T r : (考, 考 ) ↙ (老, 考 ) is a left (考, 考 )-module operator, such that the relation b ? T l (a) = T r (b) ? a (2.1) is satis?ed for every a ﹋ (老, 老), b ﹋ (考, 考 ). We denote by M (老, 考 ) the set of multipliers from 老 into 考 . Note that (2.1) implies T l = T r . In order to economize in notation, we de?ne T a := T l (a) , bT := T r (b) , T ﹋ M (老, 考 ) , so that (2.1) can be regarded as an associativity constraint: b ? T a = bT ? a ﹋ (老, 考 ) . (2.2)

Example 2.1. Let t ﹋ (老, 考 ). We de?ne Mt a := t ? a , bMt := b ? t , a ﹋ (老, 老), b ﹋ (考, 考 ). It is clear that Mt ﹋ M (老, 考 ). For every 老 ﹋ obj C , we also de?ne 1老 a = a1老 := a , a ﹋ (老, 老). It is clear that 1老 ﹋ M (老, 老); we call 1老 the identity multiplier of 老 . 4

An involution is de?ned on multipliers, by M (老, 考 ) ↙ M (考, 老) T ↙ T ? : T ? b := (b? T )? , aT ? := (T a? )? . (2.3)

In the case 老 = 考 , a composition law can be naturally de?ned: if A, A∩ ﹋ M (老, 老), then the maps (老, 老) ? a ↙ A∩ (Aa) , (老, 老) ? a ↙ (aA)A∩ , de?ne a multiplier A∩ ? A ﹋ M (老, 老). It is now clear that every A ﹋ M (老, 老) de?nes a multiplier of (老, 老) in the usual C*-algebra sense, so that M (老, 老), endowed with the above *-algebra structure, coincides with the multiplier algebra of (老, 老), with identity 1老 . Lemma 2.1. Let C be a C*-category. For every 老, 考, 而 ﹋ obj C , there are maps M (考, 而 ) ℅ (老, 考 ) ↙ (老, 而 ) , (S, t) ↙ St := lim竹 (Se考 竹) ? t (考, 而 ) ℅ M (老, 考 ) ↙ (老, 而 ) , (t∩ , T ) ↙ t∩ T := lim竹 t∩ ? (e考 竹T ) (2.4)

where (e考 竹 )竹 ? (考, 考 ) is an approximate unit. The above maps naturally extend the composition of arrows in C . Proof. By [18, Prop.2.16], every (老, 考 ) is non-degenerate as a Hilbert bimodule w.r.t. the left (考, 考 )-action and right (老, 老)-action. Thus, every t ﹋ (老, 考 ) admits factorizations t = b ? t2 = t1 ? a , t1 , t2 ﹋ (老, 考 ), a ﹋ (老, 老), b ﹋ (考, 考 ) (see [3, Prop.1.8]). Let S ﹋ M (考, 而 ). We consider the net {(Se考 竹 ) ? t}竹 , and estimate (Se考 竹 ) ? t ? (Sb) ? t2 = S (e考 竹 ? b ? b ) ? t2 t2 ≒ S (e考 竹 ? b ? b) ≒ S e考 t2 竹 ?b?b


.

考 Since by de?nition of approximate units e考 竹 ? b ? b ↙ 0 , we conclude that {(Se竹 ) ? t}竹 converges to (Sb) ? t2 ﹋ (老, 考 ). Note that the limit is unique, and that it does not depend on the choice of the approximate unit. The same argument applies for the net {t∩ ? (e考 竹 T )} , and this proves the lemma.

We can now introduce the composition law M (考, 而 ) ℅ M (老, 考 ) ↙ M (老, 而 ) S, T ↙ S ? T : (S ? T ) a := S (T a) , c (S ? T ) := (cS )T , (2.5)

a ﹋ (老, 老), c ﹋ (而, 而 ), T ﹋ M (老, 考 ), T ∩ ﹋ M (考, 而 ). Note in fact that T a ﹋ (老, 考 ), cS ﹋ (而, 考 ), thus we can apply the previous lemma and perform the compositions S (T a), (cS )T . A C*-subcategory I ? C , with arrows (老, 考 )I ? (老, 考 ), 老, 考 ﹋ objI = objC , is said to be an ideal if t ? t∩ ﹋ (而, 考 )I , t∩∩ ? t ﹋ (老, 缶 )I for every t ﹋ (老, 考 )I , t∩ ﹋ (而, 老), t∩∩ ﹋ (考, 缶 ). If I ? C is an ideal, we write I ? C . In particular, I is said to be an essential ideal if every ideal of C has nontrivial intersection with I (i.e., (老, 考 )I ﹎ (老, 考 )J = {0} for some 老, 考 ﹋ obj C , J ? C ). Proposition 2.2. Let C be a C*-category. The category M (C ) having the same objects as C , and arrows M (老, 考 ) , 老, 考 ﹋ obj C , is a unital C*-category. Moreover, the maps {(老, 考 ) ? t ↙ Mt ﹋ M (老, 考 )} de?ned in Ex.2.1 induce a C*-monofunctor I : C ?↙ M (C ) , in such a way that I (C ) is an essential ideal of M (C ) . If C is unital, then I (C ) = M (C ) . If C is an essential ideal of a unital C*-category C ∩ , then there exists a C*-monofunctor I ∩ : C ∩ ?↙ M (C ) extending I . 5

Proof. It follows from Lemma 2.1 that M (C ) is a C*-category, with units de?ned as by Ex.2.1. It is clear that I is a C*-functor; by well-known properties of multiplier algebras the maps I : (老, 老) ↙ 2 2 M (老, 老) are isometric, thus if t ﹋ (老, 考 ), then t = t? ? t = M (t? ? t) = M (t) . This ∩ implies that I is a C*-monofunctor. Let now C be a unital C*-category as above; we denote by (老, 考 )∩ the spaces of arrows of C ∩ (by de?nition of essential ideal, we identify obj C with obj C ∩ ). For every w ﹋ (老, 考 ) we de?ne a multiplier Mw ﹋ M (老, 考 ), Mw a := w ? a , a ﹋ (老, 老) bMw := b ? w , b ﹋ (考, 考 ) (note in fact that w ? a , b ? w ﹋ (老, 考 )). In this way, we obtain a C*-functor I ∩ : C ∩ ↙ M (C ), w ↙ Mw , which reduces to I for arrows in C . In the case 老 = 考 , we ?nd that I ∩ induces C*∩ algebra morphisms I老 : (老, 老)∩ ↙ M (老, 老); such morphisms are injective by [28, Prop.3.12.8]. Thus, if w ﹋ (老, 考 )∩ , then w 2 = w? ? w = Mw? ?w = Mw 2 . In the case of C*-categories with two objects, we obtain a construction for bi-Hilbertian bimodules: Corollary 2.3. Let A , B be C*-algebras, A MB a bi-Hilbertian bimodule. Then, there exists a universal bi-Hilbertian M (A) - M (B ) -bimodule M (M) with a monomorphism I : A MB ↙ M (A) M (M)M (B) , and satisfying the universal property w.r.t. bi-Hilbertian bimodules containing M as essential ideal. Example 2.2. Let X be a locally compact, paracompact space, d ﹋ N , and E ↙ X a rank d vector bundle. We denote by 忙0 E the C0 (X )-module of continuous sections of E vanishing at in?nity, and by 忙0 (E , E ) the C*-algebra of compact operators of 忙0 E . Then, 忙0 (E , E ) is a continuous bundle of C*-algebras over X , with ?bre the matrix algebra Md ; moreover, 忙0 E naturally becomes a Hilbert 忙0 (E , E )- C0 (X )-bimodule. We consider the C*-category C with objects 老, 考 , and arrows (老, 老) := C0 (X ), (考, 考 ) := 忙0 (E , E ), (老, 考 ) := 忙0 (E ), (考, 老) := 忙? 0 E (i.e., the conjugate bimodule of 忙0 (E )). Let 忙b (E , E ) denote the C*-algebra of bounded, continuous sections of the C*-bundle associated with 忙0 (E , E ), and 忙b E the module of bounded, continuous sections of E . Then, M (C ) has arrows M (老, 老) = Cb (X ), M (考, 考 ) := 忙b (E , E ), M (老, 考 ) = 忙b E (see [1, Thm.3.3]). A di?erent approach to multiplier C*-categories can be found in [19]. Since the universal property Prop.2.2 is satis?ed by the C*-categories introduced in the above-cited reference, the two constructions provide the same result.

2.2

C0 (X ) -categories.

Let C be a C*-category. For every 老 ﹋ obj C , we denote by ZM (老, 老) the centre of the C*-algebra M (老, 老) of multipliers of (老, 老). De?nition 2.4. Let C be a C*-category, X a locally compact Hausdor? space. C is said to be a C0 (X ) -category if for every 老 ﹋ obj C there is given a non-degenerate morphism i老 : C0 (X ) ↙ M (老, 老) , such that for every 考 ﹋ obj C , t ﹋ (老, 考 ) , f ﹋ C0 (X ) , the following equality holds: t i老 (f ) = i考 (f ) t . (2.6)

The set {i老 }老﹋obj C is called C0 (X )-action. It follows from (2.6) that i老 (f ) ﹋ ZM (老, 老) for every 老 ﹋ obj C , thus (老, 老) is a C0 (X )-algebra. By de?nition, every (老, 考 ) is a C0 (X )-Hilbert (考, 考 )- (老, 老)-bimodule in the sense of Sec.1.1.1. A C*-functor 灰 : C ↙ C ∩ between C0 (X )-categories is said to be a C0 (X )-functor if 灰 (t i老 (f )) = 灰 (t) i∩ 灰 (老) (f ) for every t ﹋ (老, 考 ), 老, 考 ﹋ obj C , f ﹋ C0 (X ). In the sequel, we will drop the symbol i老 , so that (2.6) simply becomes tf = f t . 6

2.2.1

The bundle structure.

