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Dissertation zur Erlangung des Doktorgradesder Mathematisch-Naturwissenschaftlichen Fakult¨atender Georg-August-Universit¨at G ¨ottingenangefertigt von A NSG AR S CHNEIDER aus F OERDE geboren in S IEG EN
G ¨ottingen, im Oktober 2007 einen Eltern und Großeltern– Jetzt habt ihr den Salat! orrede und Danksagung
Die vorliegende Arbeit ist das wesentliche Ergebnis meiner Zeit am mathema-tischen Institut der Universit¨at G ¨ottingen. Dort war ich vom Sommer 2005 biszum Herbst 2007 Promotionsstudent, unterst ¨utzt durch den dortigen Gradu-iertenkolleg Gruppen und Geometrie.Wie der Titel sagt, besch¨aftigt sich die Arbeit mit der lokalen Struktur vonT-Dualit¨atstripeln. T-Dualit¨atstripel (das T steht f ¨ur Torus) sind mathemati-sche Objekte, die sich als zweckm¨aßig erwiesen haben, T-Dualit¨at mittels to-pologischer Methoden zu beschreiben [BRS]. T-Dualit¨at selbst ist eine Dualit¨atvon Stringtheorien [Po] auf zwei verschiedenen Raumzeitmanigfaltigkeiten.Im einfachsten Falle sind diese durch eine Transformation Radius C ∗ -algebraischer Zugang beschrieben, dessen zentrale Objektegewisse C ∗ -dynamische Systeme sind, deren Dualit¨atstheorie wiederum eineandere mathematische Beschreibung von T-Dualit¨at liefert.Das Ziel dieser Arbeit ist es, einen expliziten Zusammenhang zwischendem C ∗ -algebraischen Zugang und den Resultaten ¨uber topologische T-Dualit¨atherzustellen. Dabei wird sich zeigen, daß wir dieses Ziel erreichen k ¨onnen, in-dem wir die lokale Struktur der zugrundeliegenden Objekte analysieren undmit Hilfe der gewonnenen lokalen Daten zeigen, daß in beiden F¨allen die je-weiligen, geeignet gew¨ahlten ¨Aquivalenzklassen der topologischen und C ∗ -algebraischen Objekte ¨ubereinstimmen.Ich will mich an dieser Stelle recht herzlich bei all denjenigen bedanken, diezum Gelingen und Entstehen dieser Arbeit beigetragen haben. An erster Stel-le gilt mein Dank nat ¨urlich meinem Doktorvater Herrn Ulrich Bunke, der sich(erstaunlicherweise) dazu bereit erkl¨arte, mich als Doktorand zu betreuen, undmich immer wieder auf vielf¨altige Weise gefordert und gef ¨ordert hat. Ohneihn w¨are diese Arbeit tats¨achlich nicht m ¨oglich gewesen, und da die mir auf-getragene Fragestellung im Grenzgebiet verschiedener, mathematischer Diszi-plinen liegt, konnte ich Dinge lernen, die mir unter anderen Umst¨anden sicherverwehrt geblieben w¨aren. Meinem Doktoronkel Herrn Thomas Schick undmeinem Doktorbruder Herrn Moritz Wiethaup sei hier ausdr ¨ucklich f ¨ur diezahlreichen, fruchtbaren Gespr¨ache gedankt, ohne die ich einige Dinge nichth¨atte so einfach oder schnell oder ¨uberhaupt h¨atte bewerkstelligen k ¨onnen.Dem Graduiertenkolleg Gruppen und Geometrie danke ich f ¨ur das entgegen-gebrachte Vertrauen und die großz ¨ugige Unterst ¨utzung in den letzten zweiJahren.Nachdem mir von prominenter Stelle zugetragen wurde, daß es wohl an-ebracht ist, auch denjenigen zu danken, die indirekt an dieser Arbeit betei-ligt sind, will ich manchen von meinen Lehrern und Hochschullehrern dan-ken, von denen ich (nat ¨urlich in alphabetischer Folge) besonders die HerrenHelmut Becker, Detlev Buchholz, R ¨udiger Heidersdorf, Bernhard Meyer, Karl-Henning Rehren und Wolfgang Watzlawek erw¨ahnen will, die alle ihren be-sonderen Anteil an meinem Bildungsweg hatten und haben. Herrn Friedrichvon Schiller danke ich f ¨ur die Ode an die Freude, Herrn Ralf Lindemann f ¨uralles und Herrn Jens Latsch daf ¨ur, daß er mir im ersten Semester meine Obe-ronprogramme zugeschickt hat. Zum Schluß will ich meinen lieben Eltern undmeiner lieben Großmutter daf ¨ur danken, daß diese Arbeit, nicht nur in der of-fensichtlichen Weise, ohne sie nicht h¨atte entstehen k ¨onnen.G ¨ottingen, im Oktober 2007Ansgar Schneider ontents G = R n and N = Z n . . . . . . . . . . . . . . . . . . 713.4 The Structure of the Associated C ∗ -Dynamical Systems . . . . . 79 A Some Notation and Basic Lemmata 88
A.1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88A.2 Group and ˇCech Cohomology . . . . . . . . . . . . . . . . . . . . 89A.3 The Unitary and the Projective Unitary Group . . . . . . . . . . 89A.4 Crossed Products . . . . . . . . . . . . . . . . . . . . . . . . . . . 92A.5 Some Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 • Index 97 • References 99 Introduction and Summary
String theory [Po] is based on the physical idea to describe nature not only bypoint particles but with the concept of higher dimensional objects. Althoughit is still unclear how close string theory is to physics a lot of interesting newphenomena have been observed which have led to many fruitfully new ideasand have made it a rich theory. In particular, a lot of new mathematical ideashave arose from the desire to understand the discovered structures. One ofthose is the concept of T-duality.T-duality is a duality of string theories (type IIA and IIB) on different un-derlying space-time manifolds E and b E which are (in the simplest case) re-lated by a transformation of type: radius E [BM], where the twist is given by a background fieldon E (a 3-form H called H-flux). Then T-duality must give the answer how thebackground fields transform and should lead to an isomorphism of the twistedK-theories of the underlying manifolds.We cannot give a summary of the whole subject, but we can try to point outsome of its mathematical issues. In literature there are different approachesof a mathematical understanding of T-duality. One is based on the theory of C ∗ -dynamical systems which serve a notion of T-duality using crossed prod-uct C ∗ -algebras [BHM2, MR], another is by geometric and topological means[BEM, BHM1, BS, BRS], a third using methods from algebraic geometry [BSST].We focus our attention to the first and second approach and continue to de-scribe some features of the geometric-topological side in more detail.Let us think of the manifolds E and b E as principal circle bundles which haveisomorphic quotients E / S ∼ = b E / S = : B . In [BEM] it is described in terms ofdifferential geometry how the data of the curvature F of E and of the H-flux H on E are related to the corresponding dual data b F and b H of b E . The result isthat integration of H along the fibres of E yields the dual curvature and viceversa. In the case of S -bundles E and b E we can identify the classes of thecurvatures with the realifications of the first Chern classes c , ˆ c ∈ H ( B , Z ) of the respective bundles, and there also exist integer cohomology classes h ∈ H ( E , Z ) , ˆ h ∈ H ( b E , Z ) whose realifications are h R = [ H ] and ˆ h R = [ b H ] . Inthis sense, we forgot geometry and now may only consider these topologicaldata. This is the point of view which was adopted in [BS], wherein amongother things the results of [BEM] are restated on a purely topological level. Thehigher dimensional case, where the circle S is replaced by the n -dimensionaltorus T n = ( S ) × n , i.e. E is thought of a principal T n -bundle, is described in[BHM1] in terms of differential geometry. Its topological structure is described The accent circonflexe ˆ is going to be the most overloaded symbol in this work.
