C* Continuous Fields of -Algebras Arising from 1 C* Extensions of Tensor -Categories 0 0 2 Ezio Vasselli ∗ n a J February 1, 2008 1 1 ] A Abstract O ThenotionofextensionofagivenC*-categoryC byaC*-algebraAis introduced. InthecommutativecaseA=C(Ω),theobjectsoftheexten- . h sion category areinterpretedasfiberbundlesoverΩofobjects belonging t to the initial category. It is shown that the Doplicher-Roberts algebra a m (DR-algebra in the following) associated to an object in the extension of a strict tensor C*-category is a continuous field of DR-algebras coming [ from the initial one. In the case of the category of the hermitian vector 1 bundles over Ω the general result implies that the DR-algebra of a vec- v tor bundle is a continuous field of Cuntz algebras. Some applications to 9 Pimsner C*-algebras are given. 9 0 1 1 Introduction. 0 1 Itiswellknownthattensorcategoriesplayanimportantroleinthetheory 0 ofduality. Inparticularinthebasicpaper[14]suitableabstractstrictten- / h sorC*-categoriesarecharacterizedasdualsofcompactgroups. Oneofthe t steps of the proof of that result is the association of a C*-algebra, called a the DR-algebra, to each object of the given strict tensor C*-category. m SomeoftheseC*-algebrashavebeenstudied([12,5,30,11,18])andoth- v: ersarewellknown([7]);theaim ofthisworkistostart thestudyofDR- i algebrasarising from acertainclass ofstrict tensorC*-categories, having X not trivial space of intertwiners of the identity object. Such categories r have already been studied in the context of superselection structures, in a thecasewherethefixedpointalgebraofacrossedproducthasanontriv- ial center(see [3] and related references). The present paperis organized as follows: In section 2 some basics are given about strict tensor C*-categories, DR-algebras and continuous fields of C*-algebras with a circle action. Furthermoresome preliminary results are illustrated. In section 3 we introduce a simple procedure for extending a given C*-category C by a unital C*-algebra A. The objects of the extended category CA are projections in the spatial tensor product A⊗(ρ,ρ) for someobject ρinC. WegiveinthecommutativecaseA=C(Ω)apicture oftheextension asthecategorical analogue ofafiberbundleoverΩ,and show that it inherits naturally the tensor structure by C. We also give a ∗Dipartimento di Matematica dell’Universita` di Roma “Tor Vergata”. E-mail [email protected] 1 2 PRELIMINARIES. 2 descriptionofCC(Ω)intermsofprincipalbundles,associatingtoobjectsof theextensioncategoryelementsofsuitablecohomologysets(Prop. 3.11). In section 4 we study DR-algebras associated to objects of CA. Our main result is that in the commutative case they have the structure of continuous fields of C*-algebras, whose fiber is given by DR-algebras as- sociatedtoobjectsofC (Thm. 4.6). Particularcareisgivenatthecaseof Hermitian vector bundles, whose associated DR-algebra is a continuous field of Cuntz algebras. We prove that the isomorphism between the as- sociatedDR-algebrasdoesnotimplytheisomorphismbetweenthevector bundles(Prop. 4.10), so that classification questions arise. Insection 5wereturntothecase inwhichAisnoncommutativeand investigate the existence of tensor products on our extension categories. Further structure has to be added to obtain a tensor product: we need an abstract analogue of the left action on a Hilbert C*-bimodule, so we considerendomorphismsoftheformA→A⊗(ρ,ρ)(see[14],§1.7). With