Let X be a locally compact Hausdor? space, C a C0 (X )-category, U ? X an open set. The restriction of C over U is the C*-subcategory CU having the same objects as C , and arrows the Banach spaces (老, 考 )U := CU (X )(老, 考 ) := span {f t, f ﹋ CU (X ), t ﹋ (老, 考 )} ; it is clear that CU ? C is an ideal, and that CU is a C0 (U )-category. Let now W ? X be a closed set. The restriction of C over W is de?ned as the C*-category CW having the same objects as C , and arrows the quotients (老, 考 )W := (老, 考 )/[CX ?W (X )(老, 考 )]. It is easily checked that CW is a C*-category, in fact composition of arrows and involution factorize through elements of CX ?W (X )(老, 考 ), 老, 考 ﹋ obj C . By de?nition, there is an exact sequence of C*-functors 0 ?↙ CX ?W ?↙ C ?↙ CW .
iX ?W 灰W

(2.7)

In the case in which Cb (W ) is a C0 (X )-algebra (for example, in the case of X being normal), then CW may be described as the C*-category with arrows (老, 考 ) ?X Cb (W ), where ?X is the tensor product with coe?cients in C0 (X ) de?ned as in Sec.1.1.1. Moreover, CW has an obvious structure of C (汕W )-category, where 汕W denote the Stone-Cech compacti?cation of W . De?nition 2.5. Let C be a C0 (X ) -category, x ﹋ X . The ?bre of C over x is de?ned as the restriction Cx := C{x} . The restriction C*-epifunctor 羽x : C ↙ Cx is called the ?bre functor. We denote by (老, 考 )x the spaces of arrows of Cx , x ﹋ X . Remark 2.1. Let C be a C*-category. Two objects 老, 考 ﹋ obj C are said to be unitarily equivalent if there exists u ﹋ (老, 考 ) such that u ? u? = 1考 , u? ? u = 1老 . We denote by obju C the set of unitary equivalence classes of objects of C . Let now C be a C0 (X )-category. By de?nition, the restrictions of C over open (closed) subsets of X have the same objects as C . In particular, this is true for the ?bres Cx , x ﹋ X . Now, it is clear that the ?bre functors map unitarily equivalent objects into unitarily equivalent objects. On the other side, 老, 考 may be unitarily equivalent as objects of every Cx , x ﹋ X , but this does not imply that 老, 考 are unitarily equivalent in C . Thus, the maps obju C ↙ obju Cx induced by the ?bre functors are always surjective, but not injective in general. Let F : C ↙ C ∩ be a C0 (X )-functor. Then, for every x ﹋ X there exists a C*-functor ∩ such that Fx : Cx ↙ Cx ∩ Fx ? 羽x = 羽x ?F (2.8)
∩ ∩ are the ?bre functors de?ned on C ∩ . In fact, if t, v ﹋ (老, 考 ) and 羽x (t ? v ) = 0 , : C ∩ ↙ Cx where 羽x ∩ then there exists a factorization t ? v = f w , f ﹋ Cx (X ), w ﹋ (老, 考 ), so that 羽x ? F (t ? v ) = ∩ f (x)羽x ? F (w) = 0 , and the l.h.s. of (2.8) is well-de?ned. We have the following categorical analogue of [26, Thm.2.3],[3, Cor.1.12,Prop.2.8].

Proposition 2.6. Let X be a locally compact Hausdor? space, C a C0 (X ) -category, and {羽x : C ↙ Cx }x﹋X the set of ?bre functors of C . For every 老, 考 ﹋ obj C , t ﹋ (老, 考 ) , the norm function nt (x) := 羽x (t) , x ﹋ X , is upper semicontinuous and vanishing at in?nity; moreover, t = supx﹋X 羽x (t) . Proof. Let t ﹋ (老, 考 ), 老, 考 ﹋ obj C , so that t? ? t ﹋ (老, 老). Since (老, 老) is a C0 (X )-algebra w.r.t. 1/2 the C0 (X )-action of Def.2.4, it follows from [26, Thm.2.3] that the map nt (x) = 羽x (t? ? t) , 2 ? ? x ﹋ X , is upper-semicontinuous; moreover, t = t ? t = supx 羽x (t ? t) . Let C be a C0 (X )-category. If the norm function nt is continuous for every arrow t ﹋ (老, 考 ), 老, 考 ﹋ obj C , then we say that C is a continuous bundle of C*-categories. In such a case, it follows from the above de?nition that the spaces of arrows of C are continuous ?elds of Banach spaces; 7

in particular, (老, 老) is a continuous bundle of C*-algebras for every 老 ﹋ obj C . The proof of the following lemma is trivial, thus it will be omitted. Lemma 2.7. Let X be a locally compact Hausdor? space, C a continuous bundle of C*-categories over X . Then, the closure for subobjects Cs and the additive completition C+ are continuous bundles of C*-categories. Remark 2.2. Let X be a locally compact Hausdor? space, C a continuous bundle of C*-categories over X , 老, 考 ﹋ obj C . In the de?nition of continuous ?eld by Dixmier-Douady, the vanishing-atin?nity property is not assumed for norm functions; as a consequence of this fact, if U ? X is open and W := U is the closure, then the restriction 灰W (老, 考 ) de?nes a unique continuous ?eld over U , i.e. the restriction 灰W (老, 考 )|U in the sense of [9, 10.1.7]. In order to be concise, we de?ne 灰 步 W (老, 考 ) := 灰W (老, 考 )|U . 2.2.2 Local triviality and bundle operations.

Let C? be a C*-category; we de?ne the constant bundle X C? as the C0 (X )-category having the same objects as C? , and arrows the spaces (老, 考 )X of continuous maps from X into (老, 考 ) vanishing at in?nity, 老, 考 ﹋ obj C ? (the structure of C0 (X )-category is de?ned in the obvious way). In the case in which X is compact, we consider the spaces of continuous maps from X into (老, 考 ); if t ﹋ (老, 考 ), then with an abuse of notation we will denote by t ﹋ (老, 考 )X the corresponding constant map. Let C be a C0 (X )-category, C? a C*-category. We say that C is locally trivial with ?bre C? if for every x ﹋ X there exists a closed neighborhood W ? x with a Cb (W )-isomorphism 汐W : CW ↙ W C? , in such a way that the induced map 汐W : obj C ↙ obj C? does not depend on the choice of W . The functors 汐W are called local charts. The above de?nition implies that 老? := 汐W (老), 老 ﹋ obj C , does not depend on W ; in this way, every space of arrows (老, 考 ) is a locally trivial continuous ?eld of Banach spaces with ?bre (老? , 考? ), trivialized over subsets which do not depend on the choice of 老, 考 . It is clear that we may give the analogous de?nition by using open neighborhoods, anyway for our purposes it will be convenient to use closed neighborhoods, as the corresponding local charts map unital C*-categories into unital C*-categories. Example 2.3. Let A be a C0 (X )-algebra. The C*-category with objects the projections of A , and arrows (E, F ) := {t ﹋ A : t = F t = tE } is a C0 (X )-category. Example 2.4. Let C be a unital C*-category, X a compact Hausdor? space. The extension C*category C X is de?ned as the closure for subobjects of the constant bundle X C . It turns out that C is a continuous bundle of C*-categories w.r.t. the C (X )-structure induced by X C (see [30]). Example 2.5. Let X be a locally compact Hausdor? space. Then, vect(X ) (Sec.1.1.3) is a continuous bundle of C*-categories with ?bre the category hilb of Hilbert spaces. If X is locally contractible, then vect(X ) is locally trivial; in fact, if W ? X is contractible then every vector bundle E ↙ X can be trivialized on W . If X is not locally contractible, then in general vect(X ) may be not locally trivial; in fact, it could be not possible to trivialize all the elements of vect(X ) over the same closed subset. Let X be a locally compact, paracompact Hausdor? space. We assume that there exists a locally ?nite, open cover {Xi } and, for every index i , a continuous bundle of C*-categories Ci over the closure X i , with C0 (Xij )-isofunctors 汐ij : 灰j,Xij (Cj ) ↙ 灰i,Xij (Ci ) : 汐ij ? 汐jk = 汐ik . 8

In the previous expression, 灰i,Xij denotes the restriction epifunctor of Ci over X ij . By hypothesis, the sets {obj Ci }i are in one-to-one correspondence. We want to de?ne a category C b , having the same objects as a ?xed Ck , and arrows the glueings of the continuous ?elds of Banach spaces {(汐ik (老), 汐ik (考 ))}i , 老, 考 ﹋ obj Ck , in the sense of [9, 10.1.13] (note that 汐ik (老), 汐ik (考 ) ﹋ obj Ci , thus (汐ik (老), 汐ik (考 )) is a continuous ?eld over X i ). Since in the above reference open covers are used, for every index i we consider the restriction (汐ik (老), 汐ik (考 ))|Xi , and de?ne the spaces of arrows (老, 考 )b of C b as the glueings of { (汐ik (老), 汐ik (考 ))|Xi }i w.r.t. the isomorphisms 步 i,Xij (汐ik (老), 汐ik (考 )) 汐ij : 灰 步 j,Xij (汐jk (老), 汐jk (考 )) ↙ 灰 (recall Rem.2.2). Let 老, 考 ﹋ obj C b ; by de?nition, the elements t ﹋ (老, 考 )b are in one-to-one correspondence with families (ti )i ﹋
i ∩ ∩ We de?ne an involution (ti )? := (t? i ) and a composition (ti ) ? (ti ) := (ti ? ti ), in such a way that b b C is a C*-category. Now, our de?nition does not ensure that C is a C0 (X )-category, in fact the C0 (X )-action on a continuous ?eld of Banach spaces may be degenerate in the case of X being not compact. We de?ne a C*-category C having the same objects as C b , and arrows the spaces (老, 考 ) := C0 (X )(老, 考 )b . It is clear that C is an ideal of C b . We call C the glueing of the categories Ci . By construction, C is a continuous bundle on X , satisfying the following universal property: for every index i there exists a C*-isofunctor 羽i : CXi ↙ Ci , 羽i (老) := 老i , such that ?1 羽i ? 羽j = 汐ij ;

( 汐ik (老), 汐ik (考 ) ) : 汐ij ? 灰j,Xij (tj ) = 灰i,Xij (ti ) .