8n [BRS] which we want to discuss in more detail. They introduce so-calledT-duality triples and define that a pair ( E , h ) is dual to a pair ( b E , ˆ h ) if there isa T-duality triple connecting them. The notion of T-duality triples which weare going to call topological triples (Definition 2.9) is central for this work, solet us clarify what it means that a T-duality triple connects the pairs ( E , h ) and ( b E , ˆ h ) :A T-duality triple is a commutative diagram P × B b E } } zzzzzzzzz $ $ IIIIIIIII E × B b P z z uuuuuuuuu ! ! DDDDDDDDD ∼ = κ o o P " " DDDDDDDDDD E × B b E $ $ JJJJJJJJJJ z z tttttttttt b P | | zzzzzzzzzz E % % JJJJJJJJJJJJ b E y y tttttttttttt B , (1)wherein P → E and b P → b E are principal bundles with structure group PU ( H ) ,the projective unitary group of some infinite dimensional, separable Hilbertspace H , such that both of these bundles are trivialisable when restricted tothe fibres of E → B or b E → B respectively. Moreover, the top-isomorphism κ satisfies the following local condition: Due to the triviality condition on thebundles P , b P , we can trivialise (1) over each u ∈ B such that the isomorphism κ induces a map κ ( u ) : T n × T n → PU ( H ) which implements the isomorphism.Now, PU ( H ) is an Eilenberg-McLane space of type K ( Z , 2 ) so κ ( u ) defines aclass [ κ ( u )] ∈ H ( T n × T n , Z ) . We force this class to satisfy [ κ ( u )] ∈ π + im ( pr ∗ ) + im ( pr ∗ ) , where pr : T n × T n → T n are the projections and π isthe class of the tautological line bundle over T n × T n which is π = y ∪ ˆ y + · · · + y n ∪ ˆ y n , for the generators y , . . . , y n , ˆ y , . . . , ˆ y n of H ( T n × T n , Z ) . Remark 1.1
In fact, this is not the definition of [BRS]. Firstly, they uses the moregeneral notion of twists instead of the bundles P and b P, but the category of PU ( H ) -principal bundles with homotopy classes of bundle isomorphisms as morphisms is amodel of twists. Secondly, they require the (a priori) more restrictive condition thatthe class of the bundles [ P ] ∈ H ( E , Z ) (analogously for [ b P ] ∈ H ( b E , Z ) ) lies inthe second step F H ( E , Z ) of the filtration { } ⊂ F H ( E , Z ) ⊂ F H ( E , Z ) ⊂ F H ( E , Z ) ⊂ H ( E , Z ) associated to the Leray-Serre spectral sequence. The re-quirement of triviality over the fibres of E we stated above precisely means that theclass lies in the first step F H ( E , Z ) of the filtration. However, these two conditionson the classes are equivalent as we show in Lemma 3.7. A T-duality triple which we denote by ( κ , ( P , E ) , ( b P , b E )) connects the twopairs ( E , h ) and ( b E , ˆ h ) if we have an equality of the classes [ P ] = h and [ b P ] = ˆ h .9o a T-duality triple we can associate two C ∗ -algebras, namely the C ∗ -algebraof sections Γ ( E , F ) and Γ ( b E , b F ) of the associated bundles F : = P × PU ( H ) K ( H ) and b F : = b P × PU ( H ) K ( H ) . It is the very aim of this work to understand howthese two C ∗ -algebras are related to each other. This issue turns our focus onthe C ∗ -algebraic approach to T-duality [MR, BHM2] which is based on the un-derstanding of abelian C ∗ -dynamical systems.We shortly summarise some C ∗ -algebraic background.The duality theory of abelian C ∗ -dynamical systems which has been inves-tigated for a quite long time [Pe1] is the foundation to understand T-duality by C ∗ -algebraic means. The dual of an abelian C ∗ dynamical system ( A , G , α ) , i.e. A a C ∗ -algebra with strongly continuous action α : G → Aut ( A ) of a locallycompact, abelian group G , is the crossed product C ∗ -algebra G × α A equippedwith the natural action ˆ α of the dual group b G , i.e. ( G × α A , b G , ˆ α ) becomes againa C ∗ -dynamical system (see [Pe1] or section A.4). A central result is the Takaiduality theorem (Theorem A.2) which states in particular that the bi-dual C ∗ -algebra is stably isomorphic to the original one, i.e. they are Morita equivalent.Thus, it is completely trivial to understand the structure of the bi-dual and thedifficult task is to understand the dual G × α A .In the 80s and 90s big progress has been made to understand the dual incase A is a continuous trace algebra which we assume from now on. The basicstructure theorem of Dixmier and Douady (see e.g. [Di]) says that any separa-ble, stable continuous trace algebra A is isomorphic to Γ ( E , F ) the C ∗ -algebraof sections vanishing at infinity, where E : = spec ( A ) is the spectrum of A and F → E is a locally trivial bundle with each fibre isomorphic to the compacts K ( H ) . Their isomorphism classes are classified by ˇ H ( E , U ( )) ∼ = H ( E , Z ) (cp. section A.3), and the class in H ( E , Z ) which determines the isomorphismtype of A ∼ = Γ ( E , F ) is called the Dixmier-Douady invariant of A .A first result [Pe2, RW] for an understanding the crossed product G × α A was that if G is compactly generated and the induced action of G on the spec-trum E of A is trivial, then the crossed product G × α A is isomorphic to thebalanced tensor product C ( b E ) ⊗ C ( B ) A = : p ∗ A , wherein p : b E → B is a b G -principal bundle consisting of the spaces b E : = spec ( G × α A ) and B : = E .The more general situation wherein the action of G does not fix the spec-trum E of A but has constant isotropy group N for each π ∈ E is concerned in[RR]. One of the statements therein is the following. Assume that E with theinduced action of G / N is a principal fibre bundle E → B : = E / ( G / N ) andthat the restricted action α | N of N on A is locally unitary, then there is a pull10ack diagram of principal fibre bundles E × B b E ∼ = spec ( N × α | N A ) ˆ p ( ( QQQQQQQQQQQQQ p w w nnnnnnnnnnn E : = spec ( A ) ( ( PPPPPPPPPPPPPP b E : = spec ( G × α A ) v v lllllllllllllll B ,wherein the down-right arrows have fibre G / N and the down-left arrows havefibre b N ∼ = b G / N ⊥ . ( N ⊥ is the annihilator of N which is the set of charactersof G whose restriction to N is identically 1.) Moreover, p ∗ A is isomorphic to N × α | N A and Morita equivalent to ˆ p ∗ ( G × α A ) . Thus, we have the followingschematic situation of C ∗ -algebras over their spectra p ∗ A N × α | N A y y ∼ = o o A E × B b E ˆ p % % KKKKKKKKKKK p { { G × α A y y E $ $ IIIIIIIIIII b E x x qqqqqqqqqqqqq B (2) ≀ Morita ˆ p ∗ ( G × α A ) which obviously is similar to diagram (1). The question is whether or not itis possible that both A and G × α A are separable, stable continuous trace alge-bras. This question has been answered in [ER, Thm. 6]. In particular, this is trueif α | N is point-wise unitary and the action of G / N on E × B E ∼ = spec ( N × α | N A ) is proper, e.g. G / N is compact.This finishes our summary.The approach to T-duality of [MR] considers the following set-up. Let E bea locally compact space (with certain finiteness assumptions) with an action ofthe torus T n such that E → B : = E / T n becomes a principal torus bundle, andlet h ∈ H ( E , Z ) . They concern these data as a stable continuous trace algebra Γ ( E , F ) that has Dixmier-Douady invariant h . Under which circumstances isit possible to lift the T n -action from the spectrum E to an action of R n = : G on Γ ( E , F ) ? If so, is it further possible to obtain an action whose restriction to N : = Z n is point-wise unitary for all π ∈ E ? These questions are answered in[MR, Thm 3.1]. The general answer is no, but if we make further restrictions11o the class h one achieves a positive answer. Namely, a lift α to an R n -actionexists