aa r X i v : . [ m a t h - ph ] A p r Representationsofnets of C ∗ -algebras over S Giuseppe Ruzzi and Ezio Vasselli ∗ Dipartimento di Matematica, Universit`a di Roma “Tor Vergata”,Via della Ricerca Scientifica, I-00133 Roma, Italy. [email protected] Dipartimento di Matematica, Universit`a di Roma “La Sapienza”,Piazzale Aldo Moro 5, I-00185 Roma, Italy. [email protected]
Abstract
In recent times a new kind of representations has been used to describe superse-lection sectors of the observable net over a curved spacetime, taking into account ofthe effects of the fundamental group of the spacetime. Using this notion of represen-tation, we prove that any net of C ∗ -algebras over S admits faithful representations,and when the net is covariant under Diff( S ), it admits representations covariantunder any amenable subgroup of Diff( S ). Nets of C ∗ -algebras are the basic objects of study in algebraic quantum field theory and,as well-known to the specialists, encode the basic idea that any suitable region Y of aspacetime defines an abstract C ∗ -algebra A Y , interpreted as the one generated by thequantum observables localized in Y ; from this assumption it is natural to require thatthere are inclusions morphisms Y ′ Y : A Y → A Y ′ , ∀ Y ⊆ Y ′ , which, for coherence, mustfulfil the equalities Y ′′ Y ′ ◦ Y ′ Y = Y ′′ Y , ∀ Y ⊆ Y ′ ⊆ Y ′′ ∗ Both the authors are supported by the EU network “Noncommutative Geometry” MRTN-CT-2006-0031962. Open, relatively compact and simply connected subsets of the spacetime. X the set of regions Y ⊂ X is upwarddirected under inclusion, so the pair ( A , ), A := {A Y } , := { Y ′ Y } , is indeed a netand we can construct the inductive limit ~ A := lim( A , ). In this way, families of Hilbertspace representations of A Y , Y ⊂ X , coherent with the inclusion morphisms (that wecall Hilbert space representations of the net) are obtained by considering representationsof ~ A . This point is important for the applications, because crucial physical propertiesof the quantum system described by ( A , ), like the charge structure of elementary par-ticles, are encoded by certain Hilbert space representations of the net, called sectors ([8, 9, 2]).Now, general relativity and conformal theory lead to consider spacetimes X suchthat the set of regions { Y ⊂ X } is not directed under inclusion anymore, and the abovescenario breaks down. In this case we should say, to be precise, that ( A , ) is a precosheafof C ∗ -algebras, and the search for Hilbert space representations may be vain (see [16]).In recent times a more general notion of representation has been given for ”nets” ofC ∗ -algebras over generic spacetimes X , defined in such a way that the obstacle to getcoherence is encoded by a family { U Y ′ Y } Y ⊆ Y ′ of unitaries fulfilling the cocycle relations(see [6, 3]). These, that we simply call representations , reduce to Hilbert space represen-tations when X is simply connected and maintain the properties of charge composition,conjugation and covariance under eventual spacetime symmetries, typical of the usualsectors.In a previous paper ([16]), we introduced the notion of enveloping net bundle of thegiven net of C ∗ -algebras ( A , ); this is a net of C ∗ -algebras ( A , ) such that any Y ′ Y is anisomorphism, and fulfils the universal property of lifting any representation of ( A , ). Sothe question of existence of representations is reduced to nondegeneracy of the canonicalembedding ǫ : ( A , ) → ( A , ). We call injective those nets such that ǫ is faithful.In the present work we focus on nets of C ∗ -algebras defined over S , a remarkableclass due to its applications in conformal quantum field theory ([13, 7]). We show thatany net over S is injective, so it has faithful representations. Moreover, when the net iscovariant under the action of Diff( S ), we show the existence of covariant representationsof any amenable subgroup of Diff( S ). The technique that we will use shall be the oneof approximate the set of proper intervals of S with finite subsets (roughly speaking, ananalogue of the decomposition of S as a CW-complex in the setting of partially orderedsets), then to show that the restriction of ( A , ) on these subsets is injective, and finallyto prove injectivity of the initial net performing an inductive limit. We have postponedto Appendix A some rather technical computations showing that injectivity is preservedunder inductive limits, a result which plays a key rˆole in the analysis of nets over S andthat, we hope, could play a similar role for other spacetimes too. To make the present work self-contained in this section we recall the basic properties ofthe main objects of our study, namely nets of C ∗ -algebras. All the material presentedhere appeared in [16]; the reader may pass to the next section whenever he is already2amiliar with that paper. A poset (partially ordered set) is a set endowed with an (antisymmetric, reflexive andtransitive) order relation ≤ . A poset morphism is a map f : K → K ′ such that o ≤ ˜ o implies f( o ) ≤ ′ f(˜ o ) for all o, ˜ o ∈ K , where ≤ ′ is the order relation of K ′ . A disjointnessrelation on K is a symmetric binary relation ⊥ such that˜ o ⊥ a , o ≤ ˜ o ⇒ o ⊥ a . A group G is said to be a a symmetry group for K whenever it acts by automorphisms on K , namely go ≤ g ˜ o ⇔ o ≤ ˜ o for all g ∈ G and o, ˜ o ∈ K , and we assume that, whenever K has a disjointness relation, o ⊥ a ⇔ go ⊥ ga .The classical covariant poset used in algebraic quantum field theory is the set ofdoublecones in the Minkowski spacetime, having the inclusion as order relation, thespacelike separation as the disjointness relation and the Poincar´e group as the group ofsymmetries. We shall focus in § S , of interest in lowdimensional quantum field theory.We now give a brief description of the notion of connectedness and simply connected-ness for posets and refer the reader to the paper [16] for details . A poset K is pathwiseconnected if for any pair a, ˜ a ∈ K . there are two finite sequence a , . . . a n +1 and o , . . . o n of elements of K , with a = a and a n +1 = ˜ a , satisfying the relations a i , a i +1 ≤ o i , i = 1 , . . . , n . In the sequel, we will always assume that our poset is pathwise connected . As alreadysaid, there is a notion of the first homotopy group π o ( K ) for K . The supscript o denotesthe base point in K where the homotopy is calculated, however the isomorphism classdoes not depend on the choice of o (for this reason, often we shall write π ( K ) withoutspecification of the base point). We shall say that K is simply connected whenever π ( K )is trivial.If X is a space having a subbase K of arcwise and simply connected open sets, andif K is ordered under inclusion, then there is an isomorphism π ( X ) ≃ π ( K ) ([15]).We now deal with continuous actions of symmetry groups. Let G be topologicalsymmetry group acting on a poset K . and O ( e ) denote the set of open neighbourhoodsof the identity of the group G . Then we define o ≪ a ⇐⇒ ∃ U ∈ O ( e ) , go ≤ a , ∀ g ∈ U . (2.1)Now, a topological symmetry group G of K is said to be a continuous symmetry groupof K if ∀ o ∈ K , ∃ a ∈ K , o ≪ a , (2.2) The standard way to introduce these topological notions makes use of a simplicial set associated tothe poset. We prefer do not introduce this simplicial set since it will be not explicitly used in the presentpaper. ≪ a ⇒ ∃ ˜ a ∈ K , o ≪ ˜ a ≪ a , (2.3)and o ≪ a , a ⇒ ∃ ˜ o ∈ K , o ≪ ˜ o ≪ a , a . (2.4)This condition is suited for posets arising as subbases of topological G -spaces and,roughly speaking, encodes the idea that the sets { go, g ∈ U } , U ∈ O ( e ), yield a neigh-bourhood system for o . Double cones in Minkowski spacetime and the open intervals of S are examples of posets acted upon continuously (in the above sense) by the Poincar´egroup and the Diff( S ) group respectively. In these cases it is easely seen that a ≪ o isequivalent to the condition that the closure of a is contained in o . Note, in addition, thatthe above conditions are always verified when the symmetry group G has the discretetopology. Remark 2.1.