(2.9)

(2.10)

?1 to be concise, in the previous equality we identi?ed 羽i (resp. 羽j ) with the restriction on CXij (resp. Cj,Xij ).

3

Tensor C*-categories.

Let T be a unital C*-category. As customary, we call tensor product an associative, unital C*bifunctor ? : T ℅ T ↙ T admitting an identity object 牝 ﹋ obj T , in such a way that 牝 ? 老 = 老 ? 牝 = 老 , 老 ﹋ obj T . In such a case we say that T is a tensor C*-category, and use the notation (T , ?, 牝). For brevity, we adopt the notation 老考 := 老 ? 考 , so that if t ﹋ (老, 考 ), t∩ ﹋ (老∩ , 考 ∩ ), then t ? t∩ ﹋ (老老∩ , 考考 ∩ ). Thus, if we pick 老∩ = 考 ∩ = 牝 , then we obtain that every space of arrows (老, 考 ) is a (牝, 牝)-bimodule. In particular, t ? 1牝 = 1牝 ? t = t , t ﹋ (老, 考 ). Note that (牝, 牝) is an Abelian C*-algebra with identity 1牝 . For basic properties of tensor C*-categories, we will refer to [12, 23]. For every 老 ﹋ obj T , we denote by 老 the tensor C*-category with objects 老r , r ﹋ N , and arrows (老r , 老s ); we call 老 the generating object of 老 . Duals of locally compact (quantum) groups are well-known examples of tensor C*-categories. Another example, which will be of particular interest in the present paper, is the category vect(X ), with X compact (Sec.1.1.3). The tensor product on vect(X ) is de?ned as in [2, ∫1],[20, ∫I.4], and will be denoted by ?X (no confusion should arise with the C (X )-algebra tensor product); note that (牝, 牝) = C (X ). Example 3.1. Let A be a C*-algebra. Then, the set endA is endowed with a natural structure of tensor C*-category: the arrows are given by elements of the intertwiner spaces (老, 考 ) := {t ﹋ A : t老(a) = 考 (a)t, a ﹋ A} , 老, 考 ﹋ endA , and the tensor structure is de?ned by composition of endomorphisms (see [14, ∫1]). 9

3.0.3

DR-dynamical systems.

Let 老 ﹋ obj T . Then, a universal C*-dynamical system is associated with 老 , in the following way r ?s (see [12, ∫4] for details). For every r, s ﹋ N , we consider the inductive limit Banach space O老 r s r +1 s+1 associated with the sequence ir,s : (老 , 老 ) ?↙ (老 老 ), ir,s (t) := t ? 1老 , r ﹋ N ; in particular, 0 O老 is the inductive limit C*-algebra associated with the sequence obtained for r = s . Composition k . Now, there exists a of arrows and involution induce a *-algebra structure on 0 O老 := k﹋Z O老 0 unique C*-norm on O老 such that the circle action
k z (t) := z k t , z ﹋ T , t ﹋ O老 , k﹋Z

(3.1)

extends to an (isometric) automorphic action. We denote by O老 the corresponding C*-algebra, called the Doplicher-Roberts algebra associated with 老 (DR-algebra, in the sequel). A canonical endomorphism 老? ﹋ endO老 can be de?ned, 老? (t) := 1老 ? t , t ﹋ (老r , 老s ) . (3.2)

We denote by (O老 , T, 老? ) the so-constructed C*-dynamical system. By construction, 老? ? z = z ? 老? , s z ﹋ T ; moreover, (老r , 老s ) ? (老r ? , 老? ), r, s ﹋ N , so that there is an inclusion of tensor C*-categories s 老 ?↙ 老? . We say that 老 is amenable if 老 = 老? , i.e. (老r , 老s ) = (老r ? , 老? ), r, s ﹋ N (see [23, ∫5]). The above construction is universal, in the following sense: if 羽 : 老 ↙ 考 is functor of tensor C*-categories, then there exists a C*-algebra morphism 羽 : O老 ↙ O考 , 羽 (t) := 羽 (t) , t ﹋ (老r , 老s ) (3.3)

s such that 考? ? 羽 = 羽 ? 老? and z ? 羽 = 羽 ? z , z ﹋ T . If 羽 is a C*-epifunctor, then 羽 (老r ? , 老? ) ? s r s s r ) and 羽 is a C* -epimorphism. , 考 ) = ( 考 , 老 ); if 考 is amenable, then 羽 (老r , 考? (考? ? ? ? ?

Example 3.2. Let hilb denote the tensor C*-category of ?nite dimensional Hilbert spaces; if 老 ﹋ obj hilb , then O老 is the Cuntz algebra Od , where d is the rank of 老 ([6]). If E ↙ X is a rank d vector bundle, then the associated DR-algebra is the Cuntz-Pimsner algebra OE associated with the module of continuous sections of E . It turns out that OE is a locally trivial bundle with ?bre Od (see [30, ∫4], [31, 33]). 3.0.4 Tensor categories as C (X ) -categories.

Let (T , ?, 牝) be a tensor C*-category. We de?ne X 牝 as the spectrum of the Abelian C*-algebra {f ﹋ (牝, 牝) : f ? 1老 = 1老 ? f ﹋ (老, 老) , 老 ﹋ obj T } ; (3.4)

note that (3.4) has identity 1牝 , thus X 牝 is compact. In the sequel, we will identify C (X 牝 ) with the C*-algebra (3.4). Proposition 3.1. Every tensor C*-category (T , ?, 牝) has a natural structure of tensor C (X 牝 ) category, i.e. ? is a C (X 牝 ) -bifunctor. Proof. For every 老 ﹋ obj T , we de?ne the map i老 : C (X 牝 ) ↙ (老, 老), i老 (f ) := f ? 1老 . Since i老 (1牝 ) = 1牝 ? 1老 = 1老 , it is clear that i老 is a non-degenerate C*-algebra morphism. If t ﹋ (老, 考 ), then (1考 ? f ) ? t = t ? f = t ? (1老 ? f ); thus i考 (f ) ? t = t ? i老 (f ), and T is a C (X 牝 )-category. The fact that ? preserves the C (X 牝 )-action follows from the obvious identities (1考考∩ ? f ) ? (t ? t∩ ) = (t ? f ? t∩ ) = (t ? t∩ ) ? f = (t ? t∩ ) ? (1老老∩ ? f ) , f ﹋ C (X 牝 ), t ﹋ (老, 考 ), t∩ ﹋ (老∩ , 考 ∩ ). 10

Remark 3.1. According to the notation of Sec.2.2, in the sequel we will write f t := f ?t , f ﹋ C (X 牝 ), t ﹋ (老, 考 ). Let now U ? X 牝 be an open set; if f ﹋ CU (X 牝 ), t ﹋ (老, 考 ), t∩ ﹋ (老∩ , 考 ∩ ), then f t ﹋ (老, 考 )U , and it is clear that f t ? t∩ ﹋ (老老∩ , 考考 ∩ )U . It follows from Prop.2.6 that for every x ﹋ X 牝 there exists a ?bre functor 羽x : T ↙ Tx . A structure of tensor C*-category is de?ned on Tx , by assigning 老x 考x := (老考 )x 羽x (t) ?x 羽x (t∩ ) := 羽x (t ? t∩ ) (3.5)

It is easy to prove that ?x is well-de?ned; in fact, if v ﹋ (老, 考 ), v ∩ ﹋ (老∩ , 考 ∩ ) and 羽x (v ) = 羽x (t), 羽x (v ∩ ) = 羽x (t∩ ) (i.e., t ? v ﹋ (老, 考 )X ?{x} , t∩ ? v ∩ ﹋ (老∩ , 考 ∩ )X ?{x} ), then t ? v = f w , t∩ ? v ∩ = f ∩ w∩ for some f, f ∩ ﹋ Cx (X ), so that t ? t∩ ? v ? v ∩ = f w ? v ∩ + t ? f ∩ w∩ , and 羽x (t ? t∩ ) ? 羽x (v ? v ∩ ) = 0 . This also proves that the ?bre functors 羽x , x ﹋ X , preserve the tensor product. More generally, the above argument applies for the restriction TW over a closed W ? X 牝 : a tensor product ?W : TW ℅ TW ↙ TW is de?ned, in such a way that the restriction epimorphism 灰W : T ↙ TW preserves the tensor product. We say that T is a continuous bundle of tensor C*-categories if it is a continuous bundle w.r.t. the above C (X 牝 )-category structure. As a consequence of the above considerations, we obtain a simple result on the structure of the DR-algebra associated with an object 老 ﹋ obj T . The proof is trivial, therefore it will be omitted; note that we make the standard assumption that the map t ↙ t ? 1老 , t ﹋ (老r , 老s ), r, s ﹋ N , is isometric. Proposition 3.2. Let T be a tensor C*-category (resp. a continuous bundle of tensor C*categories), 老 ﹋ obj T . Then, O老 is a C (X 牝 ) -algebra. In particular, if 老 is a (locally trivial) continuous bundle, then O老 is a (locally trivial) continuous bundle of C*-algebras. A tensor C*-category (T , ?, 牝) is said to be symmetric if for every 老, 考 ﹋ obj T there exists a unitary 汍(老, 考 ) ﹋ (老考, 考老) implementing the ?ip 汍(考, 考 ∩ ) ? (t ? t∩ ) = (t∩ ? t) ? 汍(老, 老∩ ) , (3.6)

t ﹋ (老, 考 ), t∩ ﹋ (老∩ , 考 ∩ ), and satisfying certain natural relations (see [12, ∫1]). In essence, the notion of symmetry expresses commutativity of the tensor product. It is well-known that duals of locally compact groups are symmetric tensor C*-categories. Remark 3.2. Let (T , ?, 牝, 汍) be a symmetric tensor C*-category. By de?nition, C (X 牝 ) ? (牝, 牝). On the converse, if z ﹋ (牝, 牝), 老 ﹋ obj T , then by (3.6) we ?nd z ? 1老 = 汍(老, 牝) ? (1老 ? z ) ? 汍(牝, 老); since 汍(老, 牝) = 汍(牝, 老) = 1老 , we conclude z ﹋ C (X 牝 ) and (牝, 牝) = C (X 牝 ). For every x ﹋ X 牝 , we denote by 羽x : T ↙ Tx the ?bre epifunctor associated with T as a C (X 牝 )-category. By (3.5), every 羽x preserves the tensor product; moreover, by de?ning 汍x (老x , 考x ) := 羽x (汍(老, 考 )), we obtain that every Tx is symmetric. By de?nition, the ?bre functor 羽x preserves the symmetry. Let 老 ﹋ obj T . By [12, Appendix], for every r ﹋ N there is a unitary representation of the permutation group of r objects, which we denote by Pr : 汍老 : Pr ↙ (老r , 老r ) , p ↙ 汍老 (p) . For every r ﹋ N , we introduce the antisymmetric projection P老,汍,r := 1 r! sign(p) 汍老 (p) .
p﹋Pr