if and only if h ∈ F H ( E , Z ) the first step of the filtration0 ⊂ F H ( E , Z ) ⊂ F H ( E , Z ) ⊂ F H ( E , Z ) ⊂ H ( E , Z ) associated to the Leray-Serre spectral sequence, and there exists a point-wiseunitary action if even h ∈ F H ( E , Z ) . Consequently, in the latter case thereexists a T-dual in the sense that there exists a dual space-time b E in diagram (2)which is a c Z n ∼ = T n -principal fibre bundle over B and is the spectrum of thestable continuous trace algebra R n × α Γ ( E , F ) ∼ = Γ ( b E , b F ) .In this work we are going to show that the two approaches to T-duality areessentially equivalent, i.e. there is no difference between (equivalence classesof) T-duality triples and (equivalence classes of) abelian C ∗ -dynamical systemswhich are obtained as described above. To fulfil this task we must develop atechnique which enables us to compare such different objects. The methods of[BRS] and [MR] are not applicable for such a manoeuvre as they are too lessexplicit: The explicit description of the local structure of these two differentkinds of objects can be used to describe a transformation as desired.Our method is general enough that we do not have to restrict ourselves tothe case of the groups R n and T n ∼ = R n / Z n . - We develop a theory for allsecond countable, locally compact, abelian groups G with lattice N ⊂ G , i.e. N is a discrete, cocompact subgroup.We give an overview of this work.A basic technique we use throughout the whole of this work is to lift alllocal, projective unitary families of functions (e.g. the transition functions ofa PU ( H ) -principal bundle P → E ) to unitary Borel- or L ∞ -functions in direc-tion of the fibres G / N of E → B and to think of them as new unitary (multi-plication) operators in U ( L ( G / N ) ⊗ H ) . We call this procedure Borel liftingtechnique. The technical condition we must assume is that the base B is a para-compact Hausdorff space which is locally contractible. We call spaces withthese properties base spaces, and the whole theory we develop is a theory overbase spaces.In sections 2.1 and 2.2 we introduce the notion of pairs. A pair is a G / N -principal fibre bundle E → B over a base space B and a PU ( H ) -principal fibrebundle P → E which is trivialisable over the fibres of E . We explain their localstructure and give a quick result on their classification.In section 2.3 we introduce a twisted version of ˇCech cohomology on thebase B , where the twist is given by the bundle E → B . The Borel lifting tech-nique mentioned above defines a map from (equivalence classes of) pairs intothe second twisted ˇCech cohomology (similar to the ordinary definition of thesecond ˇCech class of a PU ( H ) -bundle).In sections 2.4 we extend the same procedure to dynamical triples ( ρ , E , P ) which are pairs ( P , E ) equipped with a lift ρ of the G / N action on E to a G -action on P . Due to this action, group cohomological expressions arise in the12escription of their local structure and lead finally to a map from (equivalenceclasses of) dynamical triples to the second cohomology of a double complexwhich has one (twisted) ˇCech cohomological direction and a second group co-homological direction. A two cocycle (or two cohomology class) of this com-plex has three entries: a pure ˇCech part due to the transition functions of thepair, a group cohomological part and a third mixed term. In section 3.1 wefocus our attention to those triples for which the group cohomological entryvanishes.Section 2.5 just contains the definition of dual pairs and dual dynamicaltriples which are nothing more than pairs and dynamical triples, but as under-lying groups we take the dual group b G of G with dual lattice N ⊥ the annihilatorof N .Then in section 2.6 we introduce topological triples. Our definition is astraight forward generalisation to arbitrary locally compact, abelian groups G with lattice N from the notion of a T-duality triple ( G = R n , N = Z n ).In section 3.1 we state our first main result. We first single out a specificsubclass of dynamical triples which we call dualisable. They are those dy-namical triples for which the group cohomological entry of the associated twocohomology class (of the double complex) vanishes. We construct an explicitmap [( ρ , E , P )] [( ˆ ρ , b E , b P )] from the set of equivalence classes of dualisabledynamical triples to the set of equivalence classes of dualisable dual dynam-ical triples, and show that it is a bijection whose inverse is given by the dualmap (defined by replacing everything by its dual counterpart). In this sensedualisable dynamical triples and dual dualisable triples are in duality.Section 3.2 contains two important statements. The first is that we have amap τ ( B ) from the set of equivalence classes of dualisable dynamical triplesto the set of equivalence classes of topological triples (everything understoodover a base space B ). This map is defined by the duality theorem of section3.1, i.e. the topological triple we define consists of the pairs of two dynamicaltriples in duality. Then we try to define a map δ ( B ) in the opposite directionwhich generally fails as an obstruction occurs. However, on the subset of thosetopological triples which have a vanishing obstruction we then find a construc-tion of a whole family of dualisable dynamical triples which is associated to atopological triple. This construction, when restricted to the image of the firstmap, can be turned into an honest map, i.e. we have a preferred choice of anelement of the family, and this map is inverse to τ ( B ) .Section 3.3 is devoted to the special case of the group G = R n with lattice N = Z n . In this situation the construction of the map δ ( B ) simplifies dras-tically, the group wherein the obstruction lives vanishes. As a result, we canassociate to each topological triple a dynamical triple which is unique (up toequivalence) because the family of dynamical triples degenerates to a familyof one single element only. As it turns out the two maps τ ( B ) and δ ( B ) arebijections and inverse to each other.Moreover, the four maps mentioned above are natural in the base, so theydefine natural transformations of functors. Thus, the main result of section 3.313an be restated as follows. In the case of G = R n and N = Z n , we have acompletely explicit construction of equivalences of functorsTop ∼ = Dyn † ∼ = d Dyn † .Therein, Dyn † ( d Dyn † ) is the functor sending a base space to the set of equiva-lence classes of dualisable (dual) dynamical triples over it. Top is the functorwhich sends a base space to the set of equivalence classes of topological (T-duality) triples over it.In section 3.4 we point out that the theory developed so far is connected tothe theory of C ∗ -dynamical systems precisely as one expects. Namely, the C ∗ -dynamical systems ( G × α ρ Γ ( E , F ) , b G , b α ρ ) and ( Γ ( b E , b F ) , b G , α ˆ ρ ) are isomorphic,wherein ( ˆ ρ , b E , b P ) is the dual of ( ρ , E , P ) and ( ρ , E , P ) ( Γ ( E , F ) , G , α ρ ) is thefunctor which sends a dualisable (dual) dynamical triple to its corresponding C ∗ -dynamical system, i.e. F is the associated K ( H ) -bundle to P and α ρ is theby ρ induced action on the C ∗ -algebra of sections Γ ( E , F ) .In an appendix we put some technical lemmata and notation we are goingto use. 14 Pairs and Triples
Definition 2.1 A base space B is a topological space which is Hausdorff, paracom-pact and locally contractible.