The notion of a continuous symmetry group of a poset introduced in thepresent paper is different from that used in [16]. We prefer this new notion of continuitybecause it involves in its definition only the poset and the group. The older, instead,involved the simplicial set associated to the poset. In Appendix A.3 we shall provethat the new notion of continuity is stronger than the older one, so that all the resultsobtained in that paper continue to hold. C ∗ -algebras A net of C ∗ -algebras over the poset K is given by a family A := {A o } o ∈ K of unitalC ∗ -algebras (called the fibres ), and a family := { ˜ oo : A o → A ˜ o , o ≤ ˜ o } of unitalmonomorphisms (called the inclusion maps ) fulfilling the net relations o ′ ˜ o ◦ ˜ oo = o ′ o , o ≤ ˜ o ≤ o ′ . In the sequel we shall denote a net of C ∗ -algebras by ( A , ) K . When every ˜ oo is anisomorphism we say that ( A , ) K is a C ∗ -net bundle and, to be short, we write o ˜ o := − oo , ∀ o ≤ ˜ o . The restriction of ( A , ) K over S ⊂ K is given by the same families restrictedto elements of S and is denoted by ( A , ) S .Clearly, the definition of net can be given for other categories and in particular weshall use the one of Hilbert spaces, especially the case of Hilbert net bundles (whose netstructure is given by unitary operators).In all the cases of interest K shall be a subbase for the topology of a space, in generalnot directed, so if we would use the correct terminology in the setting of algebraictopology we should use the term precosheaf of C ∗ -algebras ; neverthless, we prefer tomaintain the usual term net , since it is standard in algebraic quantum field theory.A morphism of nets is written( π, f) : ( A , ) K → ( B , i ) P , where f : K → P is a poset morphism and π := { π o : A o → B f( o ) } is a family of unitalmorphisms such that i f(˜ o ) , f( o ) ◦ π o = π ˜ o ◦ ˜ oo , ∀ o ≤ ˜ o . When f is the identity we shall write4 instead of ( π, id K ). We say that ( π, f) is faithful on the fibres if π o is a monomorphismfor any o ; it is an isomorphism when both π and f are isomorphisms. We say that anet is trivial if it is isomorphic to the constant net ( C , id ) K , where C o ≡ F for a fixedC ∗ -algebra F and any id ˜ oo is the identity of F .The structures that we introduce in the following lines are familiar in the setting ofquantum field theory, and reflect Poincar´e (M¨obius) symmetry and Einstein causalityrespectively. If G is a continuous symmetry group of K , then we say that the net ( A , ) K is G - covariant whenever there are isomorphisms α go : A o → A go , ∀ o ∈ K , g ∈ G , suchthat α g ˜ o ◦ ˜ oo = g ˜ o,go ◦ α go , α hgo ◦ α go = α hgo , o ≤ ˜ o ∈ K , g, h ∈ G , and fulfilling the following continuity condition : if { g λ } ⊂ G is a net converging to e ,then for any o ∈ K there exists a ≫ o and an index λ a such that g λ o ≤ a , ∀ λ ≥ λ a , and k a g λ o ◦ α g λ o ( A ) − ao ( A ) k → , A ∈ A o . (2.5)If ( A , , α ) K and ( B , i, β ) K are G -covariant nets, a morphism ( π, f) : ( A , , α ) K → ( B , i, β ) K is said to be G - covariant wheneverf( go ) = g f( o ) , π go ◦ α go = β g f( o ) ◦ π o , o ∈ K , g ∈ G .
Finally, when K has a causal disjointness relation ⊥ , we say that the net ( A , ) K is causal whenever [ ao ( t ) , a ˜ o ( s ) ] = 0 , o ⊥ ˜ o , o, ˜ o ≤ a , where t ∈ A o and s ∈ A ˜ o . The importance of net bundles in the analysis of nets resides in the following fact. Let( A , ) K be a net bundle. Since the inclusion maps are isomorphism, they induce, forany o ∈ K , an action, the holonomy action , ∗ : π o ( K ) → aut A o , (2.6)of the homotopy group π o ( K ) into the fibre A o . The C ∗ -dynamical system ( A o , π o ( K ) , ∗ )is unique up to isomorphism at varying of o in K and is a complete invariant of the netbundle since the net bundle can be reconstructed (up to isomorphism) starting from theC ∗ -dynamical system. The net bundle ( A , ) K is trivial if and only if ∗ is the trivialaction. We shall refer to ( A o , π o ( K ) , ∗ ) as the holonomy dynamical system of the netbundle.On these grounds it is crucial to understand whether and how a net can be embeddedinto a net bundle.It turns out that any net of C ∗ -algebras can be embedded into a C ∗ -net bundle.To be precise, the enveloping net bundle of a net of C ∗ -algebras ( A , ) K is a C ∗ -netbundle by ( A , ) K , which comes equipped with a morphism ǫ : ( A , ) K → ( A , ) K , called5he canonical embedding , satisfying the following remarkable universal properties: givenmorphisms with values in C ∗ -net bundles,( ϕ, h) , ( θ, h) : ( A , ) K → ( C , y ) P , ( ψ, f) : ( A , ) K → ( B , ı ) S , we have (cid:26) ( ϕ, h) ◦ ǫ = ( θ, h) ◦ ǫ ⇒ ϕ = θ , ∃ ! ( ψ ↑ , f) such that ( ψ ↑ , f) ◦ ǫ = ( ψ, f) , (2.7)where ( ψ ↑ , f) is the pullback ( ψ ↑ , f) : ( A , ) K → ( B , ı ) S . (2.8)These properties characterize the enveloping net bundle, that is, it is the unique, up toisomorphism, C ∗ -net bundle satisfying the above relations, and this leads to the followingclassification: a net of C ∗ -algebras is degenerate if its enveloping net bundle vanishes,and is nondegenerate otherwise. A nondegenerate net of C ∗ -algebras is injective if thecanonical embedding is a monomorphism. Remark 2.2. (1) When K is simply connected the ( A , ) K is a trivial C ∗ -net bundlewith fibres isomorphic to the Fredenhagen universal C ∗ -algebra of ( A , ) K (see [10]).(2) If G is a continuous symmetry group, then the enveloping net bundle of a G -covariantnet is G -covariant as well.We can now state the functoriality property and its relation with injectivity: for anymorphism ( ρ, f) : ( A , ) K → ( D , k ) P there exists a morphism ( ρ, f) : ( A , ) K → ( D , k ) P which fulfils the the relation ( ρ, f) ◦ ǫ = ˜ ǫ ◦ ( ρ, f) , where ǫ and ˜ ǫ are, respectively, the canonical embeddings of the nets ( A , ) K and ( D , k ) P .This makes the assignment of the enveloping net bundle a functor. Remark 2.3. (1) Note that if ( ρ, f) is faithful on the fibres and ( D , k ) P is injective,then ( A , ) K is injective too.(2) The functor assigning the enveloping net bundle preserves inductive limits (Prop.A.4). A state of a net of C ∗ -algebras ( A , ) K is a family of states of C ∗ -algebras ω := { ω o : A o → C , o ∈ K } fulfilling ω o = ω a ◦ ao , o ≤ a . (2.9)It turns out that the set of states of a C ∗ -net bundle ( A , ) K is in one-to-one corre-spondence with the set of invariant states of the associated holonomy dynamical system( A o , π o ( K ) , ∗ ). Since in a C ∗ -dynamical system having amenable group invariant statesalways exist, we conclude that when the fundamental group of K is amenable then anynondegenerate net has states; in fact we can compose states of the enveloping net bundle6ith the canonical embedding. If ( A , , α ) K is G -covariant net, then a state of the net ϕ is said to be G - invariant whenever ϕ go ◦ α go := ϕ o , ∀ o ∈ K , g ∈ G .