(3.7)

Let 老? ﹋ endO老 be the canonical endomorphism. It follows from (3.6) that 老? is §approximately inner§, in the sense that 老? (t) = 1老 ? t = 汍老 (s, 1) t 汍老 (1, r) , t ﹋ (老r , 老s ) 11 (3.8)

(recall that t is identi?ed with t ? 1老 in O老 ). For brevity (and coherence with (3.7)), in (3.8) we used the notation 汍老 (r, s) := 汍(老r , 老s ), r, s ﹋ N . Let (T , ?, 牝, 汍) be a symmetric tensor C*-category. We denote by aut汍 T the set of C*-autofunctors 汐 of T satisfying 汐(老) = 老 , 汐(t ? t∩ ) = 汐(t) ? 汐(t∩ ) , 汐(汍(老, 考 )) = 汍(老, 考 ) , (3.9)

t ﹋ (老, 考 ), t∩ ﹋ (老∩ , 考 ∩ ), 汍(老, 考 ) ﹋ (老考, 考老). In particular, let us consider the case T = 老 for some object 老 . We note that in order to obtain the third of (3.9) it su?ces to require 汐(汍老 (1, 1)) = 汍老 (1, 1). Moreover, z ﹋ aut汍 老 for every z ﹋ T , where z is de?ned by (3.1). We denote by aut汍 O老 ? autO老 the closed group of automorphisms commuting with 老? , and leaving every 汍老 (r, s) ?xed, r, s ﹋ N . If 老 is amenable in the sense of Sec.3.0.3, then by universality of O老 we obtain an identi?cation aut汍 老 = aut汍 O老 . (3.10) In fact, every C*-autofunctor 汐 ﹋ aut汍 老 de?nes an automorphism of O老 as in (3.3); on the converse, every 汕 ﹋ aut汍 O老 de?nes by amenability a C*-autofunctor of 老 . In the sequel we will regard aut汍 老 as a topological group, endowed with the pointwise-convergence topology de?ned on aut汍 O老 . The following notion will play an important role in the sequel. De?nition 3.3. Let (T , ?, 牝, 汍) be a symmetric tensor C*-category. An embedding functor is a C*-monofunctor i : T ?↙ vect(X 牝 ) , preserving tensor product and symmetry. The following de?nition appeared in [12, ∫2]. A tensor C*-category (T , ?, 牝) has conjugates if for every 老 ﹋ obj T there exists 老 ﹋ obj T with arrows R ﹋ (牝, 老老), R ﹋ (牝, 老老), such that the conjugate equations hold: (R ? 1老 ) ? (1老 ? R) = 1老 , (R? ? 1老 ) ? (1老 ? R) = 1老 . The notion of conjugation is deeply related with the one of dimension. Let us de?ne d(老) = R? ? R ﹋ (牝, 牝) , d(老) > 0 ; (3.12)
?

(3.11)

then, it turns out that d(老) is invariant w.r.t. unitary equivalence and conjugation ([12, ∫2]). If T is symmetric, then (with some assumptions) d(老) = d1牝 for some d ﹋ N . In the case T = hilb , then d is the dimension in the sense of Hilbert spaces. For further investigations about the notion of dimension, we refer the reader to [23, 18]. Example 3.3 (Group duality in the Cuntz algebra). The following construction appeared in [11]. Let Hd be the canonical rank d Hilbert space. For every r, s ﹋ N , we denote by (H r , H s ) the Banach space of linear maps from H r into H s ; it is clear that (H r , H s ) is isomorphic to the r , r ﹋ N , and arrows (H r , H s ) is a matrix space Mdr ,ds . The category Hd with objects Hd tensor C*-category, when endowed with the usual matrix tensor product. Hd is symmetric: we r +s r +s r s have the operators 牟(r, s) ﹋ (Hd , Hd ), 牟(r, s)v ? v ∩ := v ∩ ? v , v ﹋ Hd , v ∩ ﹋ Hd . Moreover aut牟 Hd = aut牟 Od ? U(d) ([11, Cor.3.3 , Lemma 3.6]). The DR-dynamical system associated with Hd is given by (Od , T, 考d ), where Od is the Cuntz algebra endowed with the gauge action T ↙ autOd and the canonical endomorphism 考d ﹋ endOd . Let G ? U(d) be a closed group. r Then, every tensor power Hd , r ﹋ N , is a G -module w.r.t. the natural action g, 肉 ↙ gr 肉 , g ﹋ G , r gr := g ? . . . ? g , 肉 ﹋ Hd . We denote by G the subcategory of Hd with arrows the spaces of equivariant maps (H r , H s )G := {t ﹋ (H r , H s ) : gs ? t = t ? gr , g ﹋ G} . (3.13)

12

It is well-known that the dual of G (i.e., the category of ?nite dimensional representations) is recovered by extending G w.r.t. direct sums. An automorphic action G ↙ autOd can be constructed, by de?ning ? g ﹋ autOd : g (t) := gs ? t ? gr , g ﹋ G, t ﹋ (H r , H s ) . (3.14) Thus, every (H r , H s )G coincides with the ?xed-point space w.r.t. (3.14). We denote by OG the C*-subalgebra of Od generated by (H r , H s )G , r, s ﹋ N ; it turns out that OG is the ?xed-point algebra w.r.t. (3.14). Since 考d ? g = g ? 考d , g ﹋ G , we obtain that 考d restricts to an endomorphism r s 考G ﹋ endOG . It turns out that (H r , H s )G = (考G , 考G ), r, s ﹋ N ; in other terms, we ?nd G = 考G (in particular, for G = {1} , we have Hd = 考d ). Moreover, the stabilizer autOG Od of OG in Od is isomorphic to G via (3.14). Thus, G and the associated tensor C*-category G are recovered by properties of the C*-dynamical system (OG , 考G ), together with the inclusion OG ?↙ Od . Let (T , ?, 牝, 汍) be a symmetric tensor C*-category. 老 ﹋ obj T is said to be a special object if there exists d ﹋ N , V ﹋ (牝, 老d ) such that V ? ? V = 1牝 , V ? V ? = P老,汍,d (V ? ? 1老 ) ? (1老 ? V ) = (?1)d?1 d?1 1老 . (3.15)

It turns out that V is unique up to multiplication by elements of T (see [12, ∫3]). In the sequel, we will denote the above data by (老, d, V ). It is proved that the integer d coincides with the dimension of 老 ([12, Lemma 3.6]). Special objects play a pivotal role in the Doplicher-Roberts duality theory. In fact, tensor categories generated by special objects are in one-to-one correspondence with duals of compact Lie groups, in the case (牝, 牝) ? C . Since in the sequel we will make use of such concepts at a detailed level, in the next Lemma an abstract duality is summarized for special objects. Lemma 3.4. Let (老, d, V ) be a special object, and suppose (牝, 牝) ? C . Then, the following properties are satis?ed: 1. 老 is amenable, and there is a C*-monomorphism i : (O老 , 老? ) ?↙ (Od , 考d ) ; 2. the above monomorphism de?nes an embedding functor i? : 老 ?↙ Hd , in the sense of Def.3.3; 3. there is a closed group G ? SU(d) such that i? (老) = G , so that G , 老 are isomorphic as symmetric tensor C*-categories; 4. i restricts to an isomorphism (O老 , 老? ) ? (OG , 考G ) ; 5. G is isomorphic to the stabilizer of OG in Od w.r.t. the action (3.14); 6. if 汐 ﹋ aut汍 老 , then there is a unitary u ﹋ U(d) such that i ? 汐 = u ? i (where u ﹋ aut牟 Od is de?ned as in (3.14)). Proof. Point 1 is [12, Lemma 4.14]. Points 2,3,4,5 are proved in [12, Thm.4.17]. About point 6, let 汐 ﹋ aut汍 老 . Since 老? (V ) = 汍老 (d, 1)V (recall (3.8)), we obtain 老? (V ? 汐(V )) = V ? 汍老 (1, d)汐(汍老 (d, 1)V ) = V ? 汐(V ) . Thus, V ? 汐(V ) is 老? -invariant, and V ? 汐(V ) ﹋ (牝, 牝) = C (see [12, Lemma 4.13]). Now, 汐(P老,汍,d ) = P老,汍,d , so that 1= V
2

≡ V ? 汐(V ) = V

V ? 汐(V ) 13

2 汐(V ? ) ≡ P老,汍,d =1,

we conclude that V ? 汐(V ) = z老 , z老 ﹋ T . Thus, 汐(V ) = P老,汍,d 汐(V ) = V V ? 汐(V ) = z老 V . Let now 竹老 ﹋ T , 竹d 老 = z 老 . We de?ne the automorphism 汕 := 竹老 ? 汐 ﹋ aut汍 老 , in such a way that 汕 (V ) = 竹老 ? 汐(V ) = 竹d 老 z老 V = V . By [14, Cor.4.9(d)], we obtain that there is v ﹋ SU(d) such that i ? 汕 = v ? i . By de?ning u := 竹老 v ﹋ U(d) , we obtain u ? i = i ? 汐 . 3.0.5 Twisted special objects.