The category of base spaces consists of bases spaces as objects and contin-uous maps between them as morphisms. A typical class of base spaces areCW-complexes [FP, Thm. 1.3.2, Thm. 1.3.5].By G we will always denote a second countable, Hausdorff, locally compactabelian group and by N a discrete, cocompact subgroup, i.e. the quotient G / N is compact.Let H be an infinite dimensional, separable Hilbert space. Let E → B be a G / N -principal fibre bundle and P → E be a PU ( H ) -principal fibre bundle. Definition 2.2
We call the data ( P , E ) a pair over B with underlying Hilbert space H ifi ) B is a base space,ii ) the restriction of the bundle P → E to the fibres of E → B is trivialisable.
Remark 2.1
We do not require local compactness for B, because we want to developa theory that includes non locally compact spaces such as classifying spaces of groups(s. next section). Therefore E need not to be locally compact and thus need not equalthe spectrum of any (continuos trace) C ∗ -algebra such as Γ ( E , F ) the C ∗ -algebra ofbounded sections (or Γ ( E , F ) the C ∗ -algebra of sections vanishing at infinity [A sec-tion vanishing at infinity vanishes already identically on the set of points which don’thave a compact neighbourhood.]) of the associated C ∗ -bundle F : = P × PU ( H ) K ( H ) . A morphism ( ϕ , ϑ , θ ) over B from a pair P → E → B with underlying Hilbertspace H to a pair P ′ → E ′ → B with underlying Hilbert space H ′ is a commuta-tive diagram of bundle isomorphisms P ϑ / / (cid:15) (cid:15) ϕ ∗ P ′ (cid:15) (cid:15) E θ / / (cid:15) (cid:15) E ′ (cid:15) (cid:15) B = / / B , (3)wherein ϕ ∈ PU ( H , H ′ ) : = U ( H , H ′ ) /U ( ) is the class of a unitary isomor-phism H → H ′ . ϕ ∗ P ′ is the PU ( H ) -bundle with total space ϕ ∗ P ′ = P ′ , butwith PU ( H ) -action that is induced by ϕ ∗ : PU ( H ) → PU ( H ′ ) , i.e. x ′ · U : = differing from [BRS] ′ · ( ϕ ∗ ) − U , x ′ ∈ P ′ , U ∈ PU ( H ) . Pairs over a base space B and their mor-phisms form a category; composition of morphisms ( ϕ , ϑ , θ ) and ( ϕ ′ , ϑ ′ , θ ′ ) isjust component-wise composition ( ϕ ′ ◦ ϕ , ϑ ′ ◦ ϑ , θ ′ ◦ θ ) . This category is even agroupoid, i.e. every morphism is an isomorphism.This notion of morphism is well-behaved under stabilisation in the follow-ing sense. Let ( P , E ) be a pair over B with underlying Hilbert space H . Let H be any separable Hilbert space, not neccessarily infinite dimensional. ThenPU ( H ) is isomorphic to the subgroup ⊗ PU ( H ) of PU ( H ⊗ H ) and thusPU ( H ) acts on PU ( H ⊗ , H ) by left multiplication: ( U , V ) ( H ⊗ U ) V .Then the associated (stabilised) bundle P H : = PU ( H ) ⊗ P : = P × PU ( H ) PU ( H ⊗ H ) (4)is a PU ( H ⊗ H ) -principal bundle and ( P H , E ) is a pair over B with underlyingHilbert space H ⊗ H . We call two pairs ( P , E ) and ( P ′ , E ′ ) with underlyingHilbert spaces H and H ′ stably isomorphic if there exists a Hilbert space H such that the pairs ( P H , E ) and ( P ′ H , E ′ ) are isomorphic. Proposition 2.1
Two pairs over B are stably isomorphic if and only if they are iso-morphic.
Proof :
It is clear that isomorphic pairs are stably isomorphic. To prove theconverse it suffices to show that P and ϕ ∗ P H are isomorphic over E , for anisomorphism ϕ : H ∼ = H ⊗ H . To do so, we show that P and ϕ ∗ P H define thesame ˇCech class, hence the classification theorem of PU ( H ) -bundles (TheoremA.1) implies that the two bundles are isomorphic.In fact, if ζ ji : V ji → PU ( H ) are transition functions for P (here { V i } isa covering of E ) , then ζ ji : = ϕ ∗ ( ⊗ ζ ji ) are transition functions for ϕ ∗ P H .If we refine the covering such that the transition functions ζ ji lift to unitary-valued functions ζ ji (Lemma A.8), then these lifts define also lifts ζ ji for theother family of transition functions. Thus, on threefold intersections V kji thecocycle identities of the two families are perturbed by the same ˇCech 2-cocycle c kji : = ζ nji ζ nki − ζ nkj : V kji → U ( ) · , n =
1, 2.Hence, their classes agree. (cid:4)
Remark 2.2
In case of an additional structure such as a group action of G on P stableisomorphism and isomorphism are different notions when we force them to preserve theextra structure. This will be important in section 2.4.
Let us denote by Par the set valued contravariant functor that sends a basespace B to the set of stable isomorphism classes of pairs over B , i.e.Par ( B ) : = { pairs over B } (cid:14) stable isomorphism ,16nd if f : B ′ → B is a continuous map between base spaces, then pullbackdefines a map f ∗ : Par ( B ) → Par ( B ′ ) .There is a subcategory of pairs over B consisting of pairs with a fixed G / N -bundle E → B and morphisms of the form ( ϕ , ϑ , id E ) , and we call pairs ( P , E ) and ( P ′ , E ) stably isomorphic over E if there is a isomorphism of this specialform between ( P H , E ) and ( P ′ H , E ) , for a Hilbert space H . We definePar ( E , B ) : = { pairs over B with fixed E } (cid:14) stable isomorphism over E ,and for a bundle morphism E ′ (cid:15) (cid:15) θ / / E (cid:15) (cid:15) B ′ / / B we define by pullback a map θ ∗ : Par ( E , B ) → Par ( E ′ , B ′ ) , so Par ( . , .. ) be-comes a contravariant functor from the category of G / N -principal fibre bun-dles over base spaces to sets. The bundle automorphisms Aut B ( E ) of E overid B act on Par ( E , B ) by pullback, and we have a decomposition Par ( B ) ∼ = ∐ [ E ] ( Par ( E , B ) /Aut B ( E )) , wherein the disjoint union runs over all isomorphismclasses of G / N -bundles E → B . Remark 2.3
For each fixed E → B the set