The next result concerns the existence of G -invariant states and is proved in [16]: Proposition 2.4 ([16]) . Let G be amenable. Then: (i) Any G -covariant C ∗ -net bundlehaving states has G -invariant states. (ii) If π ( K ) is amenable, then any nondegenerate G -covariant net over K has G -invariant states. A representation of the net ( A , ) K is given by a pair ( π, U ), where π := { π o : A o →B ( H o ) } is a family of representations and U := { U ˜ oo : H o → H ˜ o , o ≤ ˜ o } is a family ofunitary operators fulfilling the relations U o ∈ ( π o , π ˜ o ◦ ˜ oo ) , U o ′ ˜ o ◦ U ˜ oo = U o ′ o , ∀ o ≤ ˜ o ≤ o ′ . (2.10)We call U the family of inclusion operators . We say that ( π, U ) is faithful whenever π o is faithful for any o ∈ K , and that ( π, U ) is a Hilbert space representation whenever any U ˜ oo is the identity on a fixed Hilbert space (and in this case we write ( π, )).It follows from (2.10) that the pair ( H , U ) K , H := {H o } , is a Hilbert net bundle inthe sense of the previous section. Using the adjoint action we obtain the C ∗ -net bundle( BH , ad U ) K , so that ( π, U ) can be regarded as a morphism π : ( A , ) K → ( BH , ad U ) K . Hilbert space representations correspond, in essence, to morphisms with values in trivialnets. When K is simply connected any ( BH , ad U ) K is trivial, so we have only Hilbertspace representations. When K is not simply connected it is very easy to give examplesof nets having faithful representations but no Hilbert space representations (see [16]),and this is the reason why it is convenient to use the more general definition. In algebraicquantum field theory it is customary to use Hilbert spaces representations, also becausethe usual background is the Minkowski spacetime that is simply connected. In curvedspacetimes and in S it is of interest to give results stating the existence of (possibly)faithful representations, and this is the motivation of our work.Let us focus for a moment on C ∗ -net bundles. There exists a one-to-one correspon-dence between representations ( π, U ) of ( A , ) K and covariant representations ( π o , U ∗ )of the holonomy dynamical system ( A o , π o ( K ) , ∗ ). In particular U ∗ , which is a unitaryrepresentation of the fundamental group of K , is nothing but that the holonomy of theHilbert net bundle ( H , U ) K (see 2.6). Since any C ∗ -dynamical system has faithful co-variant representations, we conclude that any C ∗ -net bundle has faithful representations.We now return to the general case in which ( A , ) K is a net of C ∗ -algebras. Usingthe pullback (see (2.8)), we see that any representation ( π, U ) of ( A , ) K extends to arepresentation ( π ↑ , U ) of ( A , ) K and this yields a one-to-one correspondence betweenrepresentations of ( A , ) K and those of its enveloping net bundle. Thus, as C ∗ -net bun-dles are faithfully represented, we conclude that a net of C ∗ -algebras is injective if, andonly if, it has faithful representations . 7et G be a continuous symmetry group of K and ( A , , α ) K a G -covariant net. A G -covariant representation of ( A , , α ) K is a representation ( π, U ) of ( A , ) K with a stronglycontinuous family Γ of unitaries Γ go : H o → H go , g ∈ G , o ∈ K , such thatΓ hgo ◦ Γ go = Γ hgo , adΓ go ◦ π o = π go ◦ α go , Γ g ˜ o ◦ U ˜ oo = U ˜ go go ◦ Γ go , for all g, h ∈ G , o ≤ ˜ o ∈ K . With the term strongly continuous we mean the followingproperty: if { g λ } ⊂ G converges to the identity, then for any o ∈ K there exists a ≫ o and an index λ a such that g λ o ≤ a for any λ ≥ λ a and k U a g λ o Γ g λ o Ω − Ω k → , ∀ Ω ∈ H a . (2.11)We have the following result (see, as usual, [16]): Proposition 2.5.
Let K be a poset with amenable fundamental group and G an amenablecontinuous symmetry group of K . Then every injective, G -covariant net of C ∗ -algebrasover K has strongly continuous G -covariant representations. S Let I be the poset formed by the set of connected, open intervals of S having closure cl ( o ) properly contained in S , ordered by inclusion; that is, o ≤ a if, and only if, o ⊆ a .The homotopy group of this poset is Z , since I is a base for the topology of S . By a net of C ∗ -algebras over S we mean a net of C ∗ -algebras over I .On I there is a natural causal disjointness relation: o ⊥ a if, and only if, o ∩ a = ∅ .Important symmetries for nets over S are given by Diff( S ) or the M¨obius subgroup.These groups act continuously on S and, hence, on the poset I as well, according to(2.2,2.3,2.4). These groups are non-amenable. However there are important amenablesubgroups: the rotation group, the semidirect product of the translations and the dila-tions.In the present section we show that any net of C ∗ -algebras over S is injective, so itadmits faithful representations. As a consequence any such a net has states and, if thenet is covariant under Diff( S ), states which are invariant under any amenable subgroupof Diff( S ).To prove injectivity, we shall follow the strategy of finding a family of finite, ”ap-proximating” subposets of I , that we call cylinders , where injectivity can be established. A strategy for proving injectivity is suggested by the analysis of inductive systems ofnets (see in appendix). Let ( A , ) K be a net of C ∗ -algebras over a poset K . Assume thatthere is a family { K α } of subsets of K satisfying the following conditions:1. the family { K α } is upward directed under inclusion and K is the inductive limitposet of { K α } (each K α equipped with the order relation inherited by K );8. the net ( A , ) K α , that is the restriction of ( A , ) K to K α , is injective for any α ;Condition 1 says that the net ( A , ) K is the inductive limit of the system ( A , ) K α .Condition 2 implies the injectivity of the inductive limit net (Theorem A.5).Although it is a hard problem, even impossible in some cases, to find the right familyof subsets of a poset where injectivity can be established, this problem, in the case of S ,can be fully solved. We briefly explain how. Given a net ( A , ) I we will find a sequence {I N } , N ∈ N , of subsets of I satisfying the condition 1. Concerning condition 2., firstwe will construct (using I N ) a finite poset P N and show that the net ( A , ) I N embeds,faithfully on the fibres, into a suitable net over P N . Secondly, we shall show that anynet of C ∗ -algebras over P N is injective. These facts imply that the restrictions ( A , ) I N are injective for any N .The proof that any net over P N is injective relies on the isomorphism between P N and an abstract poset C N , called the N -cylinder, and on the fact that any net over thisposet is injective. N -Cylinder We now introduce a class { C N , N ∈ N } of finite posets. These are of interest for tworeasons; the first one is that any net of C ∗ -algebras over some C N is injective, and thesecond one is that, as we shall explain in the following, each C N arises from a suitablesimplicial approximation of the circle.As we shall see soon C N can be seen as a lattice of N elements on a cylinder of finiteheight. To deal with periodicity we shall use the equivalence relation mod N with thefollowing convention: we choose as representative elements of the classes associated withthe equivalence relation mod N the numbers 1 , , . . . , N . So, for instance, for N = 4 wehave, (0) = 4, ( − = 3, (5) = 1, etc... .Using this convention, elements of the N - cylinder C N are pairs ( i, l ) with i, l ∈{ , . . . , N } . We shall think of C N as a matrix whose rows and columns are indexed by l and i respectively. The order relation is defined inductively, as follows: given an element( i, l ) of the l -row, with l < N , it has only two majorants in the ( l + 1)-row, given by( i, l ) < ( i, l + 1) , ( i, l ) < (( i − N , l + 1) . (3.1)Finally, the relation among ( i, l ) and that of the ( l + t )-rows with t > N = 4, the poset C is represented by the following diagram,(4 ,