a technical assumption (namely the existence of a unique C*-norm over (ρ,ρ)⊗ AextendingtheC*-normsofthefactorsforeachobjectρinC) alg weconstructatensorproductontheextensioncategory(Prop.5.3). Asan application of the abstract results we characterize Hilbert C*-bimodules of the form X = E ⊗Z A, where Z denotes the center of A and E is the moduleofsectionsofavectorbundleoverspect(Z),intermsofproperties of the Pimsner C*-algebra associated to X (Prop. 5.7). Some of theresults in section 4 havebeen proved independently(and previously) by J.E. Roberts [34]. 2 Preliminaries. 2.1 DR-Algebras. Adetailedexpositionoftensorcategoriescanbefoundin[24];in[14]strict tensor C*-categories with conjugates are considered and illustrated by examples, andthenotion ofDR-algebraisintroduced(in bothreferences theterm monoidalis usedinstead oftensor). More recentdevelopements on strict tensor C*-categories can befound in [23]; we refer to thispaper for the definition of conjugates. ToeachobjectρofastricttensorC*-categoryC iscanonically associ- ated a C*-algebra O , whose construction we briefly recall. We consider ρ for each k in Z theBanach space Ok definedby theinductivelimit ρ ...−×→1ρ (ρr,ρr+k)−×→1ρ (ρr+1,ρr+k+1)−×→1ρ ... having assumed that tensoring on the right with the identity arrow 1 ρ is an isometric map. Note that O0 is a C*-algebra. The direct sum ρ 0O :=⊕ Ok,withproductgivenbycompositionofarrowsandinvolution ρ k ρ induced by the analogue categorical structure, is a *-algebra. On 0O is ρ naturally defineda circle action, assigning to each z in T the map ϕ (t ):=zkt (2.1) z k k wheret ∈Ok. SuchastructureiscalledaZ-gradedC*-algebra. Itcanbe k ρ shownthatthereexistsauniqueC*-normon0O extendingtheC*-norm ρ 2 PRELIMINARIES. 3 onO0 suchthat(2.1)extendstoanautomorphicaction;thecompletition ρ O is called the DR-algebra associated to ρ. The isomorphism class of ρ O does not change as ρ varies in its unitary equivalence class (we say ρ that ρ is unitarily equivalent to σ if there exists u ∈ (ρ,σ) such that u∗◦u = 1 ,u◦u∗ = 1 ): in fact, for each suitable r,k ∈ Z we have the ρ σ Banach space isomorphisms ϕr,k : (ρr,ρr+k) → (σr,σr+k) u t 7→ u⊗r+ktu∗⊗r (2.2) (in the following we will adopt this notation also when, more generally, u◦u∗ 6=1 )realizinganisomorphismofZ-gradedC*-algebrasandhencea σ gradingpreservingisomorphismattheleveloftheassociatedC*-algebras. In particular for ρ=σ the algebra O is acted upon by the group U of ρ ρ unitary arrows in (ρ,ρ). O is also naturally equipped with a canonical ρ endomorphismρobtainedbytensoringontheleftwiththeidentityarrow 1 . Notethat ρ commutes with theaction of U . ρ ρ A similar cobnstruction can be done in the more general setting of strict semitensbor C*-categories (see [11]: roughly speaking, at the level of arrows, only the operation of tensoring on the right with an identity arrow is allowed), for example the category of finitely generated Hilbert bimodules. ExamplesofDR-algebrasaretheCuntzalgebras O ,associated ton- n dimensional Hilbert spaces with n<∞ (at infinite dimension the Cuntz algebra O∞ is strictly contained in the DR-algebra associated to the Hilbert space, see [6]), the Pimsner C*-algebras [29, 11] associated to Hilbertbimodulesandthealgebras O associated torepresentationsofa G compact Lie group G [12]. 