Par ( E , B ) has a natural group structure.If [( P , E )] and [( P ′ , E )] are two classes of pairs, then we let [( P , E )] + [( P ′ , E )] : =[( P ⊗ P ′ , E )] , wherein P ⊗ P ′ is the PU ( H ⊗ H ′ ) -bundle which is associated to the PU ( H ) × PU ( H ′ ) -bundle P × E P ′ ,P ⊗ P ′ : = ( P × E P ′ ) × PU ( H ) × PU ( H ′ ) PU ( H ⊗ H ′ ) . The unit element is given by the class of a trivial bundle and the inverse of [( P , E )] isgiven by the class [( P , E )] of the complex conjugate bundle P which is as space thebundle P but has the action ( x , U ) x · U . U is here the complex conjugate (notthe adjoint) of U ∈ PU ( H ) (which may be defined by identifying H = l ( N ) andtaking the complex conjugate matrix of u = ( u ij ) i , j ∈ N , for U = Ad ( u ) ).In this way we just mimic the group structure of ˇ H ( E , U ( )) , i.e. the classifica-tion map Par ( E , B ) → ˇ H ( E , U ( )) is turned into a group homomorphism. An automorphism of a pair is a morphism from a pair onto itself. Thegroup of automorphisms of a pair is denoted by Aut ( P , E ) . It becomes a topo-logical group when equipped with the initial topology of the forgetful mapAut ( P , E ) → PU ( H , H ′ ) × Map ( P , P ) which sends a morphism ( ϕ , ϑ , θ ) to ( ϕ , ϑ ) , wherein U ( H , H ′ ) has the strong topology, i.e. the topology of point-wise convergence. Map ( P , P ) has the compact open topology.Since G / N is a commutative group mappings of the form θ z : E ∋ e e · z ∈ E , z ∈ G / N , are bundle morphisms. They give rise to a subgroup See Theorem A.1. ( P , E ) which consist of all morphisms ( id H , ϑ , θ z ) . Let Aut ( P , E ) denotethe subgroup consisting of morphisms ( id H , ϑ , id E ) , then we find a short exactsequence of topological groups → Aut ( P , E ) → Aut ( P , E ) → G / N → By A we denote the group of automorphisms of the trivial pair over a point. Inparticular, a = ( ϕ , ϑ , θ ) ∈ A makes G / N × PU ( H ) ϑ / / (cid:15) (cid:15) G / N × PU ( H ) (cid:15) (cid:15) G / N θ = + z / / (cid:15) (cid:15) G / N (cid:15) (cid:15) ∗ / / ∗ commute, for some z ∈ G / N . It is immediate that there is an isomorphism A ∼ = G / N ⋉ Map ( G / N , PU ( H )) ⋊ PU ( H ) of topological groups, wherein G / N ⋉ Map ( G / N , PU ( H )) ⋊ PU ( H ) is the semi-direct product with multiplication ( y , η , u ) · ( z , ζ , v ) : = ( y + z , ( z · η · v ) ζ , uv ) .The action on the continuous functions Map ( G / N , PU ( H )) is ( z · η · v )( x ) : = v − η ( x + z ) v .Let ( P , E ) be any pair over B with underlying Hilbert space H . B is a basespace, and so we can choose a covering { U i | i ∈ I } of B of open sets such thatfor each U i there is a commutative diagram U i × G / N × PU ( H ) h i ∼ = / / (cid:15) (cid:15) q − ( p − ( U i )) / / (cid:15) (cid:15) P q (cid:15) (cid:15) U i × G / N k i ∼ = / / (cid:15) (cid:15) p − ( U i ) / / (cid:15) (cid:15) E p (cid:15) (cid:15) U i = / / U i ⊂ / / B , (5)with bundle isomorphisms k i , h i . We refer to such a covering as an atlas U • = { ( U i , k i , h i ) | i ∈ I } consisting of the charts ( U i , k i , h i ) . The transition fromone chart to another is described by a set of transition functions. For a pair18his consists of two families of continuous functions g ij : U ij → G / N and ζ ij : U ij → Map ( G / N , PU ( H )) which appear in h − j ◦ h i : U ji × G / N × PU ( H ) → U ij × G / N × PU ( H ) . ( u , z , U ) ( u , g ji ( u ) + z , ζ ji ( u )( z ) U ) It follows that on threefold intersections U ijk the relations g ki ( u ) = g kj ( u ) + g ji ( u ) and ζ ki ( u )( z ) = ζ kj ( u )( g ji ( u ) + z ) ζ ji ( u )( z ) ∈ PU ( H ) (6)are valid; equivalently, the family of functions a ij : = g ij × ζ ij : U ij → G / N ⋉ Map ( G / N , PU ( H )) = : A ⊂ A satisfies the ˇCech 1-cocycle condition a ij ( u ) a jk ( u ) = a ik ( u ) . Now, let EA → BA be the universal A -principal fibre bundle. We call the associated pair P univ (cid:15) (cid:15) : = EA × A ( G / N × PU ( H )) E univ (cid:15) (cid:15) : = EA × A G / NB univ : = BA the universal pair . Indeed we can choose a CW-model for BA such that theuniversal pair is a pair in the sense of Definition 2.2. Its name is due to thefollowing universal property. Proposition 2.2
The space B univ classifies pairs over (pointed) CW-complexes, i.e.if B is a (pointed) CW-complex, then [ B , B univ ] ∼ = Par ( B ) , wherein the left hand side is the set of (pointed) homotopy classes of maps B → B univ . Proof :
We already observed that the transition functions of a pair define aˇCech class [ a .. ] = [ g .. × ζ .. ] ∈ ˇ H ( B , A ) . This class is independent of the chosenatlas. Now, let ( ϕ , ϑ , θ ) : ( P , E ) → ( P ′ , E ′ ) be an isomorphism of pairs, then thetransition functions a ′ .. for ( P ′ , E ′ ) can be turned into a ˇCech A -cocycle inducedby ϕ ∗ : PU ( H ) ∼ = PU ( H ′ ) , and this class depends on the isomorphism class ofthe pair only. Conversely, the associated pairs of isomorphic A -principal bun-dles are isomorphic, and each isomorphism class arises.Thus, pairs and A -principal bundles over B have the same isomorphismclasses, but for a (pointed) CW-complex B the latter isomorphism class is givenby homotopy classes of maps to BA . (cid:4) .3 Pairs and Twisted ˇC ech Cohomology Let M and G be abelian (pre-)sheaves on a space B and assume that M is a right G module. We are going to twist the ˇCech coboundary operator of M by a ˇCech G U • = { U i | i ∈ I } of B and let g ∈ ˇ Z ( U • , G ) .Then for ϕ ∈ ˇ C n − ( U • , M ) , n =
1, 2, . . . , we define δ g ϕ : = δϕ + g ⋆ ϕ ,wherein δ is the ordinary ˇCech coboundary operator and ( g ⋆ ϕ ) i ... i n : = ( − ) n − (cid:0) ϕ i ... i n − − ϕ i ... i n − · g i n − i n (cid:1) | U i in .The choice of the sign ( − ) n − is such that the last term of δϕ and the first termof g ⋆ ϕ cancel. We obtain a sequence0 / / ˇ C ( U • , M ) δ g / / ˇ C ( U • , M ) δ g / / ˇ C ( U • , M ) δ g / / · · · Lemma 2.1 ( ˇ C • ( U • , M ) , δ g ) is a cochain complex. Proof :
We have to show that the square of δ g vanishes. Let ϕ ∈ ˇ C n − ( U • , M ) ,then δ g δ g ϕ = δδϕ + δ ( g ⋆ ϕ ) + g ⋆ δϕ + g ⋆ ( g ⋆ ϕ )= δ ( g ⋆ ϕ ) + g ⋆ δϕ + g ⋆ ( g ⋆ ϕ ) and therefore ( δ g δ g ϕ ) i ... i n + = n + ∑ k = ( − ) k ( g ⋆ ϕ ) i ...ˆ i k ... i n + +( − ) n (cid:0) n ∑ k = ( − ) k ϕ i ...ˆ i k ... i n − n ∑ k = ( − ) k ϕ i ...ˆ i k ... i n · g i n i n + (cid:1) +( − ) n ( g ⋆ ϕ ) i ... i n − ( − ) n ( g ⋆ ϕ ) i ... i n · g i n i n + = n − ∑ k = ( − ) k ( − ) n − ( ϕ i ...ˆ i k ... i n − ϕ i ...ˆ i k ... i n · g i n i n + )+( − ) n ( − ) n − ( ϕ i ... i n − − ϕ i ... i n − · g i n − i n + )+( − ) n + ( − ) n − ( ϕ i ... i n − − ϕ i ... i n − · g i n − i n )+( − ) n (cid:0) n ∑ k = ( − ) k ϕ i ...ˆ i k ... i n − n ∑ k = ( − ) k ϕ i ...ˆ i k ... i n · g i n i n + (cid:1) +( − ) n ( − ) n − (cid:0) ϕ i ... i n − − ϕ i ... i n − · g i n − i n (cid:1) − ( − ) n ( − ) n − (cid:0) ϕ i ... i n − − ϕ i ... i n − · g i n − i n · g i n i n + (cid:1) = See A.2 g ij + g jk = g ik . (cid:4) By the last lemma we have well-defined cohomology groups ˇ H n ( U • , M , g ) for n =