4) (1 ,
4) (2 ,
4) (3 ,
4) (4 , , O O (1 , O O c c GGGGGGGG (2 , O O c c GGGGGGGG (3 , O O c c GGGGGGGG (4 , O O c c GGGGGGGG (4 , O O (1 , O O c c GGGGGGGG (2 , O O c c GGGGGGGG (3 , O O c c GGGGGGGG (4 , O O c c GGGGGGGG (4 , O O (1 , O O c c GGGGGGGG (2 , O O c c GGGGGGGG (3 , O O c c GGGGGGGG (4 , O O c c GGGGGGGG where, for simplicity, the first column is the repetition of the last. The order relationis represented by an arrow from the smaller element to the greater one. So, C N has N maximal elements, those belonging to the N -row, and N minimal elements, thosebelonging to the 1-row.The rest of the section is devoted to proving that any net of C ∗ -algebras ( A , ) C N admits a faithful representation, a property equivalent to injectivity (see § A ( i,N ) . Take a cardinal κ greater than the cardinality of any such analgebra. Let ρ i denote the tensor product of the universal representation of the algebra A ( i,N ) and of 1 κ . Then define π ( i,l ) := ρ i ◦ ( i,N )( i,l ) , l = 1 , , . . . , N . (3.2)In words, the representation of the algebras associated with elements of the i -columnis obtained by restricting ρ i to such algebras. In particular π ( i,N ) = ρ i . In the case of N = 4 we will label the columns of the above diagram as follows ρ ρ ρ ρ ρ (4 ,
4) (1 ,
4) (2 ,
4) (3 ,
4) (4 , , O O (1 , O O c c GGGGGGGG (2 , O O c c GGGGGGGG (3 , O O c c GGGGGGGG (4 , O O c c GGGGGGGG (4 , O O (1 , O O c c GGGGGGGG (2 , O O c c GGGGGGGG (3 , O O c c GGGGGGGG (4 , O O c c GGGGGGGG (4 , O O (1 , O O c c GGGGGGGG (2 , O O c c GGGGGGGG (3 , O O c c GGGGGGGG (4 , O O c c GGGGGGGG
Since universal representations and the inclusion maps are faithful, any representation π ( i,l ) is faithful. 10e now define the inclusion operators. We proceed by defining the inclusion oper-ators from a l -row to ( l − l = N ; the others will be obtained bycomposition. To this end we note that the maximal element ( i, N ) has two minorantsin the ( N − i, N −
1) and (( i + 1) N , N − as the inclusion operatorfrom ( i, N −
1) and ( i, N ), because these two elements belong to the same column i .Concerning the inclusion operator from (( i + 1) N , N −
1) to ( i, N ), the representations π ( i,N ) ◦ ( i,N ) (( i +1) N ,N − and π (( i +1) N ,N − are unitarily equivalent, because they areunitarily equivalent to tensor product of the universal representation of the algebra A ( i +1) N ,N − and 1 κ (see [4]). So, there is a unitary operator V i, ( i +1) N such that V i, ( i +1) N π (( i +1) N ,N − = π ( i,N ) ◦ ( i,N ) (( i +1) N ,N − V i, ( i +1) N . (3.3)So we take V i, ( i +1) N as inclusion operator from (( i + 1) N , N −
1) to ( i, N ); the reason wayit is labelled only by the column indices will become clear soon. Given an element ( i, l ),with 1 < l < N , consider the minorants ( i, l −
1) and (( i + 1) N , l − as the inclusion operator from ( i, l −
1) and ( i, l ), since they belong to thesame column. But the important fact is that we may take the same operator V i, ( i +1) N satisfying equation (3.3) as inclusion operator from (( i + 1) N , l −
1) to ( i, l ). In fact byusing equation (3.3) and Definition (3.2) we have V i, ( i +1) N π (( i +1) N ,l − == V i, ( i +1) N ρ ( i +1) N ◦ (( i +1) N ,N ) (( i +1) N ,l − = V i, ( i +1) N ρ ( i +1) N ◦ (( i +1) N ,N ) (( i +1) N ,N − ◦ (( i +1) N ,N −
1) (( i +1) N ,l − = V i, ( i +1) N π (( i +1) N ,N − ◦ (( i +1) N ,N −
1) (( i +1) N ,l − = π ( i,N ) ◦ ( i,N ) (( i +1) N ,l − V i, ( i +1) = π ( i,N ) ◦ ( i,N ) ( i,l ) ◦ ( i,l ) (( i +1) N ,l − V i, ( i +1) = π ( i,l ) ◦ ( i,l ) (( i +1) N ,l − V i, ( i +1) , where we have used the relations (( i + 1) N , N − > (( i + 1) N , l −
1) and ( i, N ) > ( i, l )for N > l >
1. Hence V i, ( i +1) N π (( i +1) N ,l − = π ( i,l ) ◦ ( i,l ) (( i +1) N ,l − V i, ( i +1) . (3.4)11his choice, in the case of C , corresponds to the diagramme ρ ρ ρ ρ ρ (4 ,
4) (1 ,
4) (2 ,
4) (3 ,
4) (4 , , O O (1 , O O V c c GGGGGGGG (2 , O O V c c GGGGGGGG (3 , O O V c c GGGGGGGG (4 , O O V c c GGGGGGGG (4 , O O (1 , O O V c c GGGGGGGG (2 , O O V c c GGGGGGGG (3 , O O V c c GGGGGGGG (4 , O O V c c GGGGGGGG (4 , O O (1 , O O V c c GGGGGGGG (2 , O O V c c GGGGGGGG (3 , O O V c c GGGGGGGG (4 , O O V c c GGGGGGGG
Finally, consider a generic inclusion ( i k , l k ) > ( i , l ). This inclusion can be obtained bya composition ( i , l ) < ( i , l ) < · · · < ( i k − , l k − ) < ( i k , l k ) , (3.5)of the generators (3.1). Given such a composition we define the inclusion operator V ( i k ,l k )( i ,l ) := V ( i k ,l k )( i k − ,l k − ) V ( i k − ,l k − )( i k − ,l k − ) · · · V ( i ,l )( i ,l ) , (3.6)where the inclusion operators for the generators of C N are defined according to the aboveprescriptions. However note that for l k ≥ l + 2 the inclusion ( i k , l k ) > ( i , l ) can beobtained by different compositions of the generators (3.1). For instance, for C , theinclusion (3 , < (2 ,
3) can be obtained either(3 , < (3 , < (2 , , or (3 , < (2 , < (2 , . However, the definition (3.6) does not depend on the chosen composition. Because inany such composition the following ordered sequence of transitions from a column to thepreceding one ( mod N ) must be present: i → ( i − N → · · · → ( i k + 1) N → i k , (3.7)and no other. So the difference between two compositions of inclusions leading to( i k , l k ) > ( i , l ) depends on how inclusions preserving the column index are insertedbetween the elements of the sequence. However the inclusion operator, associated to in-clusions which preserve the column index, is the identity. Since the inclusion operatorsassociated with inclusions of the form ( i, l ) < (( i − N , l + 1) depend only on the columnindex, the definition (3.6) does not depend on the chosen path. Thus the pair ( π, V ) isa faithful representation of ( A , ) C N (see 2.10), and in conclusion we have the following Proposition 3.1.
Any net of C ∗ -algebras over C N is injective. .3 Finite approximations of S , and injectivity Following the strategy outlined in § ∗ -algebras ( A , ) I over S is injective. The subsets I N . Let { x n } be a dense sequence of points in S . Define I N := ∪ Ni =1 I x i , N ∈ N , (3.8)where I x := { o ∈ I | x cl ( o ) } , for a point x of S , and cl ( o ) denotes the closure of theinterval o . Note that I x is, as a subposet of I , upward directed. For N ≥ I N is a base of neighbourhoods for the topology of S , so its homotopy group is Z .The family {I N } satisfy condition 1. outlined in § I N ⊂ I N +1 for any N ∈ N . So given a net of C ∗ -algebras ( A , ) I considerthe restrictions ( A , ) I N for any N , and for any inclusion N ≤ M define (cid:26) i M,N ( o ) := o , o ∈ I N ,ι M,No ( A ) := A , o ∈ I N , A ∈ A o , giving unital monomorphisms ( ι M,N , i M,N ) : ( A , ) I N → ( A , ) I M . Lemma 3.2.