2.2 Continuous Fields of C*-algebras with a circle action. The following definition for continuous fields of Banach spaces and C*- algebras appears in [20]. Although less general, it has the advantage of being less technical in comparison with the original one [9, 10]. The two definitionscoincideforacompactbasespace,thatisthecaseinwhichwe are interested. Definition 2.1. A continuous field of Banach spaces over a locally com- pactHaussdorffspaceΩisapair(X,π :X →X ),whereX isaBanach ω ω space and (π ) is a family of surjective Banach space maps indexed by ω ω thepointsofΩ. TheBanachspacesX arecalledthefibersofX inω. For ω each x inX the family(x :=π (x)) iscalled the vector field associated ω ω ω to x. We require the following properties: (1) the norm function ω7→kx k belongs to C (Ω) for each x∈X ω 0 (2) for each x in X, kxk=supkx k ω ω (3) X is a C (Ω)-module w.r.t. pointwise multiplication by continuous, 0 vanishing at infinity maps over Ω. 3 EXTENSIONS OF C*-CATEGORIES. 4 The corresponding definition for continuous fields of C*-algebras is analogous. If (A,A → A ) is a continuous field of C*-algebras then ω A is called the C*-algebra of the field. For further basic notions and terminology (morphisms, local triviality) see thereferences above. We now give a characterization of locally trivial continuous fields of C*-algebras with constant fiber A ≡ A in terms of the cohomology set ω H1(Ω,autA) (see [17] for the definition), where the group of automor- phisms autA is endowed with the pointwise convergence topology. These classical ideas werefirststated in [10], §26 forcontinuousfieldshavingas fiberthecompact operators, butthesameargumentworksinthegeneral case. The proof of the next theorem, that we omit, is based on usual arguments of transition functions for fiber bundles and makes use of a clutching lemma ([9] 10.1.13). The hypothesis of connectedness is nec- essary to fix the fiber, and the compactness of Ω can be replaced with paracompatness using thedefinition by Dixmier and Douady. Theorem 2.2. Let Ω be a compact connected Haussdorff space and A a C*-algebra. Then there is a bijective correspondence between the set of isomorphism classes oflocallytrivialcontinuous fieldsofC*-algebras with fiber A over Ω and the cohomology set H1(Ω,autA). By a general result ([32]) we have H1(Sn,G) = πn−1(G)/π0(G) for each topological group G, where Sn denotes the n-sphere and π (G) is 0 the set of connected components of G. Thus the classification of locally trivialcontinuousfieldsofC*-algebrasoverthen-spheresisequivalentto the homotopy theory of the group of automorphisms of the fiber. There areseveralresultsinthisdirectioninthecaseofAF-algebras[27,36,28]1. The next corollaries will be used and illustrated by explicit examples in thesequel. Corollary 2.3. With the notation of the previous theorem, let (A,α) be a C*-dynamical system over a topological group G. Then α induces a map from H1(Ω,G) into the set of isomorphism classes of locally trivial continuous fields of C*-algebras with fiber A. By the previous corollary, the existence of the automorphic action (2.1) and the well known isomorphism H1(Ω,T) ≃ H2(Ω,Z) we deduce the following result: Corollary 2.4. Let A be a C*-algebra carrying a strongly continuous T- action by automorphisms (as in particular a DR-algebra). Then each el- ement of the Cech cohomology group H2(Ω,Z) defines a locally trivial continuous field over Ω with fiber A. 3 Extensions of C*-Categories. Let C be a C*-category and A a C*-algebra with identity 1. Given a pair ρ,σ of objects in C we consider thespace of arrows (ρ,σ); it has the following structure of a Hilbert (σ,σ)–(ρ,ρ)–bimodule: 1Theauthor thanks P.Goldsteinfordrawinghisattention tothesereferences. 