Note that V ? V ∩ ﹋ (牝, 牝), thus V is endowed with a natural structure of right Hilbert (牝, 牝)-module. It is clear that VV ? may be identi?ed with the C*-algebra of compact, right (牝, 牝)-module operators on V , and that P老,汍,d is the identity of V . The map i : (牝, 牝) ↙ K (V ), i(f ) := f ? P老,汍,d de?nes a left (牝, 牝)-action on V ; note that (3.16(3)) implies that i is surjective. From (3.16(3)) we also obtain that V is full, in fact V ? ? V = V ? V ? = fV ? P老,汍,d , V ﹋ V , fV ﹋ (牝, 牝), so that every positive element of (牝, 牝) appears as the square of the norm of an element of V . Moreover, i(f )V = f ? V = 0 , V ﹋ V , implies f ? V ? V = 0 , i.e. f fV = 0 , fV := V ? V ﹋ (牝, 牝); since at varying of V in V we obtain all the positive elements of (牝, 牝), we ?nd that f = 0 , and the left action i is injective. We conclude that V is a bi-Hilbertian (牝, 牝)-bimodule. Lemma 3.5. If existing, the Hilbert (牝, 牝) -bimodule V is unique. Moreover, there exists a unique up-to-isomorphism line bundle L老 ↙ X 牝 such that V is isomorphic to the module of continuous sections of L老 . Proof. The generalised Serre-Swan theorem proved in [8] implies that V is the module of continuous sections of a Hilbert bundle L老 ↙ X 牝 ; since K (V ) ? C (X 牝 ), we conclude that L老 has rank one, i.e. it is a line bundle. We now pass to prove the unicity; we consider a ?nite set of generators ? d ? {Vi } ? V , so that i Vi ? Vi = 1牝 . If W ? (牝, 老 ) is another (牝, 牝)-bimodule i Vi ? Vi = P老,汍,d , satisfying (3.16) and W ﹋ W , then W = P老,汍,d ? W . This implies W = i Vi ? (Vi? ? W ), with Vi? ? W ﹋ (牝, 牝), thus W ﹋ V . This proves W = V . The set of transition maps associated with L老 ↙ X 牝 de?nes a T -cocycle in H 1 (X 牝 , T). The isomorphism H 1 (X 牝 , T) ? H 2 (X 牝 , Z) allows one to give the following de?nition. 14

Lemma 3.4 implies that tensor C*-categories generated by special objects correspond to duals of compact Lie groups G ? SU(d) , in the case (牝, 牝) ? C . In particular, by performing the embedding 老 ?↙ Hd , we obtain that V corresponds to a normalized generator of the totally antisymmetric tensor product _d Hd . In the case in which (牝, 牝) is nontrivial, the notion of special object is too narrow to cover all the interesting cases. For example, let us consider the category vect(X ), where X is a compact Hausdor? space: if E ↙ X is a rank d vector bundle, then it is well-known that the totally antisymmetric tensor product _d E is a line bundle, in general non-trivial. The cohomological obstruction to get triviality of _d E is encoded by the ?rst Chern class of E . In the particular case in which _d E is trivial, then we may ?nd a (normalized) section, which plays the right role in the de?nition of special object. Since our duality theory will be modeled on vect(X ), it becomes natural to generalize the notion of special object, in such a way to include the above-described case. Let T be a symmetric tensor C*-category. A twisted special object is given by a triple (老, d, V ), where 老 ﹋ obj T , d ﹋ N , and V ? (牝, 老d ) is a closed subspace such that for every V, V ∩ ﹋ V , f ﹋ (牝, 牝), ? ? f ?V =V ?f ﹋V (V ? ? 1老 ) ? (1老 ? V ∩ ) = (?1)d?1 d?1 (V ? ? V ∩ ) ? 1老 (3.16) ? VV ? := span {V ∩ ? V ? : V, V ∩ ﹋ V老 } = (牝, 牝) ? P老,汍,d .

De?nition 3.6. The Chern class of a twisted special object (老, d, V ) is the unique element c(老) ﹋ H 2 (X 牝 , Z) associated with L老 . By de?nition, usual special objects are exactly those with Chern class c(老) = 0 ; in such a case, V is generated as a (牝, 牝)-module by an isometry V ﹋ (牝, 老d ). Let now W ? X 牝 ; we denote by 灰W : T ↙ TW the restriction epimorphism; by using the argument of Rem.3.2, for every 老, 考 ﹋ obj T we de?ne 汍W (老, 考 ) := 灰W (汍(老, 考 )), and conclude that (TW , ?W , 灰W (牝), 汍W ) is symmetric. Corollary 3.7. For every x ﹋ X 牝 , there exists a closed neighborhood W ? X 牝 , W ? x , such that 灰W (老) is a special object. Proof. It su?ces to pick a closed neighborhood W trivializing L老 : this implies the existence of a normalized section of L老 |W , which appears as an element VW ﹋ 灰W (V ) ? 灰W (牝, 老d ) such that ? ? VW ? VW = 灰W (1牝 ), VW ? VW = 灰W (P老,汍,d ). Let A be a C*-algebra. Twisted special objects in endA have been studied in [32, ∫4], and have been called special endomorphisms. Lemma 3.8. Let (老, d, V ) be a twisted special object. Then 老 is amenable, and 老 is a continuous bundle of tensor C*-categories with base space X 牝 . For every x ﹋ X 牝 , there exists a compact Lie group G(x) ? SU(d) such that the ?bre 老x is isomorphic to G(x) . Proof. As ?rst, we prove that 老 is amenable. Let {Xi } be an open cover trivializing L老 , with a subordinate partition of unity {竹i } . We denote by 老i the restriction of 老 on the closure X i , and by 灰i : 老 ↙ 老i the restriction epifunctor. Since L老 |Xi is trivial, we ?nd that (老i , d, Vi ) is a special object, where Vi ﹋ 灰i (V ) ? (牝, 老d i ) is a suitable partial isometry. By [12, Lemma 4.14] we ?nd that s s , 老 ) = (老r 老i is amenable, i.e. (老r i,? i,? i , 老i ), r, s ﹋ N . This implies that 灰i induces a C*-epimorphism s r s 灰i : (O老 , 老? ) ↙ (O老i , 老i,? ). Let t ﹋ (老r ? , 老? ); for every index i , we consider 灰i (t) ﹋ (老i,? , 老i,? ). s r s r s Since 老i is amenable, we conclude that 灰i (t) ﹋ (老r , 老 ). By construction, 灰 : ( 老 , 老 ) ↙ ( 老 , i i i i 老i ) is the restriction on X i of the continuous ?eld of Banach spaces (老r , 老s ); by the Tietze theorem [9, 10.1.12], there is ti ﹋ (老r , 老s ) such that 灰i (ti ) = 灰i (t). Now, we have t = i 竹i ti ; since r s r s 竹 t ﹋ ( 老 , 老 ), we conclude that t ﹋ ( 老 , 老 ), and 老 is amenable. We now prove that 老 is a i i i continuous bundle. Since 老 is a twisted special object, we ?nd that 老? ﹋ endO老 is a quasi-special endomorphism in the sense of [32, Def.4.10], with the additional property that the permutation symmetry [32, Def.1.1,∫4.1] is satis?ed. Thus, by applying [32, Thm.5.1,Cor.5.2], we ?nd that O老 is a continuous bundle, with ?bres isomorphic to OG(x) , x ﹋ X , where each G(x) ? SU(d) is a compact Lie group. The same results also imply that 老 is a continuous bundle with ?bres G(x) , x ﹋ X. Let (老, d, V ) be a twisted special object. Then, V appears as a Hilbert (牝, 牝)-bimodule in O老 , so that an inner endomorphism 糸 ﹋ endO老 is de?ned, in the sense of [31, ∫3]. In explicit terms, if {肉l } ? V is a ?nite set of generators for V , then 糸 (t) :=
l ? 肉l t肉l , t ﹋ O老 .

(3.17)

Since z (肉l ) = z d 肉l , z ﹋ T , we ?nd that 糸 commutes with the circle action, i.e. 糸 ? z = z ? 糸 , k k z ﹋ T . Thus, in particular we ?nd 糸 (O老 ) ? O老 , k ﹋ Z.

15

3.0.6

Continuity of tensor C*-categories.