Given a net of C ∗ -algebras ( A , ) I over S , then(i) (cid:8) ( A , ) I N , ( ι M,N , i M,N ) (cid:9) N is an inductive system of nets of C ∗ -algebras whose link-ing morphisms ( ι M,N , i M,N ) are monomorphisms.(ii) ( A , ) I is isomorphic to the inductive limit of (cid:8) ( A , ) I N , ( ι M,N , i M,N ) (cid:9) N .Proof. ( i ) easily follows from the definition of inductive system (A.2). ( ii ) Once we haveshown that I is the inductive limit poset of ( I N , i M,N ) N (see § A.1), the proof follows fromthe definition of ( A , ) I N and from the universal property of inductive limits Prop.A.3,To this end it is enough to observe that since { x n } is, by assumption, dense in S , forany o ∈ I there exist N o ∈ N such that o ∈ I N for any N ≥ N o . In fact, by density,there exists N o such that x N o ∈ S \ cl ( o ); hence o ∈ I N o .So what remains to be shown is that the nets ( A , ) I N are injective for any N . Wefirst introduce some suited finite approximations of S . Finite approximations of S : the poset P N . Starting from the sequence { x n } introduced in the previous step, we construct for any N ∈ N a finite poset P N associatedto the first N elements x , . . . , x N of the sequence.The definition of the poset P N is notably simplified if we assume that the points x , . . . , x N are ordered as x i < x i +1 , i = 1 , . . . N − , S . This does not affect the generality of the proof ofthe injectivity of the net ( A , ) I N since I N does not depend on the order of the points x , . . . , x N .Given the ordered N -ple x , . . . , x N , the elements of the poset P N are the openintervals ( x i , x k ), for i, k = 1 , . . . , N , having, with respect to the clockwise orientation, x i as initial extreme and x k as final extreme respectively, ordered under inclusion. Inthis way, each ( x i , x i ) is the interval S \ { x i } and hence is maximal; on the other hand,the intervals ( x i , x i +1 ), i = N , and ( x N , x ), have contiguous extreme points and henceare minimal.To handle the periodicity with respect to clockwise rotations we use classes mod N ,in the following way: for each n ∈ N we denote its class mod N by ( n ) N ∈ , . . . , N (here we use the convention of § x ( n ) N ∈ S . Moreover, weintroduce the length function assigning to each ( x i , x k ) ∈ P N the positive integer ℓ i,k := ♯ { j ∈ { , . . . , N } : ( x ( j ) N , x ( j +1) N ) ⊆ ( x i , x k ) } , where ♯ stands for the cardinality. In this way each minimal element of P N has length1 and each maximal element has length N . Note that the length of an interval can beeasily calculated from the indices, ℓ i,k = ( k − i ) N . Any interval ( x i , x k ) of length ℓ i,k < N has only two majorants among the intervals oflength ℓ i,k +1, namely ( x i , x ( k +1) N ) and ( x ( i − N , x k ). Finally, note that with our conven-tions the points of { x , . . . , x N } belonging to ( x i , x k ) are x ( i +1) N , x ( i +2) N , . . . , x ( i + ℓ ik − N . Lemma 3.3.
The poset P N is isomorphic to the cylinder C N .Proof. We define a mapping f : P N → C N as follows:f( x i , x k ) := ( i, ℓ ik ) , i, k ∈ { , , . . . N } . This map has inverse f ′ ( i, ℓ ) := ( x i , x ( i + ℓ ) N ) , ( i, ℓ ) ∈ C N , and is thus bijective. To prove that f is order preserving it suffices to compute, for ℓ i,k < N f( x i , x ( k +1) N ) = ( i, ℓ i,k + 1) > ( i, ℓ i,k ) = f( x i , x ( ℓ i,k + i ) N ) = f( x i , x k ) , and f( x ( i − N , x k ) = (( i − N , ℓ i,k + 1) > ( i, ℓ i,k ) = f( x i , x ( ℓ i,k + i ) N ) = f( x i , x k ) . So f is an isomorphism and the proof follows.14 nducing nets over P N . We now show that the net ( A , ) I induces a net over P N . Inthe next paragraph we shall see that the restrictions ( A , ) I N embed, faithfully on thefibres, into such a nets.For any i, k = 1 , , . . . , N , let I ( i,k ) N := (cid:8) o ∈ I | cl ( o ) ⊂ ( x i , x k ) (cid:9) . (3.9)According to this definition ( x i , x k ) / ∈ I ( i,k ) N and I ( i,k ) N ⊂ I N . The poset I ( i,k ) N is upwarddirected with respect to inclusion; so {A o , ao } I ( i,k ) N is an inductive system, and we definethe inductive limit C ∗ -algebra b A ( i,k ) := lim −→{A o , ao } I ( i,k ) N . (3.10)The algebras b A ( i,k ) are associated with the interval ( x i , x k ) of P N . So we have definedthe fibres of a net over P N . We now define the inclusion maps. Let J ( i,k ) o : A o → b A ( i,k ) be the canonical embedding for o ∈ I ( i,k ) N . This is a unital monomorphism satisfying therelations J ( i,k ) o ◦ oa = J ( i,k ) a , a ≤ o. (3.11)Note that if ( x i , x k ) ⊆ ( x j , x s ) then I ( i,k ) N ⊆ I ( j,s ) N . We then define, for a ∈ I ( i,k ) N , b ( j,s )( i,k ) ( J ( i,k ) a ( A )) := J ( j,s ) o ◦ oa ( A ) , A ∈ A a , (3.12)where we take some o ∈ I ( j,s ) N with a ≤ o since I ( i,k ) N is directed. We easily find, applying(3.11), that our definition does not depend on the choice of o ≥ a . It turns out that b ( j,s )( i,k ) extends to a unital monomorphism from b A ( i,k ) into b A ( j,s ) ; applying (3.12), weimmediately find that b ( j,s )( i,k ) fulfills the net relations b ( j,s )( i,k ) ◦ b ( i,k )( m,r ) = b ( j,s )( m,r ) , ( x m , x r ) ⊆ ( x i , x k ) ⊆ ( x j , x s ) , therefore the system ( b A , b ) P N is a net of C ∗ -algebras. Lemma 3.4.
The net ( b A , b ) P N is injective for any N .Proof. This follows by Proposition 3.1 and Lemma 3.3.
Conclusion: the embedding of the nets ( A , ) I N We now are ready to prove thatany net over S is injective. Theorem 3.5.
Any net of C ∗ -algebras ( A , ) I over S is injective. In particular, thenet ( A , ) I N embeds, faithfully on the fibres, into ( b A , b ) P N for any N .Proof. Injectivity of ( A , ) I follows from Lemma 3.2 and Theorem A.5, once we haveshown that ( A , ) I N is injective for any N . To this end, as observed in Remark 2.3,it will be enough to show that ( A , ) I N embeds, faithfully on the fibres, into ( b A , b ) P N P N is a quotient of I N . Definef( o ) := ( x i , x k ) , o ∈ I N (3.13)if (cid:26) cl ( o ) ⊂ ( x i , x k ) and x ( i +1) N , x ( i +2) N , . . . , x ( i + ℓ ik − N ∈ cl ( o ) , (3.14)where the points listed in the above equation are those x , x , . . . , x N that belong to( x i , x k ). It is clear that f is order preserving and surjective, i.e. it is an epimorphism.As a second step, we define a unital morphism faithful on the fibres from ( A , ) I N intothe injective net ( b A , b ) P N , and this will suffice to conclude the proof (by functoriality).Given o ∈ I N , define η o ( A ) := J f( o ) o ( A ) , A ∈ A o . (3.15)So η o : A o → b A f( o ) is a unital monomorphism. Given a ≤ o , by (3.15) and (3.12), wehave η o ◦ oa = J f( o ) o ◦ oa = b f( o )f( a ) ◦ J f( a ) o , and can conclude that ( η, f) : ( A , ) I N → ( b A , b ) P N is a unital morphism faithful on thefibres, completing the proof.Using this theorem we deduce the following result for Diff( S )-covariant nets. Corollary 3.6.