3 EXTENSIONS OF C*-CATEGORIES. 5 l,s7→l◦s s,m7→s◦m  hs,ti :=s∗◦t∈(ρ,ρ) (3.1)  hs,tiR:=s◦t∗ ∈(σ,σ) L where s,t ∈ (ρ,σ), m∈ (ρ,ρ), l ∈ (σ,σ). As for example in [21] we construct the exterior tensor product A⊗(ρ,σ), that is a right Hilbert C*-module over the spatial tensor product A⊗(ρ,ρ). In order for more concise notation we write (ρ,σ)A :=A⊗(ρ,σ), while the symbol (ρ,σ)A a will indicate the algebraic tensor product. We also adopt the Sweedler notationx:=x ⊗x ,x ∈A,x ∈(ρ,σ)forelementsof(ρ,σ)A. Wenow 1 2 1 2 definea composition law (σ,τ)A×(ρ,σ)A→(ρ,τ)A a a a extendingnaturally theone definedon C: yx:=y x ⊗(y ◦x )∈(ρ,τ)A (3.2) 1 1 2 2 a where y ∈(σ,τ)A, x∈(ρ,σ)A. With an abuse of notation we also define a a an involution ∗:(ρ,σ)A→(σ,ρ)A a a by setting (x ⊗x )∗ :=x∗⊗x∗. 1 2 1 2 The following lemma shows that the extended composition and involu- tion are bounded, hence defined on the closures of the algebraic tensor products. Lemma 3.1. Let x∈(ρ,σ)A,y∈(σ,τ)A. Then a a (1) kxk2 =kx∗xk=kxx∗k (2) kyxk≤kykkxk (3) kxk=kx∗k Proof. (1) Itissimplythedefinitionofthe2-normin(ρ,σ)AasaHilbert C*-bimodule. (2) It suffices to observe that x∗(ky∗yk−y∗y)x is positive as an element of (ρ,ρ)A. (3) The isometry of the involution is obtained by embedding (ρ,σ)Aviathecompositionlawintothespaceofthebounded(ρ,ρ)A-linear mapsofrightHilbertC*-modulesfrom(ρ,ρ)A into(ρ,σ)A: ifx∈(ρ,σ)A then x∗ is the adjoint of x as Hilbert C*-module map. In order to define the extension category we consider the category CA havingthesameobjectsofC andarrows(ρ,σ)A. Thepreviouslemma impliesthatCAisaC*-category;wecloseforsubobjectsandobtaininthis weay a C*-category CA whose objects are the projections e∈(ρ,ρ)A. We e 3 EXTENSIONS OF C*-CATEGORIES. 6 call CA the extension of C by A. By definition, the spaces of intertwiners in CA are given by (e,f):= x∈(ρ,σ)A:xe=fx=x wheree∈(ρ,ρ)A,f ∈(σ,σn)A. Alsobydefinition (e,fo) is aHilbert (e,e)- (f,f)-bimodule. Inparticularthefollowing bimoduleactionofthecenter Z of A is defined: ζx:=(ζ⊗1 )x=xζ :=x(ζ⊗1 ), σ ρ where ζ ∈Z, x∈(e,f). In the case in which A is non unital we define CA := CA+, where A+ := A⊕C. This definition is motivated by the fact that spaces of intertwiners of an object of a C*-category have tobe unital C*-algebras. Proposition 3.2. Let C be a C*-category, A a unital C*-algebra. Then (1) CA has subobjects; (2) If C have direct sums, then so does CA. Proof. Thefirstassertionisobvious. Forthesecondone,lete∈(ρ,ρ)A,f ∈ (σ,σ)A;then1 =v◦v∗+w◦w∗,1 =v∗◦v,1 =w∗◦wforsomeobject τ ρ σ τ in C and we can define the isometries v := 1⊗v·e, w := 1⊗w·f. e f Thus the projection g :=v v∗+w w∗ ∈(τ,τ)A is a direct sum of e and e e f f f. Before we state some simple functorial properties of the above con- struction let us introduce the following terminology: a functor between two C*-categories is said to be a C*-functor if commutes with the invo- lutions and the maps induced on the spaces of arrows are bounded and linear. Proposition 3.3. The operation C,A→CA of assigning an extension to aC*-categorydependscovariantlyontheC*-algebraandcovariantly(con- travariantly) on the category, corresponding to covariant or contravariant C*-functors