The following notion is a generalization of the analogue in [12, ∫3]. We say that a symmetric tensor C*-category (T , ?, 牝, 汍) is T-specially directed if every object 老∩ ﹋ obj T is dominated by a twisted special object 老 ﹋ obj T , i.e. there exist orthogonal partial isometries Si ﹋ (老∩ , 老ni ), ? i = 1, 2, . . . , m , with 1老∩ = i Si ? Si . If T has direct sums, subobjects and conjugates, then T is T-specially directed, with the additional property that every 老∩ is dominated by a (non-twisted) special object (see the proof of [12, Thm.3.4]). Theorem 3.9. Let (T , ?, 牝, 汍) be a symmetric tensor C*-category. Then, C (X 牝 ) coincides with (牝, 牝) . If T is T-specially directed, then T is a continuous bundle of symmetric tensor C*-categories over X 牝 . Proof. In Rem.3.2, we veri?ed that C (X 牝 ) = (牝, 牝), thus it remains to prove that T is a continuous bundle. At this purpose, we consider 老, 考 ﹋ obj T , t ﹋ (老, 考 ), and prove the continuity of the 1/2 norm function nt (x) := 羽x (t) , x ﹋ X 牝 . Since nt (x) = 羽x (t? ? t) , with t? ? t ﹋ (老, 老), it su?ces to verify the continuity of nt only for arrows that belong to (老, 老), 老 ﹋ obj T . By Lemma 3.8, the norm function nt is continuous for every t ﹋ (老r , 老s ), r, s ﹋ N , where 老 is a twisted special object. Since there is an obvious inclusion 老n ?↙ 老 , the norm function remains continuous for arrows between tensor powers of twisted special objects. This implies that if we consider the full C*-subcategory T sp of T with objects tensor powers of twisted special objects, then T sp is a continuous bundle of C*-categories over X 牝 . By Lemma 2.7, we conclude that (T sp )s,+ is a continuous bundle of C*-categories. Since T is T -specially directed, we ?nd that every 老∩ ﹋ obj T is an object of (T sp )s,+ , in fact 老∩ is the direct sum of subobjects of 老ni , i = 1, . . . , m , where 老 is a twisted special object. We conclude that the norm function of 老∩ is continuous, and the theorem is proved. Remark 3.3. An analogue of the previous theorem has been proved by P. Zito ([36, ∫3]), in the setting of 2 -C*-categories with conjugates. In the above-cited result, no symmetry property of the tensor product is assumed. It is not di?cult to prove that the bundle structure constructed by Zito coincides with the one of Thm.3.9, in the common case of symmetric tensor C*-categories with conjugates. Remark 3.4. The ?bre epifunctors 羽x , x ﹋ X 牝 , considered in the previous theorem preserve symmetry and tensor product, thus every ?bre Tx has conjugates; moreover, 羽x (牝, 牝) ? C for every x ﹋ X . By closing each Tx w.r.t. subobjects and direct sums, we obtain a tensor C*-category satisfying the hypothesis of [12, Thm.6.1], which turns out to be isomorphic to the dual of a compact group G(x) . In particular, for each pair 老, 考 ﹋ obj T , the ?bres of the continuous ?eld (老, 考 ) are the ?nite-dimensional vector spaces (H老,x , H考,x )G(x) of G(x) -equivariant operators between Hilbert spaces H老,x , H考,x , x ﹋ X 牝 . 3.0.7 Local Triviality.

In the next de?nition, we give a notion of local triviality for a symmetric tensor C*-category, compatible with the bundle structure of Thm.3.9. Let (T? , ?? , 牝? , 汍? ) be a symmetric tensor C*category, X a compact Hausdor? space; then, the constant bundle X T? (Sec.2.2) has a natural structure of symmetric tensor C*-category,
∩ ∩ X (t ?X ? t )(x) := t(x) ?? t (x) , 汍? (老, 考 ) := 汍? (老, 考 ) ,

(3.18)

16

t, t∩ ﹋ (老, 考 )X , 老, 考 ﹋ obj T? . Let now (T , ?, 牝, 汍) be a symmetric tensor C*-category. The ?bres Tx , x ﹋ X 牝 , are symmetric tensor C*-categories having the same set of objects as T , in such a way that the ?bre epifunctors 羽x : T ↙ Tx induce the identity map obj T ↙ obj Tx √ obj T (Rem.2.1); moreover, 羽x preserves tensor product and symmetry (Rem.3.2), and every Tx , x ﹋ X 牝 , has simple identity object. De?nition 3.10. Let (T , ?, 牝, 汍) be a symmetric tensor C*-category, endowed with the natural C (X 牝 ) -category structure. T is said to be locally trivial if it is locally trivial in the sense of Sec.2.2.2, with the additional property that the local charts preserve tensor product and symmetry. In explicit terms, there is a symmetric tensor C*-category (T? , ?? , 牝? , 汍? ) with simple identity object, and a cover {Xi ? X 牝 } of closed neighborhoods with C*-isofunctors 羽i : TXi ↙ Xi T? satisfying ? ? 羽i (老) = 羽j (老) ∩ i 羽i (t ? t∩ ) = 羽i (t) ?X (3.19) ? 羽i (t ) ? Xi 羽i (汍(老, 考 )) = 汍? (老, 考 ) where 老, 考 ﹋ obj T , t ﹋ (老, 考 ), t∩ ﹋ (老∩ , 考 ∩ ). If X is a compact Hausdor? space, we denote by sym(X, T? ) (3.20)

the set of isomorphism classes of locally trivial, symmetric tensor C*-categories with ?bres isomorphic to T? , and such that the space of intertwiners of the identity object is isomorphic to C (X ). Let (老? , ?? , 牝? , 汍? ) be a symmetric tensor C*-category with generating object 老? , such that (牝? , 牝? ) ? C . If (T , ?, 牝, 汍) ﹋ sym(X, 老? ), then T has the same objects as 老? ; in order to avoid confusion, we will denote by 老 the object of T corresponding to 老? , so that T is generated by the tensor powers of 老 , i.e. T = 老 . In the following lines, we regard the topological group aut汍? 老? (3.9) as the §structure group§ for a locally trivial, symmetric tensor C*-category. Let K be a topological group. A K -cocycle is given by a pair ({Xi } , {gij }), where {Xi } is a ?nite open cover of X and gij : Xij ↙ K are continuous maps such that gij (x)gjk (x) = gik (x), x ﹋ Xijk . We say that cocycles ∩ }) are equivalent if there are continuous maps uil : Xi ﹎ Xl∩ ↙ K such ({Xi } , {gij }), ({Xl∩ } , {glm ∩ ∩ that gij (x)ujm (x) = uil (x)glm (x), x ﹋ Xij ﹎ Xlm . The set of equivalence classes of K -cocycles is 1 1 denoted by H (X, K ). It is well-known that H (X, K ) classi?es the principal K -bundles over X ([17, Chp.4]). Lemma 3.11. Let 老? be amenable. Then, there is a one-to-one correspondence sym(X, 老? ) ? H 1 (X, aut汍? 老? ) . Proof. Let (老, ?, 牝, 汍) ﹋ sym(X, 老? ). Then, there are local charts 羽i : 老Xi ↙ Xi 老? , where {Xi ? X } is a cover of closed neighborhoods. Let Xij = ? ; then, every 羽i restricts in a natu?1 ; ral way to a local chart 羽i,ij : 老Xij ↙ Xij 老? . We de?ne 汐ij : Xij 老? ↙ Xij 老? , 汐ij := 羽i,ij ? 羽j,ij by (3.19), we obtain 汐ij (t ?? ij t∩ ) = 汐ij (t) ?? ij 汐ij (t∩ ) , 汐ij (汍? ij (r, s)) = 汍? ij (r, s) ,
∩ r s s t ﹋ (老r ? , 老? )Xij , t ﹋ (老? , 老? )Xij . Thus, 汐ij ﹋ aut汍? (Xij 老? ). Now, the DR-algebra associated with 老? (regarded as an object of Xij 老? ) is the trivial ?eld C (Xij ) ? O老? ; we denote by 羽x : C (Xij ) ? O老? ↙ O老? the evaluation epimorphism assigned for x ﹋ Xij . By (3.10), we may regard
∩ ∩

X

X

X

X

17

汐ij as an element of aut老? ,汍? (C (Xij ) ? O老? ). In particular, 汐ij is a C (Xij )-automorphism, thus we may regard 汐ij as a continuous map 汐ij : Xij ↙ aut老? ,汍? O老? , [汐ij (x)](t) := 羽x ? 汐ij (1ij ? t) ,
?1 = where t ﹋ O老? , x ﹋ Xij , and 1ij ﹋ C (Xij ) is the identity. Now, the obvious identity 羽i,ijk ? 羽k,ijk ?1 ?1 羽i,ijk ? 羽j,ijk ? 羽j,ijk ? 羽k,ijk implies 汐ik = 汐ij ? 汐jk (over Xijk ); thus, by applying (3.10), we conclude that the set {汐ij } de?nes an aut汍? 老? -cocycle. By choosing another set of local charts ∩ ∩ ∩ 羽h : 老 ↙ Yh 老? , we obtain a cocycle 汕hk := 羽h,hk ? (羽k,hk )?1 which is equivalent to {汐ij } , in fact ?1 ?1 ∩ ? 羽i . If (老∩ , ?∩ , 牝∩ , 汍∩ ) is a symmetric 汕hk = Vhi ? 汐ij ? Vkj , Vhi : Xi ﹎ Yh ↙ aut汍? 老? , Vhi := 羽h

tensor C*-category with a C*-isofunctor F : 老 ↙ 老∩ , then {羽i ? F } is a set of local charts associated ?1 with 老∩ ; thus, 汐ij = (羽i ? F ) ? (F ?1 ? 羽j ) is a cocycle associated with 老∩ . In other terms, we de?ned an injective map i? : sym(X, 老? ) ?↙ H 1 (X, aut汍? 老? ) . On the converse, let {汐ij } be an aut汍? 老? -cocycle associated with a ?nite cover {Xi } of closed neighborhoods. As a preliminary remark, we consider the obvious structure of symmetric tensor C*-category (3.18) on the trivial bundle W 老? , W ? X closed. With such a structure, every 汐ij de?nes a C*-autofunctor 汐ij : Xij 老? ↙ Xij 老? preserving tensor product and symmetry, and such that 汐ij (老? ) = 老? . We consider the set of symmetric tensor C*-categories {(Xi 老? , ?i , 牝i , 汍i )} (where ?i , 牝i , 汍i are de?ned according to (3.18)), and the C*-category T obtained by glueing every Xi 老? w.r.t. the C*-isofunctors 汐ij *s. By construction, the objects of T are the tensor powers of 老? ; the spaces of arrows of T are de?ned according to (2.9). We de?ne a tensor product on T , by posing r +s s , r, s ﹋ N , and 老r ? 老? := 老? ∩ (ti )i ? (t∩ (3.21) i )i := (ti ?i ti )i
s where the families {ti ﹋ C (Xi , (老r ? , 老? ))} , preserves the tensor product, we ?nd r s t∩ i ﹋ C (Xi , (老? , 老? ))
∩ ∩

satisfy (2.9). Since each 汐ij

灰i,Xij (ti ? t∩ i)