Any
Diff( S ) -covariant net of C ∗ -algebras over S has H -invariantstates and strongly continuous H -covariant representations for any amenable subgroup H of Diff( S ) .Proof. Since any net of C ∗ -algebras over S is injective and the homotopy group of S is amenable, the proof follows by Prop.2.4 and Prop.2.5.We stress that, by Prop.2.5, the covariant representations of the previous corollaryinduce strongly continuous H -representations in the sense of (2.11). We list some topics and questions arising from the present paper.1. A problem which deserves a further investigation is whether any net over S admitsfaithful Hilbert space representations. This is a stronger condition than injectivity andis related with the existence of a proper ideal of the fibres of the enveloping net bundle[16].2. An interesting question is whether it is possible to construct representations of netsover S directly from the representations of the associated nets over N -cylinders § N -cylinders have some symmetries that might be in-herited by the induced representations of the nets over S .16. Having shown that any net over S is injective, it would be interesting to understandthe rˆole of representations, carrying a nontrivial representation of the fundamental groupof S , in chiral conformal quantum field theories and, in particular, to relate them tothe new superselection sectors introduced in [7], since they carry the same topologicalcontent.4. It is hoped that the method used in the present paper to prove injectivity of nets over S can be used for other manifolds. This point is the object of a work in progress. A Inductive limits
The constructions that can be made in the category of C ∗ -algebras can be also madein the category of nets of C ∗ -algebras. The direct sum of two nets, for instance, hasfibres and inclusion maps given by, respectively, the direct sum of the fibres and of theinclusion maps of the two nets. On the same line the tensor product is defined. In thisappendix, however, we focus on a single construction which will turns out to be veryuseful for analysing injectivity: the inductive limit. We show that the category of netsof C ∗ -algebras has inductive limits. The functor assigning the enveloping net bundleturns out to preserve inductive limits, and this implies that inductive limits preserveinjectivity. A.1 Basic properties A inductive system of nets of C ∗ -algebras is given by the following data: an upwarddirected poset Λ (we shall denote the elements of Λ by Greek letters α, β , etc..., andthe order relation by (cid:22) ); a family of nets of C ∗ -algebras ( A α , α ) K α , with α ∈ Λ, anda family of unital morphisms ( ψ ασ , f ασ ) : ( A σ , σ ) K σ → ( A α , α ) K α for σ (cid:22) α , with f ασ injective, such that( ψ ασ , f ασ ) ◦ ( ψ σδ , f σδ ) = ( ψ αδ , f αδ ) , δ (cid:22) σ (cid:22) α . (A.1)We shall denote such an inductive system by (cid:8) ( A α , α ) K α , ( ψ σα , f σα ) (cid:9) Λ , (A.2)and call ( ψ σα , f σα ) the linking morphisms of the system.We stress that do not assume in the definition of inductive system that the linkingmorphisms are monomorphisms, but we only require that f ασ is a monomorphism ofposets.Note that by the definition of morphisms of nets, for any α (cid:22) σ , ψ σαa ◦ αae = σ f σα ( a )f σα ( e ) ◦ ψ σαe , e ≤ α a , (A.3)17here ≤ α is the order relation of K α . Moreover, it is worth observing that an inductivesystem of posets { K α , f σα } Λ is associated with our inductive system of nets. In thefollowing we show that any inductive system of nets of C ∗ -algebras has an inductivelimit, which turns out to be a net of C ∗ -algebras over the inductive limit poset.First of all we explain the inductive limit poset of { K α , f ασ } Λ . This is a poset K ,with order relation denoted by ≤ , such that for any α ∈ Λ there is a monomorphism F α : K α → K satisfying the following properties:(1) F α ◦ f αβ = F β for any β (cid:22) α ;(2) K = ∪ α ∈ Λ F α ( K α ) ;(3) for any other poset K ′ with a family of morphisms H α : K α → K ′ such thatH α ◦ f αβ = H β there exists a unique morphism H : K → K ′ such that H ◦ F α = H α for any α ∈ Λ.Existence of the inductive limit poset can be proved as a consequence of the more generalconstruction of colimits in the setting of small categories (see [14, § III.3]).Now, given o ∈ K , we considerΛ o := { α ∈ Λ : o ∈ F α ( K α ) } ⊆ Λ . Clearly Λ o is not empty because of property (2) of the inductive limit poset. Lemma A.1. If K is the inductive limit poset of ( K α , f σα ) Λ , then the following prop-erties hold: (i) Λ o is an upper set of Λ for any o ∈ K and, as a subposet of Λ , it isupward directed. (ii) For any o ∈ K there is a map Λ o ∋ α o α ∈ K α such that f σα ( o α ) = o σ , α (cid:22) σ ;F α ( o α ) = o , α ∈ Λ o . (A.4) Proof. ( i ) Given α ∈ Λ o the element a ∈ K α satisfying F α ( a ) = o is unique because F α is injective, so we denote this element by o α . Now, given α ∈ Λ o , and β ∈ S with α (cid:22) β then β ∈ Λ o , since F β (f βα ( o α )) = F α ( o α ) = o , so Λ o is an upper set of Λ. Thus Λ o isupward directed because Λ is. ( ii ) has been proved in the course of proving ( i ).We now construct the inductive limit of (cid:8) ( A α , α ) K α , ( ψ σα , f σα ) (cid:9) Λ . Given o ∈ K ,using the properties of the function Λ o ∋ α o α and applying (A.4), (A.3), we seethat {A αo α , ψ σαo α } Λ o is an inductive system of C ∗ -algebras over Λ o . So we can define theC ∗ -inductive limit A o := lim −→A αo α . (A.5)Now, it is worth recalling some facts about inductive limits (see [5]). First of all, thereis a unital morphism ψ αo : A αo α → A o such that ψ σo ◦ ψ σαo α = ψ αo , α (cid:22) σ . (A.6) An upper set P of Λ is a subset such that if σ ∈ Λ and there is α ∈ P such that σ (cid:22) α , then σ ∈ P . ∗ -algebra A ′ o defined by A ′ o := [ α ∈ Λ o ψ αo ( A αo α ) (A.7)is a dense ∗ -subalgebra of A o for any o ∈ K . Finally, the norm k · k ∞ of the inductivelimit satisfies, for any A ∈ A αa , the relation k ψ αo ( A ) k ∞ = inf σ (cid:23) α k ψ σαo α ( A ) k = lim σ (cid:23) α k ψ σαo α ( A ) k , (A.8)because the norm k ψ σαo α ( A ) k is monotone decreasing in σ .The correspondence A : K ∋ o → A o , where A o is the unital C ∗ -algebra defined byequation (A.5), is the fibre of the inductive limit net over K . What is yet missing arethe inclusion maps. Given o, ˜ o ∈ K with o ≤ ˜ o , we first define the inclusion map ˜ oo on the ∗ -algebra A ′ o . Afterwards we shall prove that these maps can be isometricallyextended to all of A o . Given o, ˜ o as above, take α ∈ Λ o and choose σ ∈ Λ ˜ o such that σ (cid:23) α , and, for any A ∈ A αa , define ˜ oo ( ψ αo ( A )) := ψ σ ˜ o ◦ σ ˜ o σ o σ ◦ ψ σαo α ( A ) . (A.9)This definition does not depend on the choice of σ ∈ Λ ˜ o with σ (cid:23) α . In fact, take γ ∈ Λ ˜ o with γ (cid:23) σ . Relations (A.6), (A.3),(A.4), and (A.1) give ψ σ ˜ o ◦ σ ˜ o σ o σ ◦ ψ σαo α = ψ γ ˜ o ◦ ψ γσ ˜ o σ ◦ σ ˜ o σ o σ ◦ ψ σαo α = ψ γ ˜ o ◦ γ f γσ (˜ o σ )f γσ ( o σ ) ◦ ψ γσo σ ◦ ψ σαo α = ψ γ ˜ o ◦ γ ˜ o γ o γ ◦ ψ γαo α . By the definition of A ′ o ˜ oo maps from A ′ o into A ′ ˜ o since Λ o is upward directed. We nowprove that these maps satisfy all the properties of inclusion maps. Lemma A.2.