on C. Proof. Let C be a C*-category, φ : A → B a C*-algebra morphism. We constructapullback C*-functor φ∗ :CA→CB. Nowthereexists,foreach object ρ in C, theC*-algebra morphism φ⊗id :(ρ,ρ)A→(ρ,ρ)B; (ρ,ρ) wedefineφ∗(e):=φ⊗id(ρ,ρ)(e). Now,as φ⊗id(ρ,ρ)(x) ≤kxkfor each x in (ρ,ρ)A, we definethebounded linear maps (cid:13) (cid:13) (cid:13) (cid:13) φ∗ :(ρ,σ)A→(ρ,σ)B by setting φ∗(a⊗s):=φ(a)⊗s, By definition φ∗(yx) = φ∗(y)φ∗(x), so that φ∗(e,f) ⊂ (φ∗(e),φ∗(f)). Let now F : C → C be a C*-functor; we want to define a C*-functor 1 2 F :CA→CA. WeobservefirstthatforeachobjectρinC theC*-algebra 1 2 1 morphism 3 EXTENSIONS OF C*-CATEGORIES. 7 id ⊗F :(ρ,ρ)A→(Fρ,Fρ)B A is induced,so we define FAe:=(id ⊗F)(e). A Again, as id ⊗F is bounded,we find that themaps A FA(x ⊗x ):=x ⊗F(x ), 1 2 1 2 where x := x ⊗x ∈ (ρ,σ)A, can be extended to (ρ,σ)A. It is triv- 1 2 a ial to check that composition and involution are preserved. The proof for contravariant C*-functors follows the same lines by using C*-algebra antihomomorphisms. Corollary 3.4. IfChasdirectsums,everypullbackC*-functorφ∗ :CA→ CA preserves direct sums in CA. Proof. Let e,f be objects in CA. With the notation used in the proof of Prop. 3.2, it suffices to verify that φ∗(veve∗ +wewe∗) is a direct sum of φ∗(e),φ∗(f). Inordertoemphasizeatopological point ofviewwewill adoptinthe followingthenotationCΩ toindicatetheextensionofC byacommutative C*-algebra A=C(Ω); in thesame spirit we write (ρ,σ)Ω :=(ρ,σ)C(Ω) ≃ C(Ω,(ρ,σ)) for each pair of objects ρ,σ in C. Let ϕ : Ω′ → Ω be a continuous map of compact Haussdorff spaces. Applying the previous proposition wegetapullbackfunctorϕ∗ :CΩ→CΩ′. Ifϕistheinclusion {ω} ֒→ Ω we call the associated functor the fiber functor of CΩ over ω and use the notation ω∗ :CΩ →Cω. ω∗ associates to objects and arrows of CΩ, regarded as continuous maps, their evaluation in ω. Note that by definition Cω is isomorphic to the closure for subobjects of C. Proposition 3.3failsformultifunctors. LetinfactF :C ×C →C be 1 1 2 a C*-functor; proceeding with the same argument we could try to define a map FA :(ρ,σ)A×(ρ′,σ′)A →(F(ρ,ρ′),F(σ,σ′))A a a a by theexpression FA(x ⊗x ,x′ ⊗x′):=x x′ ⊗F(x ,x′). (3.3) 1 2 1 2 1 1 2 2 Inordertoverifythatthismappreservesthecomposition,letx∈(ρ,σ)A, a x′∈(ρ′,σ′)A, y∈(σ,τ)A, y′∈(σ′,τ′)A. Then we have a a a FA(yx,y′x′)=y x y′x′ ⊗F(y ,y′)◦F(x ,x′); 1 1 1 1 2 2 2 2 however FA(y,y′)FA(x,x′)=y y′x x′ ⊗F(y ,y′)◦F(x ,x′) 1 1 1 1 2 2 2 2 sothatunlessAiscommutativethereisnonaturalwaytodefineFA. We specialize for the moment our discussion to the commutative case, but will return to thegeneral case in thesubsequent sections. 3 EXTENSIONS OF C*-CATEGORIES. 8 Proposition 3.5. LetC beaC*-category, ΩacompactHaussdorffspace. Then (1) Each C*-functor F :C ×C →C induces a C*-functor FΩ :CΩ× 1 1 2 1 CΩ→CΩ. 1 2 (2) CΩ is a strict tensor C*-category if C itself is so; the space of in- tertwiners of the identity object ι := 1⊗1 in CΩ is (ι ,ι ) ≃ Ω ι Ω Ω C(Ω)⊗(ι,ι), where ι is the identity object in C. (3) CΩ is symmetric if C is symmetric. (4) If C has conjugates, then so does CΩ. Proof. The computation above shows that the map (3.3) preserves the composition of arrows in CΩ. Using commutativity of C(Ω) it is also verifiedthatFΩ preservestheinvolution. Now,thankstotheC*-identity, in order to verify that FΩ is bounded, it suffices to show this just in the casewherethedomainobjectcoincideswiththecodomain. Wenotethat