= 灰j,Xij (ti ) ?i 灰j,Xij (t∩ i) = 汐ij (灰j,Xij (tj )) ?j 汐ij (灰j,Xij (t∩ j )) ) , = 汐ij ? 灰j,Xij (tj ?j t∩ j

so that the l.h.s. of (3.21) is actually an arrow in T ; thus, the tensor product on T is well-de?ned. For the same reason, the operators
s 汍(r, s) := ( 1Xi ? 汍? (老r ? , 老? ) )i

de?ne a symmetry on T . We denote by T := (老, ?, 牝, 汍) the symmetric tensor C*-category obtained in this way. By construction, 老 is equipped with a set of local charts 羽i : 老Xi ↙ Xi 老? such that ?1 汐ij = 羽i,ij ? 羽j,ij (see (2.10)), thus the map i? is also surjective. Example 3.4 (The Permutation category.). Let (老? , ?? , 牝? , 汍? ) be a symmetric tensor C*-category s r r with (牝? , 牝? ) ? C . We assume that (老r ? , 老? ) = {0} if r = s ﹋ N , and that every (老? , 老? ) is generated as a Banach space by the permutation operators 汍? 老? (p), p ﹋ Pr . It is clear that in this case the automorphism group aut汍? 老? reduces to the identity. Let d ﹋ N denote the dimension of 老? ; by [12, Lemma 2.17] we ?nd that 老? is isomorphic as a symmetric tensor C*-category to U(d) r r r (see Ex.3.3). Thus, (老r ? , 老? ) ? (Hd , Hd )U(d) , r ﹋ N . Let 老 ﹋ sym(X, 老? ); since aut汍? 老? = {id} , from the previous lemma we conclude 老 ? X 老? . We denote by PX,d ? X U(d) the unique (up to isomorphism) element of sym(X, U(d)). 18

3.0.8

Special categories.

Let G ? U(d) be a closed group. We denote by N G the normalizer of G in U(d) , and by QG := N G/G the quotient group, so that we have an epimorphism p : N G ↙ QG . (3.22)

The inclusion G ? U(d) implies that OU(d) ? OG . We denote by aut(Od , OG ) ? aut牟 Od the group of automorphisms of Od leaving OG globally stable, and coinciding with the identity on OU(d) . From [11, Cor.3.3], we conclude that aut(Od , OG ) is isomorphic to a subgroup of U(d) , acting on Od according to (3.14). Theorem 3.12. Let (老, d, V ) be a special object, and suppose (牝, 牝) = C . Then, there is a closed group G ? SU(d) with a group isomorphism aut汍 老 ? QG . Let 羽 : aut(Od , OG ) ↙ autOG be the map assigning to 汐 ﹋ aut(Od , OG ) the restriction on OG ; then, there is a commutative diagram of group morphisms p / QG (3.23) NG
b b 羽

 aut(Od , OG )

 / aut牟 OG

The vertical arrows of the above diagram are group isomorphisms. Proof. By Lemma 3.4, there are isomorphisms aut汍 老 ? aut牟 G ? aut牟 OG ; moreover, for every 汐 ﹋ aut汍 老 there exists u ﹋ U(d) such that u ? i = i ? 汐 . Now i(O老 ) = OG , so that the previous equality implies that u ﹋ autOd restricts to an automorphism of OG ; moreover, for every g ﹋ G we ?nd u ? g ? u? (t) = u ? g(u? (t)) = u ? u? (t) = t (3.24) (we used u? (t) ﹋ OG , and g ﹋ autOG Od ). Thus, we conclude that u ? g ? u? ﹋ autOG Od , i.e. ugu? = g ∩ for some g ∩ ﹋ G ; in other terms, u ﹋ N G . Moreover, it is clear that ug ? i = u ? i = i ? 汐 , thus u ﹋ N G is de?ned up to multiplication by elements of G . Moreover, note that u|OG = v |OG , u, v ﹋ N G , implies uv ? ﹋ autOG Od , i.e. uv ? ﹋ G . On the converse, if u ﹋ N G , t ﹋ OG , then g ? u(t) = u ? g ∩ (t) = u(t) for some g ∩ ﹋ G , and this implies u ﹋ aut(Od , OG ). We conclude that the map 羽 (u) ↙ p(u), u ﹋ aut(Od , OG ), is an isomorphism. De?nition 3.13. A special category is a locally trivial, symmetric tensor C*-category (老, ?, 牝, 汍) with ?bre (老? , ?? , 牝? , 汍? ) , such that 老? is a special object. We emphasize the fact that a special category is locally trivial in the sense of Def.3.10, thus the local charts preserve tensor product and symmetry. This also implies that (牝? , 牝? ) ? C . Theorem 3.14. Let (老, ?, 牝, 汍) be a special category with ?bre (老? , ?? , 牝? , 汍? ) . Then, 老 is amenable. There exists d ﹋ N and a unique up to isomorphism compact Lie group G ? SU(d) such that 老? ? G . Thus, O老 is a locally trivial bundle of C*-algebras with ?bre OG , and there is a one-toone correspondence sym(X 牝 , 老? ) ? H 1 (X 牝 , QG) . (3.25) Proof. It follows from point (3) of Lemma 3.4 that the ?bre of 老 is isomorphic to (G, ?, 牝? , 牟), where G ? SU(d) is a compact Lie group unique up to isomorphism. Note that G is amenable s (see Ex.3.3). Let now 老? ﹋ endO老 be the canonical endomorphism, so that (老r , 老s ) ? (老r ? , 老? ), 19

r, s ﹋ N . We consider a cover of closed neighborhoods {Xi } with local charts 羽i : 老 ↙ Xi G , and a 步 i } . Since G is amenable, every Xi G is amenable; thus, if partition of unity {竹i } subordinate to { X ?1 s r s r s r s t ﹋ (老r , 老 ), then 羽 ( 竹 t ) ﹋ C ( X , ( 考 , 考 i i 0 i ? ? G G )) = C0 (Xi , (H , H )G ). Since 羽i (C0 (Xi , (H , H )G )) is contained in (老r , 老s ), we conclude that 竹i t ﹋ (老r , 老s ). Thus t = i 竹i t ﹋ (老r , 老s ), and 老 is amenable. By Thm.3.12 and Lemma 3.11 we obtain (3.25). The assertions about O老 follow from Prop.3.2. For every T ﹋ sym(X, G), we denote by QT ﹋ H 1 (X, QG) the unique-up-to-isomorphism principal QG -bundle associated with T . The map (3.25) has to be intended as an isomorphism between sets endowed with a distinguished element, in the sense that Q(X G) coincides with the trivial principal QG -bundle X ℅ QG . Some particular cases follow. If QG is Abelian, then H 1 (X, QG) is a sheaf cohomology group ([16, I.3.1]); this allows one to de?ne a group structure on sym(X, G). Let SY be the suspension of a compact Hausdor? space Y , and QG arcwise connected. By classical arguments ([17, Chp.7.8]), we have an isomorphism sym(SY, G) ? [Y, QG], where [Y, QG] is the set of homotopy classes of continuous maps from Y into QG . In particular, if Y = S n is the n -sphere (i.e. X = S n+1 ), then sym(S n+1 , G) is isomorphic to the homotopy group 羽n (QG). Proposition 3.15. Let (老, ?, 牝, 汍) be a special category. Then, there are d ﹋ N and V ? (牝, 老d ) such that the triple (老, d, V ) de?nes a twisted special object. Proof. We take d as the dimension of the ?bre 老? in the sense of (3.12). Let us consider the totally d d antisymmetric projection P老,汍,d ﹋ (老d , 老d ). To be concise, we also write P牟,d := PHd ,牟,d ﹋ (Hd , Hd ); if 羽i : 老 ↙ Xi G is a local chart in the sense of (3.19), then 羽i (P老,汍,d ) = P牟,d , in fact 羽i (汍老 (p)) = 牟(p), d d d p ﹋ Pd . Now, P牟,d ﹋ (Hd , Hd )G and has rank one, in fact P牟,d = S ? S ? , where S ﹋ (牝? , Hd )G ? d ?1 ?1 satis?es (S ? 1d ) ? (1d ? S ) = (?1) d 1d ([11, Lemma 2.2]). We de?ne V := V ﹋ (牝, 老d ) : V = P老,汍,d ? V .