Given an inductive system (cid:8) ( A α , α ) K α , ( ψ σα , f σα ) (cid:9) Λ of nets of C ∗ -algebras the following assertions hold:(i) The triple ( A , ) K is a net of C ∗ -algebras.(ii) If the system is composed of C ∗ -net bundles, then ( A , ) K is a C ∗ -net bundle.Proof. ( i ) By definition ˜ oo : A ′ ˜ o → A ′ o is a unital morphism such that ˆ o ˜ o ◦ ˜ oo = ˆ oo , o ≤ ˜ o ≤ ˆ o . We now prove that ˜ oo are isometries. To this end, given α ∈ Λ o , take σ ∈ Λ ˜ o with σ (cid:23) α . By (A.8), (A.3) and (A.1) we have k ˜ oo ◦ ψ αo ( A ) k ∞ = inf γ (cid:23) σ k ψ γσ ˜ o σ ◦ σ ˜ o σ o σ ◦ ψ σαo α ( A ) k = inf γ (cid:23) σ k γ ˜ o γ o γ ◦ ψ γσo σ ◦ ψ σαo α ( A ) k = inf γ (cid:23) σ k γ ˜ o γ o γ ◦ ψ γαo α ( A ) k = inf γ (cid:23) σ k ψ γαo α ( A ) k = k ψ αo ( A ) k ∞ , γ ˜ o γ o γ are isometries. So ˜ oo admits an isometric extension to a mapping from A o into A ˜ o .( ii ) We study the image of ˜ oo . First of all we show that the following relations hold forevery α ∈ Λ o , ˜ oo ◦ ψ αo ( A ) = ψ α ˜ o ◦ α ˜ o α o α ( A ) , A ∈ A αo α ; (A.10)in fact, given σ ∈ Λ ˜ o with σ (cid:23) α , by A.9 and (A.6) we have ˜ oo ◦ ψ αo ( A ) = ψ σ ˜ o ◦ σ ˜ o σ o σ ◦ ψ σαo α ( A )= ψ σ ˜ o ◦ ψ σα ˜ o α ◦ α ˜ o α o α ( A ) = ψ α ˜ o ◦ α ˜ o α o α ( A ) . Since every α ˜ o α o α : A αo α → A α ˜ o α is an isomorphism, the previous relation gives ˜ oo ◦ ψ αo ( A αo α ) = ψ α ˜ o ( A α ˜ o α ). So, ˜ oo ( A ′ o ) = A ′ ˜ o , and this, in turns, implies that ˜ oo extends toan isomorphism from A o to A ˜ o , completing the proof.Finally, we have the following result. Proposition A.3.
Let (cid:8) ( A α , α ) K α , ( ψ σα , f σα ) (cid:9) Λ be an inductive system of nets of C ∗ -algebras. Then for each α ∈ Λ there is a unital morphism (Ψ α , F α ) : ( A α , α ) K α → ( A , ) K such that(i) (Ψ σ , F σ ) ◦ ( ψ σα , f σα ) = (Ψ α , F α ) for any α (cid:22) σ ;(ii) The image Ψ αo α ( A αo α ) , as α varies in Λ o , is dense in A o for any o ∈ K ;(iii) If there is a net of C ∗ -algebras ( B , y ) K and a collection of morphisms (Φ α , F α ) :( A α , α ) K α → ( B , y ) K , α ∈ Λ , such that (Φ σ , F σ ) ◦ ( ψ σα , f σα ) = (Φ α , F α ) , α (cid:22) σ ,then there exists a unique morphism Φ : ( A , ) K → ( B , y ) K such that Φ ◦ (Ψ α , F α ) =(Φ α , F α ) for any α ∈ Λ .Proof. Define Ψ αa := ψ α F α ( a ) , a ∈ K α . (A.11)By relation (A.10) we haveΨ αa ◦ αae = ψ α F α ( a ) ◦ αae = F α ( a )F α ( e ) ◦ ψ α F α ( e ) = F α ( a )F α ( e ) ◦ Ψ αe , and this proves that (Ψ α , F α ) : ( A α , α ) K α → ( A , ) K is a unital morphism.In this way ( i ) and ( ii ) are easy to prove. ( iii ) follows from the universal property ofinductive limits of C ∗ -algebras: in fact, if α ∈ Λ o then Φ αo α : A αo α → B o is a collection ofmorphisms satisfying the relation Φ σo σ ◦ ψ σαo α = Φ αo α , by the universal property of the inductive limit A o there is a unique morphism Φ o : A o → B o such that Φ o ◦ Ψ αo α = Φ αo α . So let Φ denotes the collection Φ o , with o ∈ K .20e now prove that Φ is a morphism of nets. Take o, ˜ o ∈ K and α ∈ Λ o , σ ∈ Λ ˜ o as inDefinition (A.10). Then we haveΦ ˜ o ◦ ˜ oo ◦ Ψ αo α = Φ o ◦ Ψ σ ˜ o σ ◦ σ ˜ o σ o σ ◦ ψ σαo α = Φ o ◦ Ψ σ ˜ o σ ◦ ψ σα ˜ o α ◦ α ˜ o α o α = Φ σ ˜ o σ ◦ ψ σα ˜ o α ◦ α ˜ o α o α = Φ α ˜ o α ◦ α ˜ o α o α = y ˜ oo ◦ Φ αo α = y ˜ oo ◦ Φ o ◦ Ψ αo α for any α . Density implies that Φ : ( A , ) K → ( B , y ) K is a morphism, and it is clear thatΦ ◦ (Ψ α , F α ) = (Φ α , F α ). Finally, uniqueness follows in a similar fashion.Given a net of C ∗ -algebras satisfying the properties of the previous proposition, aninductive system (cid:8) ( A α , α ) K α , ( ψ σα , f σα ) (cid:9) Λ , will be called the inductive limit of thesystem and, from now on, will be denoted by lim −→ ( A α , α ) K α . The property ( iii ) of theproposition is the universal property of inductive limits. A.2 Injectivity of inductive limits
By functoriality, to any inductive system of nets there corresponds an inductive systemof enveloping net bundles. The functor of taking the enveloping net bundle commuteswith inductive limits. As a consequence, injectivity is preserved under inductive limitsindicating a strategy for analyzing injectivity.Consider an inductive system of nets of C ∗ -algebras (cid:8) ( A α , α ) K α , ( ψ σα , f σα ) (cid:9) Λ . Let( A α , α ) K α be the enveloping net bundle of ( A α , α ) K α , and let ǫ α : ( A α , α ) K α → ( A α , α ) K α be the canonical embedding. By the universal property of the enveloping net bundle, forany α (cid:22) σ there is a unique morphism ( ψ σα , f σα ) : ( A α , α ) K α → ( A σ , σ ) K σ satisfyingthe relation. ( ψ σα , f σα ) ◦ ǫ α = ǫ σ ◦ ( ψ σα , f σα ) . (A.12)This implies ( ψ σα , f σα ) ◦ ( ψ αβ , f αβ ) = ( ψ σβ , f σβ ) , β (cid:22) α (cid:22) σ , (A.13)hence { ( A α , α ) K α , ( ψ σα , f σα ) } Λ is an inductive system of C ∗ -net bundles; the system ofthe enveloping net bundles .Note that even if the linking morphisms of the original system are monomorphism,in general the linking morphisms of the system of enveloping net bundles may not bemonomorphisms. This depends on the injectivity of the nets of the original system.Let lim −→ ( A α , α ) K α be the C ∗ -net bundle inductive limit of the system of the envelop-ing net bundles, and denote by (Ψ α , F α ) be the embedding of the nets ( A α , α ) K α intothis limit. We now show that this limit is nothing but the enveloping net bundle of thenet lim −→ ( A α , α ) K α . 21 roposition A.4. Let { ( A α , α ) K α , ( ψ σα , f σα ) } Λ be an inductive system. Then the in-ductive limit lim −→ ( A α , α ) K α is isomorphic to the enveloping net bundle of the inductivelimit lim −→ ( A α , α ) K α .Proof. Let ( A , ) K be the enveloping net bundle of lim −→ ( A α , α ) K α and ǫ : lim −→ ( A α , α ) K α → ( A , ) K denote the canonical embedding. We start by showing that ( A , ) K embeds intolim −→ ( A α , α ) K α . To this end, we use the universal property of inductive limits of nets(Proposition A.3. iii ): given α ∈ Λ, note that (Ψ α , F α ) ◦ ǫ α : ( A α , α ) K α → lim −→ ( A α , α ) K α is a morphism such that(Ψ α , F α ) ◦ ǫ α ◦ ( ψ αβ , f αβ ) = (Ψ α , F α ) ◦ ( ψ αβ , f αβ ) ◦ ǫ β = (Ψ β , F β ) ◦ ǫ β , (A.14)so, by the universal the property of the inductive limit lim −→ ( A α , α ) K α , there exists amorphism θ : lim −→ ( A α , α ) K α → lim −→ ( A α , α ) K α such that θ ◦ (Ψ α , F α ) = (Ψ α , F α ) ◦ ǫ α , α ∈ Λ . (A.15)On the other hand, since lim −→ ( A α , α ) K α is a C ∗ -net bundle, the universal property ofthe enveloping net bundle ( A , ) K says that there is a unique morphism Θ : ( A , ) K → lim −→ ( A α , α ) K α such that Θ ◦ ǫ = θ. (A.16)We now prove that there is a morphism in the opposite direction. Consider the mor-phisms ǫ ◦ (Ψ α , F α ) : ( A α , α ) K α → ( A , ) K , α ∈ S .