0, 1, 2, . . . and any open cover U • = { U i | i ∈ I } of B . And as in theuntwisted case we define the twisted cohomology groups ˇ H n ( B , M , g ) of B bypassing to the limit ˇ H n ( B , M , g ) : = lim V • ˇ H n ( V • , M , g | ) ,wherein the limit runs over all refinements V • of U • . To be precise, considera refinement V • = { V k | k ∈ K } of U • = { U i | i ∈ I } with refinement map ι : K → I , i.e. V k ⊂ U ι ( k ) then we define ( ι ∗ g ) kl : = g ι ( k ) ι ( l ) | V kl and similarly ( ι ∗ ϕ ) k ... k n : = ϕ ι ( k ) ... ι ( k n ) | V k kn to obtain a cochain map ι ∗ : ( ˇ C n ( U • , M ) , δ g ) → ( ˇ C n ( V • , M ) , δ ι ∗ g ) . This construction defines a functor from the category of cov-erings with refinement maps as morphisms to the category of cochain com-plexes (and after taking homology to the category of graded abelian groups).The category of coverings is filtered in the following sense: ( i ) Any two coverings have a common refinement, i.e. for any two objects ( U • , I ) and ( V • , K ) there is a third object ( W • , L ) with morphisms ( U • , I ) → ( W • , L ) and ( V • , K ) → ( W • , L ) . ( ii ) Any two refinement maps become equal finally, i.e. for any to mor-phisms ι : ( U • , I ) → ( V • , K ) and κ : ( U • , I ) → ( V • , K ) there exists an object ( W • , L ) and morphisms ι ′ : ( V • , K ) → ( W • , L ) and κ ′ : ( V • , K ) → ( W • , L ) suchthat κ ′ ◦ κ = ι ′ ◦ ι .Due to ( i ) and ( ii ) the limit lim V • ˇ H n ( V • , M , g | ) is independent of the choiceof the first covering and independent of the refinement maps.So far, in our construction we referred explicitly to a choice of a cocycle g ∈ ˇ Z ( U • , G ) , but up to isomorphism ˇ H n ( B , M , g ) depends only on the classof g . In fact, let V • = { V k | k ∈ K } and U • = { U i | i ∈ I } be open coveringsof B , and let g ′ ∈ ˇ Z ( V • , G ) and g ∈ ˇ Z ( U • , G ) represent the same element inˇ H ( B , G ) . If W • = { W m | m ∈ M } is a common refinement with refinementmaps ι : M → I and κ : M → K then there are r m ∈ G ( W m ) such that g ′ κ ( m ) κ ( n ) | W mn = r m | W mn + g ι ( m ) ι ( n ) | W mn − r n | W mn . They give rise to the followingdiagram of cochain complexes ( ˇ C • ( U • , M ) , δ g ) ι ∗ (cid:15) (cid:15) ( ˇ C • ( V • , M ) , δ g ′ ) κ ∗ (cid:15) (cid:15) ( ˇ C • ( W • , M ) , δ ι ∗ g ) ( ˇ C • ( W • , M ) , δ κ ∗ g ′ ) ∼ = r o o wherein r is defined by ( r ϕ ) k ... k n : = ϕ k ... k n · r k n | V k kn , for ϕ ∈ ˇ C n ( V • , M ) .One easily derives δ ι ∗ g ( r ϕ ) = r δ κ ∗ g ϕ , for ϕ ∈ ˇ C • ( V • , M ) , i.e. r is a cochainmap and even an isomorphism. Thus the corresponding cohomology groups21re isomorphic and this isomorphism passes to the limit. Therefore the twistedˇCech groups ˇ H n ( B , M , [ g ]) are well defined for the class [ g ] ∈ ˇ H ( B , G ) up tothe considered isomorphism.We now consider the relation between pairs and twisted ˇCech cohomology.One should note at this point that, just as in the untwisted case, the first ˇCech“group“ ˇ H ( B , M , g ) is a well-defined set even in case the sheaf M is just a sheafof groups and not necessarily abelian. In that case the additive (commutative)relation of being cohomologous ζ ji ∼ ζ ji + ( δ g η ) ji is replaced by the multiplica-tive relation ζ ji ∼ η j | U ij · g ji ζ ji η − i | U ij , for ζ ∈ ˇ Z ( U • , M , g ) , η ∈ ˇ C ( U • , M ) .The next proposition is then just a reformulation of what we already observedin Proposition 2.2. Proposition 2.3
Let B be a base space. Let M be the sheaf of topological groupson B defined by M ( U ) : = C ( U , Map ( G / N , PU ( H ))) , and let G : = G / N, i.e.G / N ( U ) : = C ( U , G / N ) , for U ⊂ B. G / N acts on M in the obvious way, i.e.by translation in the arguments. Then the first twisted cohomology classifies pairsover B, i.e. we have a bijection ˇ H ( B , M , g ) ∼ = Par ( E , B ) , if the class [ g ] ∈ ˇ H ( B , G / N ) is the class for the bundle E. Proof :
Let g .. , ζ .. be the transition functions of a pair over B for an atlas U • . So g ∈ ˇ Z ( U • , G / N ) , and the crucial point is to observe that equation (6) is equiv-alent to δ g ζ = and therefore each pair defines an element in ˇ H ( B , M , g ) . Infact, this is well defined, because if g ′ .. , ζ ′ .. are transition functions for anotheratlas then (after choosing a common refinement) the two classes match underthe isomorphism r , i.e. r ζ ′ is cohomologous to ζ . Similarly, an isomorphismof pairs leads to cohomologous cocycles. Conversely, any ζ ∈ ˇ Z ( U • , M , g ) defines an associated pair, and if ζ ′ ∈ ˇ Z ( V • , M , g ) defines the same class as ζ .. then the two associated pairs are isomorphic. Since each class arises in such away the assertion is proven. (cid:4) Let g .. , ζ .. be the transition functions of a pair over B . Since G / N is com-pact and B paracompact we can apply Lemma A.7 and Lemma A.8 for thefamily of transition functions ζ ij : U ij → Map ( G / N , PU ( H ) . I.e. we can finda refined atlas { V k | k ∈ K : = I × B } , V k ⊂ U i if k = ( i , x ) , such that on itstwofold intersections V kl the restricted transition functions lift to continuousfunctions ζ kl : V kl → Bor ( G / N , U ( H )) . These lifts are unique up to contin-uous functions V kl → Bor ( G / N , U ( )) . Let us denote by g lk the restriction g ji | V lk in case l = ( j , y ) , k = ( i , x ) ∈ I × B . On threefold intersections thefunction V klm ∋ u ζ kl ( u )( g lm ( u ) + ) won’t be continuos as a function toBor ( G / N , U ( H )) in general, but it will as a function to L ∞ ( G / N , U ( H )) . So L ∞ ( G / N , U ( H )) has the weak topology. See equation (51) in section A.3. ψ mlk : V mlk → L ∞ ( G / N , U ( )) such that ζ ml ( u )( g lk ( u ) + z ) ζ lk ( u )( z ) = ζ mk ( u )( z ) ψ mlk ( u )( z ) · ⇔ δ g ζ = ψ , (7)and it follows that δ g ψ =
1. The functions ψ klm therefore define a twisted ˇCech2-cocycle ψ ... ∈ ˇ Z ( V • , L ∞ ( G / N , U ( )) , g ) . Proposition 2.4
The construction of ψ ... defines a homomorphism of groupsPar ( E , B ) → ˇ H ( B , L ∞ ( G / N , U ( )) , g ) if [ g ] ∈ ˇ H ( B , G / N ) is the class of the bundle E → B. Proof :
We must check that the class [ ψ ... ] is independent of all choices. In fact,if ζ ji and ζ ′ ji are different choices of lifts of ζ ji , they differ by a scalar function s ji = ζ − ji ζ ′ ji and ψδ g s is the cocycle obtained from ζ ′ ji , so the class [ ψ ] is noteffected. It is also easy that the class does not change under the choice of theatlas or by considering a pair stably isomorphic to the first one. (cid:4) We do not achieve the statement that the above homomorphism is injectiveor surjective, so we are far from classifying pairs by this map.