It is clear that V is a vector space. Moreover, V has a natural structure of a Hilbert C (X 牝 )bimodule, by considering the actions f, V ↙ V ? f , f ? V , f ﹋ C (X 牝 ) = (牝, 牝). We now consider 步 i ; if V ﹋ V , then a partition of unity {竹i } ? (牝, 牝), subordinate to the open cover X 羽i (竹i V ) = 羽i (竹i P老,汍,d ? V ) = P牟,d ? 羽i (竹i V ) . Thus, there exists fi ﹋ C (X 牝 ) such that 羽i (竹i V ) = fi S ; if V ∩ ﹋ V , with 羽i (竹i V ∩ ) = fi∩ S , then 竹i V ? ? V ∩ = fi? fi∩ . This implies
?1 ? ∩ 竹i (V ? ? 1老 ) ? (1老 ? V ∩ ) = 羽i (fi fi (S ? ?? 1d ) ? (1d ?? S )) = (?1)d?1 d?1 fi? fi∩ 1d = 竹i (?1)d?1 d?1 (V ? ? V ∩ ) ? 1老 . 1/2 1/2 1/2 1/2 1/2

By summing over the index i , we obtain (3.16(2)). In the same way, the equality 羽i (竹i V ∩ ? V ? ) = fi∩ fi? P牟,d implies (3.16(3)). Let u ﹋ N G . Since G ? SU(d) , we ?nd det(ug ) = det(u) for every g ﹋ G . This means that the determinant factorizes through a morphism detQ : QG ↙ T . By functoriality of H 1 (X, ﹞ ), a map detQ,? : H 1 (X, QG) ↙ H 1 (X, T) ? H 2 (X, Z) is induced. Corollary 3.16. Let (老, ?, 牝, 汍) be a special category with associated QG -cocycle Q老 ﹋ H 1 (X, QG) . Then, 老 has Chern class c(老) = detQ,? (Q老) . 20

Proof. For every y ﹋ QG , we denote by y ﹋ aut牟 G ? aut牟 OG the associated autofunctor de?ned according to (3.23). We adopt the same notation if y : X ↙ QG is a continuous map (so that, y is a continuous aut牟 G -valued map). Let Q := ({Xi } , {yij }) be a QG -cocycle associated with 老 . ?1 = yij ), then it follows from the If 羽i : 老 ↙ Xi G are local charts associated with Q (i.e., 羽i ? 羽j d d d proof of Prop.3.15 that 羽i (V ) = C (Xi ) ? _ Hd , where _ Hd := P牟,d Hd is the rank one Hilbert space of totally antisymmetric vectors. Now, if u ﹋ U(d) then u(S ) = det u ﹞ S for every S ﹋ _d Hd ; so that, y (S ) = detQ (y ) ﹞ S , y ﹋ QG . Thus, for every V ﹋ C (Xij ) ? _d Hd we ?nd
?1 羽i ? 羽j (V ) = yij (V ) = detQ (yij ) ﹞ V .

This implies that the line bundle associated with V has transition maps detQ (yij ) : Xij ↙ T . Example 3.5. Let us consider the case G = SU(d) , so that QG = T and (3.22) is the determinant map det : U(d) ↙ T . By Thm.3.14, we obtain sym(X, SU(d)) = H 1 (X, T) ? H 2 (X, Z) , and the map detQ,? de?ned in Cor.3.16 is the identity. The previous considerations imply that elements of sym(X, SU(d)) are labeled by pairs (d, 汎 ), d ﹋ N , 汎 ﹋ H 2 (X, Z). We denote by Td,汎 the generic element of sym(X, SU(d)), and by (老, d, V ) the twisted special object generating Td,汎 . By Cor.3.16, the class 汎 is the ?rst Chern class of the line bundle L(汎 ) ↙ X associated with V . The arrows of Td,汎 are generated by the symmetries 汍老 (r, s), r, s ﹋ N , and elements of V , in the same way as the dual SU(d) is generated by the ?ip operators 牟(r, s) and the totally antisymmetric d isometry S ﹋ (牝, Hd ) (see Ex.3.3, [11, Lemma 3.7])

4

A classi?cation for certain OG -bundles.

We recall the reader to the notation of Ex.3.3. Let G ? SU(d) be a compact group, and 老 ﹋ s sym(X, G). Since 老 is amenable (Lemma 3.8, Prop.3.15) we ?nd 老? = 老 , i.e. (老r , 老s ) = (老r ? , 老? ), r, s ﹋ N . By Prop.3.2, the DR-algebra O老 is a locally trivial continuous bundle with ?bre OG (an OG -bundle, for brevity). The structure group of O老 is clearly given by aut牟 OG ? QG , in the sense that O老 admits a set of transition maps taking values in aut牟 OG ? autOG . On the converse, let A be an OG -bundle with structure group aut牟 OG . We consider an aut牟 OG -cocycle ({Xh }n h , {汐hk }) associated with A , such that each Xh is a closed neighborhood. We denote by 汐x hk ﹋ aut牟 OG the evaluation of 汐hk over x ﹋ Xhk . By construction, there is a one-to-one correspondence between elements a ﹋ A and n -ples (ah ) ﹋
n h

(C (Xh ) ? OG ) : ah (x) = 汐x hk (ak (x)) , x ﹋ Xhk .

(4.1)

x Now, by de?nition of aut牟 OG we have 考G ? 汐x hk = 汐hk ? 考G , x ﹋ Xhk . Let us denote by 牝h the identity automorphism on C (Xh ); by the above considerations, for every a ﹋ A the n -ple ((牝h ? 考G )(ah )) satis?es (4.1), and de?nes an element of A , denoted by 老(a). An immediate check shows that the map {a ↙ 老(a)} is a C (X )-endomorphism of A ; we call 老 the canonical endomorphism of A . By construction, we have a set of local charts

羽h : (A, 老) ↙ ( C (Xh ) ? OG , 牝h ? 考G ) , 羽h (a) := ah , so that in particular
r s 羽h (老r , 老s ) = C (Xh ) ? (考G , 考G ) , r, s ﹋ N .

(4.2)

21

Since 考G = G , the tensor category 老 ? endX A (de?ned as in Ex.3.1) is an element of sym(X, G). Moreover, A is the DR-algebra associated with 老 , in fact the set {(老r , 老s )}r,s is total in A (this r s is easily veri?ed by using (4.2), and the fact that {(考G , 考G )}r,s is total in OG ). Thus, there is a unique (up-to-equivalence) QG -cocycle associated with (A, 老), namely Q(A, 老) := Q老 ﹋ H 1 (X, QG) . Let A , A∩ be OG -bundles with structure group aut牟 OG . We denote by 老 ﹋ endX A , 老∩ ﹋ endX A∩ 2 2 the canonical endomorphisms, and by 汍 ﹋ (老2 , 老2 ), 汍∩ ﹋ (老∩ , 老∩ ) the ?ip operators de?ned by the associated symmetries. Moreover, ? (resp. ?∩ ) denotes the tensor product in endX A (resp. endX A∩ ), and 牝 ﹋ autX A (resp. 牝∩ ﹋ autX A∩ ) the identity automorphism. By recalling (3.10), we obtain a translation of Thm.3.14 in terms of C*-algebra bundles: Proposition 4.1. With the above notation, the following are equivalent: 1. Q(A, 老) = Q(A∩ , 老∩ ) ﹋ H 1 (X, QG) ; 2. there is a C (X ) -isomorphism 汐 : (A, 老) ↙ (A∩ , 老∩ ) , with 汐(汍) = 汍∩ ; 3. there is an isomorphism 汐? : (老, ?, 汍, 牝) ↙ (老∩ , ?∩ , 汍∩ , 牝∩ ) . We recall the reader to Ex.3.5, and brie?y discuss the case G = SU(d) . We consider Td,汎 ﹋ sym(X, SU(d)), d ﹋ N , 汎 ﹋ H 2 (X, Z); moreover, we denote by Od,汎 the DR-algebra associated with the generating twisted special object (老, d, V ). Now, Ex.3.4 implies that the C*-subalgebra O汍 ? Od,汎 generated by {汍老 (r, s)}r,s is isomorphic to C (X ) ? OU(d) . Let us consider the (graded) 0 = O汍 , we ?nd that 糸 restricts to an endomorphism 糸 ﹋ endOd,汎 de?ned as in (3.17); since Od,汎 endomorphism of O汍 , that we regard as an endomorphism 耳 ﹋ end C (X ) ? OU(d) . By Ex.3.5 it follows that V is the module of sections of the line bundle L(汎 ) ↙ X with ?rst Chern class 汎 , and that Od,汎 is generated by O汍 and V ; thus, (3.17) implies that Od,汎 is a crossed product Od,汎 ? C (X ) ? OU(d) ?耳
L(汎 )

N,

in the sense of [31, ∫3]. In the case in which X is the 2-sphere S 2 , it is well-known that H 2 (S 2 , Z) ? Z . By Prop.4.1, the set of isomorphism classes of OSU(d) -bundles with structure group aut牟 OSU(d) and base space S 2 is labeled by Z ; in particular, 0 ﹋ Z corresponds to the trivial bundle C (S 2 ) ? OSU(d) .

5

Outlooks.

A duality theory for special categories will be the next step w.r.t. the present work. In explicit terms, our aim is to prove a generalization of Lemma 3.4 (i.e., [12, Thm.4.17]): instead of the dual G , our model category will be the one of tensor powers of a vector bundle E ↙ X , with arrows morphisms equivariant w.r.t. a group bundle G ↙ X . The G -action on the tensor powers E r , r ﹋ N , will be de?ned according to the gauge-equivariant K -theory introduced in [25, ∫1], so that G will play the role of a dual object. An important step to prove such a duality is to ?nd an embedding functor in the sense of Def.3.3. As outlined in the introduction of the present paper, existence and unicity of the embedding functor (and the dual object G ) are not ensured, in contrast with the case (牝, 牝) ? C . In particular, nonisomorphic dual objects may be associated with the same special category, in the case in which more than an embedding functor exists. 22

We will use the cohomological classi?cation (3.25) to provide a complete description of such a phenomenon in geometrical terms. Given a special category T ﹋ sym(X, G) and the associated principal QG -bundle QT ﹋ H 1 (X, QG), our tasks will be the following: 1. determine which are the geometrical properties required for QT in order to ?nd an embedding functor i : T ?↙ vect(X ); 2. in the case in which there exists the embedding i , give a characterization of E and the dual object G in terms of geometrical properties of QT , and determine the dependence of E , G on i . These questions have a natural translation in terms of C*-algebra bundles and C*-dynamical systems. If (A, 老) is a C*-dynamical system as in Sec.4, one could ask whether there exists a vector bundle E ↙ X with associated DR-dynamical system (OE , T, 考 ) (see Ex.3.2), such that there is a C (X ) - monomorphism 耳 : (A, 老) ?↙ (OE , 考 ). The existence of an embedding functor for 老 is equivalent to the existence of 耳 , and the obstrution for the existence can be encoded by a cohomological invariant. Acknowledgements. The author wishes to thank an anonymous referee for many precious remarks on a previous version of the present work, in particular for the references [19, 8, 4].

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