The universal property of the enveloping net bundle ( A α , α ) K α implies that there aremorphisms ( χ α , F α ) : ( A α , α ) K α → ( A , ) K , intertwining the canonical embeddings( χ α , F α ) ◦ ǫ α = ǫ ◦ (Ψ α , F α ) , (A.17)and compatible with the inductive structures of the enveloping net bundles( χ α , F α ) ◦ ( ψ ασ , f ασ ) = ( χ σ , F σ ) . (A.18)By the universal property of inductive the limit lim −→ ( A α , α ) K α , there is a unique mor-phism Θ ′ : lim −→ ( A α , α ) K α → ( A , ) K satisfying the equationΘ ′ ◦ (Ψ α , F α ) = ( χ α , F α ) , α ∈ Λ . (A.19)We now prove that Θ is the inverse of Θ ′ . First, using equations (A.16), (A.15) (A.19)and (A.17), we note that for any α ∈ Λ,Θ ′ ◦ Θ ◦ ǫ ◦ (Ψ α , F α ) = Θ ′ ◦ θ ◦ (Ψ α , F α )= Θ ′ ◦ (Ψ α , F α ) ◦ ǫ α = ( χ α , F α ) ◦ ǫ α = ǫ ◦ (Ψ α , F α ) ,
22o Θ ′ ◦ Θ ◦ ǫ = ǫ . By (2.7) we conclude that Θ ′ ◦ Θ is the identity automorphism of( A , ) K .Conversely, equations (A.19), (A.17), (A.16) and (A.15), for any α ∈ Λ, implyΘ ◦ Θ ′ ◦ (Ψ α , F α ) ◦ ǫ α = Θ ◦ ( χ α , F α ) ◦ ǫ α = Θ ◦ ǫ ◦ (Ψ α , F α ) = θ ◦ (Ψ α , F α )= (Ψ α , F α ) ◦ ǫ α , thus (2.7), implies that Θ ◦ Θ ′ ◦ (Ψ α , F α ) = (Ψ α , F α ) for any α , and Θ ◦ Θ ′ is the identityautomorphism of lim −→ ( A α , α ) K α .We now prove the main result of the present section. Theorem A.5.
Let { ( A α , α ) K α , ( ψ σα , f σα ) } Λ be an inductive system of net of C ∗ -algebras. If the linking morphisms are monomorphisms and the nets ( A α , α ) K α are allinjective, then the inductive limit lim −→ ( A α , α ) K α is an injective net.Proof. It is enough to prove that the morphism θ : lim −→ ( A α , α ) K α → lim −→ ( A α , α ) K α , defined in the previous proposition, is a monomorphism. To this end, considering o ∈ K and A ∈ A αo α , and applying (A.15), Proposition A.3. i and (A.12), we find k θ o ◦ Ψ αo α ( A ) k ∞ = k Ψ αo ◦ ǫ αo α ( A ) k ∞ = inf σ ≥ α k ψ σαo α ◦ ǫ αo α ( A ) k = inf σ ≥ α k ǫ σo σ ◦ ψ σαo α ( A ) k = k A k , as both ǫ σo σ and ψ σαo α are isometries. Since ∪ α Ψ αo α ( A αo α ) is dense in A o (Proposition A.3. ii ),we conclude that θ o is an isometry for any o ∈ K ; hence θ is a monomorphism. A.3 On the continuity condition
We prove that the notion of a continuous symmetry group G of a poset K as given in thepresent paper implies that introduced in [16]. We shall use the simplicial set associatedto the poset and the corresponding notion of homotopy equivalence of paths. For allthese notions and related results we refer the reader to the cited paper. Lemma A.6.
Let G be a continuous symmetry group of a poset K . Then for any path p : o → a there ˜ o, ˜ a ∈ K , with o ≤ ˜ o and a ≤ ˜ a , and an open neighbourhood U of theidentity e of G such that ga ≤ ˜ a and go ≤ o and (˜ aa ) ∗ p ∗ (˜ oo ) ∼ (˜ ag ( a )) ∗ gp ∗ (˜ og ( o )) , g ∈ U . roof. We give a proof by induction. Let b a 1-simplex. By continuity of the action of G there is O ∈ K such that | b | ≪ O . Note in particular that the faces of the 1-simplexsatisfy ∂ b, ∂ b ≪ O . Let V be the neighbourhood of identity of G associated to | b | ≪ O .Then ( O, ∂ b ) ∗ b ∗ ( O, ∂ b ) ∼ ( O, g ( ∂ b )) ∗ g ( b ) ∗ ( O, g ( ∂ b )) , g ∈ V . (A.20)In fact, note that all the elements of the poset involved in the above relation are smallerthan O for any g ∈ V . Then homotopy equivalence follows because any upward directedposet is simply connected.Assume that the above relation holds for paths which are composition of n p : o → a be such a path and let b a 1-simplex such that ∂ b = o .By hypothesis there are o ≤ ˜ o and a ≤ ˜ a , and an open neighbourhood W of the identity e of G such that ga ≤ ˜ a and go ≤ o and(˜ aa ) ∗ p ∗ (˜ oo ) ∼ (˜ ag ( a )) ∗ g ( p ) ∗ (˜ og ( o )) , g ∈ W . (A.21)Let O and V be as in the equation (A.20). Since ˜ o, O ≫ o , there exists o ′ such that o ≪ o ′ ≪ ˜ o, O . Let V ′ be the neighbourhood of the indentity of G associated to o ≪ o ′ .If U := V ∩ W ∩ V ′ , then the equation (A.20) and (A.21) are verified for any g ∈ U .Furthermore since o ≤ o ′ ≤ ˜ o , we have (˜ o, o ′ ) ∗ ( o ′ o ) ∼ (˜ o, o ). Since homotopy equivalenceis stable under composition, we have (˜ oo ) ∗ (˜ oo ′ ) ∼ ( o ′ o ). This and equation (A.21) yield(˜ aa ) ∗ p ∗ ( o ′ o ) ∼ (˜ ag ( a )) ∗ g ( p ) ∗ ( o ′ g ( o )) , g ∈ U . (A.22)The same argument applied to equation (A.20) yields( o ′ , o ) ∗ b ∗ ( O, ∂ b ) ∼ ( o ′ , g ( o )) ∗ g ( b ) ∗ ( O, g ( ∂ b )) , g ∈ U , (A.23)(recall that o = ∂ b ). The composition of the left hand sides of the equations (A.22)(A.23) gives (˜ aa ) ∗ p ∗ ( o ′ o ) ∗ ( o ′ , o ) ∗ b ∗ ( O, ∂ b ) ∼ (˜ aa ) ∗ p ∗ b ∗ ( O, ∂ b ) , (A.24)while the composition of the right hand sides gives(˜ ag ( a )) ∗ g ( p ) ∗ ( o ′ g ( o )) ∗ ( o ′ , g ( o )) ∗ g ( b ) ∗ ( O, g ( ∂ b )) ∼ (˜ ag ( a )) ∗ g ( p ∗ b ) ∗ ( O, g ( ∂ b )) (A.25)for any g ∈ U . Finally, the equations (A.22),(A.23),(A.24) and (A.25) give(˜ aa ) ∗ p ∗ b ∗ ( O, ∂ b ) ∼ (˜ ag ( a )) ∗ g ( p ∗ b ) ∗ ( O, g ( ∂ b )) , g ∈ U , completing the proof.
Aknowledgements.
We gratefully acknowledge the hospitality and support of the GraduateSchool of Mathematical Sciences of the University of Tokyo, where part of this paper has beendeveloped, in particular Yasuyuki Kawahigashi for his warm hospitality. We also would like tothank all the operator algebra group of the University of Roma “Tor Vergata”, Sebastiano Carpiand Fabio Ciolli, for the several fruitful discussions on the topics treated in this paper. eferences [1] H. Araki. Mathematical theory of quantum fields.
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