Let P → E → B be a pair. The quotient map G ∋ g gN ∈ G / N induces a G action on E . Definition 2.3 A decker is just a continuous action ρ : P × G → P that lifts theinduced G-action on E such that ρ ( . , g ) : P → P is a bundle automorphism, for allg ∈ G. The existence of deckers can be a very restrictive condition on the bundle P → E . (See e.g. Prop. 2.6 below.) In fact, they need not exist and need not tobe unique in general, but the play a central r ˆole in what follows, therefore weintroduced an extra name.In context of C ∗ -dynamical systems, i.e. C ∗ -algebras with (strongly contin-uous) group actions, concretely, in context of the equivariant Brauer group sev-eral notions of equivalence of actions occur [CKRW]. In particular, the notionsof isomorphic actions, stably isomorphic actions and exterior equivalent ac-tions are combined to the notion of stably outer conjugate actions. We slightlymodify these notion for our purposes. However, we postpone the definitionuntil we made ourselves familiar with the local structure of dynamical triples. Definition 2.4 A dynamical triple ( ρ , P , E ) over B is a pair ( P , E ) over B togetherwith a decker ρ : P × G → P. ( ρ , P , E ) be a dynamical tripel over B . Proposition 2.5 i ) If we define ρ τ ( g ) : = ρ ( . , g ) : P → P, we obtain a diagramof topolgical groups / / N / / ρ τ | N (cid:15) (cid:15) G ρ τ (cid:15) (cid:15) / / G / N / / = (cid:15) (cid:15) / / Aut ( P , E ) / / Aut ( P , E ) / / G / N / /
0. (8)
Conversely, if B is locally compact then a commutative diagram (8) defines adecker.ii ) Locally, i.e. after choosing charts U i , a decker defines a family of continuouscocycles µ i : U i → Z ( G , Map ( G / N , PU ( H ))) such that on twofold inter-sections U ij ∋ u the transition functions of the pair and the cocycles are relatedby µ i ( u )( g , z ) = ζ ji ( u )( z + gN ) − µ j ( u )( g , g ji ( u ) + z ) ζ ji ( u )( z ) . (9) Conversely, any family of cocycles { µ i } i ∈ I that fulfils eq. (9) determines aunique decker. Proof : i ) The origin of the diagram is obvious. For the converse, it is suf-ficient to prove the result locally because the action of G on F preservescharts. Explicitly, over a chart ( U i , k i , h i ) the action of g ∈ G is h ∗ i ( φ ( g )) : U i × G / N × PU ( H ) → U i × G / N × PU ( H ) . ( u , z , U ) ( u , gN + z , µ ′′ i ( g )( u , z ) U ) If B is locally compact the exponential law (Lemma A.4) ensures that allfunctions µ ′ i : ( u , g , z ) µ ′′ i ( g )( u , z ) are jointly continuous. ii ) Locally, Lemma A.4 ensures that the transposed functions µ i : U i → Map ( G × G / N , PU ( H )) made out of µ ′ i are well defined. The cocyclecondition µ i ( u )( g + h , z ) = µ i ( u )( g , z + hN ) µ i ( u )( h , z ) and the validity of (9) are immediate as well as the converse statement. (cid:4) It should be mentioned at this point that equation (9) (and its unitary ver-sion we consider later) is quite powerful as turns out. A first application isgiven in the next proposition. It is a complete answer to the existence of deck-ers in the case of N =
0, i.e. G = G / N . The result is well-known, but we statea proof using (9) for the convenience of the reader.24 roposition 2.6 Assume N = and let P q → E p → B be a pair. Then a decker existsif and only if P ∼ = p ∗ P ′ for a PU ( H ) -bundle P ′ → B. Proof : If P ∼ = p ∗ P ′ for some PU ( H ) -bundle P ′ → B we obtain a decker byacting on the first entry of the fibered product p ∗ P ′ = E × B P ′ .Conversely, if a decker is given then we define a family ζ ′ ji : U ji → PU ( H ) by ζ ′ ji ( u ) : = µ j ( u )( g ji ( u ) , 0 ) − ζ ji ( u )( ) , for u ∈ U ji ⊂ B . This is well definedsince G = G / N .Claim 1 : { ζ ′ ji } j , i ∈ I are transition functions for a PU ( H ) -bundle P ′ → B .Proof : Let u ∈ U i ∩ U j ∩ U k . Then ζ ′ kj ( u ) ζ ′ ji ( u )= µ k ( u )( g kj ( u ) , 0 ) − ζ kj ( u )( ) µ j ( u )( g ji ( u ) , 0 ) − ζ ji ( u )( ) ( ) = µ k ( u )( g kj ( u ) , 0 ) − µ k ( u )( g ji ( u ) , g kj ( u ) + ) − | {z } ζ kj ( u )( + g ji ( u )) ζ ji ( u )( ) | {z } cocy.cond. = µ k ( u )( g ji ( u ) + g kj ( u ) , 0 ) − ( ) = ζ ki ( u )( )= ζ ′ ki ( u ) .Claim 2 : P ∼ = p ∗ P ′ .Proof : The bundle q ′ : p ∗ P ′ → E has transition functions p ∗ ζ ′ ji : p − ( U ij ) ∼ = U ij × G / N ∋ ( u , z ) ζ ′ ji ( u ) .We define an isomorphism f : P → p ∗ P ′ locally f i : = f | q − ( p − ( U i )) by P ⊃ q − ( p − ( U i ) f i −→ q ′− ( p − ( U i )) ⊂ p ∗ P ′ ↓∼ = ↓∼ = U i × G / N × U ( H ) −→ U i × G / N × U ( H )( u , z , U ) ( u , z , µ i ( u )( z , 0 ) − U ) .This is in fact a well defined global isomorphism since form eq. (9) it followsfor G = G / N and u ∈ U i ∩ U j µ i ( u )( z , 0 ) − = ζ ji ( u )( ) − µ j ( u )( z , g ji ( u )) − ζ ji ( u )( z )= ζ ji ( u )( ) − µ j ( g ji ( u ))( u , 0 ) µ j ( u )( z + g ji ( u ) , 0 ) − ζ ji ( u )( z )= ζ ′ ji ( u ) − µ j ( u )( z + g ji ( u ) , 0 ) − ζ ji ( u )( z ) .Thus the local definition of f is independent of the chosen chart. (cid:4) We now introduce the notions of equivalence we mentioned earlier.25 efinition 2.5