themaps7→F(s,1 )definesamorphism(ρ,ρ)→(F(ρ,σ),F(ρ,σ)),that σ we extendto amorphism FΩ:(ρ,ρ)Ω→(F(ρ,σ),F(ρ,σ))Ω tensoring by ρ theidentityonC(Ω). Thus,forx∈(ρ,ρ)Ω,theinequality FΩ(x) ≤kxk ρ holds. Now for x∈(ρ,ρ)Ω,y∈(σ,σ)Ω we have a a (cid:13) (cid:13) (cid:13) (cid:13) FΩ(x,y)=FΩ(x,1 )FΩ(1 ,y)=FΩ(x)FΩ(y) σ ρ ρ σ so that FΩ(x,y) ≤kxkkyk. ThuswecanextendFΩ to(cid:13)(cid:13)(ρ,σ)Ω a(cid:13)(cid:13)nddefine,foreachpaire,f inCΩ,the (cid:13) (cid:13) objectFΩ(e,f)inCΩ thatcanbeexpressedconvenientlybythefollowing 2 formal expression FΩ(e,f):=e f ⊗F(e ,f ). 1 1 2 2 Wepassnowto provethesecond point. Wedefinethetensor producton CΩ as theC*-functor ef :=e f ⊗e ×f 1 1 2 2 induced bythe tensor product on C;with an abuse of notation, we write x×y:=x y ⊗x ×y 1 1 2 2 toindicatethetensorproductofarrowsinCΩ. By(1),theonlyproperties we have to prove are strict associativity and the existence of an identity object. Strict associativity follows by using the strict associativity of the tensor product of arrows in C. For the symmetry,we define θ (e,f):=1⊗θ(ρ,σ)·e×f Ω (in the previous expression we regarded e,f as the identity arrows of (e,e),(f,f)); the properties of symmetry are verified with direct compu- tations. For the existence of conjugates, we observe that CΩ as defined at thebeginningof thissection obviously hasconjugates. Then we apply [23], Th. 2.4, where it is shown that each subobject of an oebject having conjugates itself has conjugates. 3 EXTENSIONS OF C*-CATEGORIES. 9 The following result emphasizes a geometrical interpretation of CΩ. Roughly speaking we can say that while a C*-category is the categorical analogueofaC*-algebra, CΩ isanexampleofthecategorical analogueof a continuousfield of C*-algebras. Proposition 3.6. Let C bea C*-category closed forsubobjects, Ωacom- pact Haussdorff space. Then (1) Each object e in CΩ defines, via the fiber functors ω∗ : CΩ → C, a family {e } of objects in C, called the fibers of e; ω ω∈Ω (2) The involution on CΩ is defined fiberwise, i.e. ∗◦ω∗ = ω∗◦∗ for each ω∈Ω; (3) Foreache,f objectsinCΩ,thespaceofarrows(e,f)definesalocally trivial continuous field of Banach spaces {(e,f),ω∗ :(e,f)→(eω,fω)}, (3.4) over Ω, with fibers the spaces of intertwiners (e ,f ) of the fibers of ω ω e,f in C. (4) If C is strict tensor the tensor product defined on CΩ induces on the spaces of arrows morphisms of continuous fields of Banach spaces. Proof. The first and second pointsare obvious. For thethirdpoint, note thate∈(ρ,ρ)Ω,f ∈(σ,σ)Ω canberegardedascontinuousfunctionsω7→ e(ω),f(ω). ThenweconsiderinCthesubobjectseω :=ω∗(e),fω :=ω∗(f) definedbye(ω),f(ω),andprovethat(3.4)isacontinuousfieldofBanach spaces. If x ∈ (e,f) ⊂ (ρ,σ)Ω then the continuous function ω 7→ x(ω) is defined; we put xω := ω∗(x) ∈ (eω,fω) and observe that (eω,fω) is isometrically isomorphic as a Banach space to (e(ω),f(ω)) ⊂ (ρ,σ) via the map s7→w sv∗, where s∈(e ,f ),v∗v =1 ,v v∗ =e ,w∗w = ω ω ω ω ω ω eω ω ω ω ω ω 1 ,w w∗ =f . Thus fω ω ω ω kx k=kx(ω)k (3.5) ω andthefunction ω7→kx kis continuous. Theequality(3.5) implies also ω that kxk = supkx k. We already know that (e,f) is a Banach C(Ω)- ω ω module, so haveproved that (3.4) is a continuous field of Banach spaces. Wepostponetheproofoflocaltrivialityuntilthenextlemma. Finally,we haveto provethat the map x,y7→x×y,x∈(e,e′), y∈(f,f′) preserves thefibers,i.e. ω∗(x×y)=ω∗(x)×ω∗(y),butthisisobviousregardingx,y as continuous map and recalling thedefinition of thefiber functor. Lemma 3.7 (Local Triviality). Let e,f be objects in CΩ. Then (1) If the maps ω7→e(ω),f(ω) are constant there exist objects ρ ,σ in 0 0 C such that there is an isomorphism of continuous fields of Banach spaces (e,f)≃(ρ ,σ )Ω; 0 0 (2) Eachpointω inΩadmitsaneighborhoodU suchthatthereisaniso- morphismofcontinuousfieldsofBanachspaces (e,f)| ≃(e ,f )U. U ω ω 3 EXTENSIONS OF C*-CATEGORIES. 10 Proof. Wedefineρ :=e ,σ :=f . Byhypothesisthereexistisometries 0 ω 0 ω v ∈ (ρ,ρ ),w ∈ (σ,σ ) intertwining e,f with the units arrows 1 ,1 . 0 0 ρ0 σ0 Thus the map x 7→ wxv∗,x ∈ (e,f), defines the isomorphism between (e,f) and (ρ ,σ )Ω. For the second point, since e ∈ (ρ,ρ)Ω,f ∈ (σ,σ)Ω, 0 0 foreachω thereexistsaclosedneighborhoodU suchthatke(ω)−e(ω′)k, kf(ω)−f(ω′)k ≤ δ < 1,ω′ ∈ U. This implies that e| ,f| and the U U constant maps e := e(ω),f := f(ω) are Murray-Von Neumann equiv- U U alent as elements of the C*-algebras (ρ,ρ)U,(σ,σ)U. Applying (1), since e ,f are constant maps, there must be an isomorphism (e,f)| ≃ U U U (e ,f )U. ω ω Some examples of extension categories follow. Example 3.1. ThecategoryoffinitedimensionalHilbertspacesisaten- sor C*-category if endowed with the usual tensor product. To get an equivalent strict tensor C*-category we proceed as in [14] and consider thecategoryHilbwithobjectsthepositiveintegersn∈Nandarrowsthe spaces ofn×m matrices (n,m):=M . Thetensor producton objects n,m isthemultiplication,whileonarrowsitisdefinedbythe”lexicographical” product for matrices (see for example [14] §1.4). The extension HilbΩ is thecategory ofprojective, finitelygeneratedC(Ω)-moduleshence,bythe Swan-Serre theorem [33], equivalent to the category of Hermitian vector bundlesoverΩ. Inthenoncommutativecasewegetthefinitelygenerated projective (right) Hilbert A-modules. Example 3.2. LetGbethecategoryofunitary,finitedimensional,con- tinuous representations of a compact group G. Each object (n,u) in G is a topological groubp map G → Un into the unitary group of the n- dimensional matrices. An object in GΩ is a projection eu in Mn⊗C(Ωb) that commutes with the action of G induced by u and, given (n,u), (m,v) objects in G, the intertwinerbs t ∈ (e ,f ) are continuous func- u v tions from Ω into the space of n×m matrices and satisfy the relations t = teu = fvt,vgtb= tug for each g ∈ G. Hence we get the category of G-HermitianvectorbundleswithtrivialactionofGoverthebasespaceΩ ([2]§1.6,[14]§1.6). Thenoncommutativeanalogueisgivenbythefinitely generatedprojectiveHilbertA-modulescarryingaunitaryrepresentation of G. Example 3.3. We can generalize the previous example by considering thecategoryoffinitedimensional(co)representationsofaunitalHopfC*- algebra(A,∆),i.e. acompactquantumgroupinthesenseofWoronowicz (seeforexampletheintroductorypaper[25]andrelatedreferences). This is a strict tensor C*-category having direct sums, subobjects and conju- gates. Theobjects are given by pairs (n,u), where u∈M ⊗A satisfies n ∆(u )= u ⊗u , lm lk km k X while an arrow x from (n,u) to (m,v) is an n× m matrix such that v(1⊗x)=(1⊗x)u. Anobject intheextension categoryoverΩisthena HermitianvectorbundleE carryinganinjectivecomoduleactionbyA,in thesenseexplainedbelow: thetensorproductE⊗Aisnaturallyendowed withthestructureofavectorA-bundleinthesenseof[26]. Thenavector bundlemorphism ϕ :E →E⊗A isinduced,definedfiberwise bysetting u