Two deckers ρ , ρ ′ : G × P → P on a pair ( P , E ) are called exteriorequivalent if the continuous function c : P × G → P which is defined by ρ ( . , − g ) ◦ ρ ′ ( . , g ) = c ( . , g ) : P → P, for g ∈ G, is locally of the following form: Thereexists an atlas U • such that for each chart ( U i , k i , h i ) there is a continuous map c i : U i × G × G / N → U ( H ) which satisfies the three conditions (E0) ( h − i ◦ c ( , g ) ◦ h i )( u , z , U ) = ( u , z , Ad ( c i ( u , g , z )) U ) ∈ U i × G / N × PU ( H ) , (E1) c j ( u , g , g ji ( u ) + z ) = ζ ji ( u )( z ) (cid:0) c i ( u , g , z ) (cid:1) ∈ U ( H ) and (E2) c i ( u , h + g , z ) = µ i ( u )( g , z ) − (cid:0) c i ( u , h , z + gN ) (cid:1) c i ( u , g , z ) ∈ U ( H ) , for the transition functions g ji , ζ ji of the pair and the cocycles µ i of the decker ρ asabove. It is clear that, if one has given a family of continuous unitary functions { c i } which satisfies (E0), (E1) and (E2) for the cocycles { µ i } of a decker ρ , the familyof cocycles µ ′ i ( u )( g , z ) : = µ i ( u )( g , z ) Ad ( c i ( u , g , z )) defines an exterior equivalent decker ρ ′ to ρ .Let c τ : G → Aut ( P , E ) be defined by c τ ( g ) : = c ( . , g ) . It satisfies thecocycle condition c τ ( g + h ) = c τ ( g ) · ρ τ ( h ) c τ ( h ) ,where · is the right action of Aut ( P , E ) on Aut ( P , E ) given by conjugation.This right action lifts to the sections Γ ( E , P × PU ( H ) U ( H )) of the to P associatedU ( H ) -bundle in the following diagram, i.e. the vertical map is Aut ( P , E ) -equivariant, Γ ( E , P × PU ( H ) U ( H )) (cid:9) Aut ( P , E ) (cid:15) (cid:15) G c τ / / c τ q o n l k i h f e c b ` Aut ( P , E ) ∼ = / / Γ ( E , P × PU ( H ) PU ( H )) ,Therein the associated bundles are both obtained by the the conjugate actionof PU ( H ) on the respective groups. Now, conditions (E0) - (E2) imply that wecan lift c τ to a unitary cocycle c τ : G → Γ ( E , P × PU ( H ) U ( H )) , i.e. c τ ( g + h ) = c τ ( g ) · ρ τ ( h ) c τ ( h ) Similar to Proposition 2.5 this global statement is an equivalent formulation ofexterior equivalence if the base B is locally compact. Proposition 2.7
Let B be locally compact, and let ρ and ρ ′ be deckers on a pair ( P , E ) . These two deckers are exterior equivalent if and only if c τ as defined above liftsto a unitary cocycle c τ . Here and in what follows we will always use the notation ζ ( c ) , ζ ∈ PU ( H ) , c ∈ U ( H ) , for theaction of PU ( H ) on U ( H ) by conjugation.
26e do not give a detailed proof of this fact, it is again just an application ofthe exponential law for locally compact spaces.The next statement gives an important example of exterior equivalent deck-ers.
Example 2.1
Let ρ be a decker on an arbitrary pair ( P , E ) , and let v : P → P be abundle automorphism. Then the conjugate decker ρ ν is exterior equivalent to ρ if theclass [ v ] ∈ ˇ H ( E , U ( )) of the automorphism vanishes. Proof :
Let us denote by µ i the cocycles of the decker ρ which satisfy (9) ona chosen atlas { U i } i ∈ I . Because the class of v vanishes, we can assume with-out restriction that it is locally implemented by unitary functions v i : U i → Map ( G / N , U ( H )) such that ζ ji ( u )( z )( v i ( u )( z )) = v j ( u )( g ji ( u ) + z ) .Locally, ρ ( , − g ) ◦ ρ ν ( , g ) is given by µ i ( u )( − g , z + gN ) Ad ( v i ( u )( z + gN )) µ i ( u )( g , z ) Ad ( v i ( u )( z ) − )= µ i ( u )( g , z ) − Ad ( v i ( u )( z + gN )) µ i ( u )( g , z ) Ad ( v i ( u )( z ) − )= Ad ( c i ( u , g , z )) ,for c i ( u , g , z ) : = µ i ( u )( g , z ) − (cid:0) v i ( u )( z + gN ) (cid:1) v i ( u )( z ) − ∈ U ( H ) . So condi-tion (E0) is satisfied. We check that the conditions (E1) and (E2) also holds. Infact, ζ ji ( u )( z ) (cid:0) c i ( u , g , z ) (cid:1) = ζ ji ( u )( z ) (cid:16) µ i ( u )( g + h , z ) − ) (cid:0) v i ( u )( z + gN + hN ) (cid:1) v i ( u )( z ) − (cid:17) ( ) = µ j ( u )( g , g ji ( u ) + z ) − (cid:16) ζ ji ( u )( z + gN ) (cid:0) v i ( u )( z + gN ) (cid:1)(cid:17) ζ ji ( u )( z ) (cid:0) v i ( u )( z ) (cid:1) − [ ν ]= = c j ( u )( g , g ji ( u ) + z ) which proves (E1), and c i ( u , g + h , z )= µ i ( u )( g + h , z ) − (cid:0) v i ( u )( z + gN + hN ) (cid:1) v i ( u )( z ) − = µ i ( u )( g , z ) − (cid:16) µ i ( u )( h , z + gN ) − (cid:0) v i ( u )( z + gN + hN ) (cid:1)(cid:17) v i ( u )( z ) − = µ i ( u )( g , z ) − (cid:16) µ i ( u )( h , z + gN ) − (cid:0) v i ( u )( z + gN + hN ) (cid:1) v i ( u )( z + gN ) (cid:17) µ i ( u )( g , z ) − (cid:0) v i ( u )( z + gN ) (cid:1) v i ( u )( z ) − which proves (E2). (cid:4) Between two dynamical triples we introduce a notion of equivalence basedon exterior equivalence.Two dynamical triples ( ρ , P , E ) and ( ρ ′ , P ′ , E ′ ) are isomorphic if there isa morphism ( ϕ , ϑ , θ ) of the underlying pairs such that ρ = ϑ ∗ ρ ′ . The triples27re outer conjugate if there is a morphism ( ϕ , ϑ , θ ) of the underlying pairssuch that ρ and ϑ ∗ ρ ′ are exterior equivalent on ( P , E ) . Furthermore, we call thetriples stably isomorphic (respectively stably outer conjugate ) if the triples ( ⊗ ρ , PU ( H ) ⊗ P , E ) and ( ⊗ ρ ′ , PU ( H ) ⊗ P ′ , E ′ ) are isomorphic (resp. outerconjugate).We can arrange these notions in a diagram of implications.isomorphismof dyn. triples + (cid:11) (cid:19) outer conjugationof dyn. triples (cid:11) (cid:19) stable isomorphismof dyn. triples + stable outer conjugationof dyn. triplesThe following example shall illustrate an important feature of the notion ofstably outer conjugation. Example 2.2
Let ( ρ , P , E ) be a dynamical triple, and let λ G : G → U ( L ( G )) bethe left regular representation of G. Then the two triples ( ρ , P , E ) and (( Ad ◦ λ G ) ⊗ ρ , PU ( L ( G )) ⊗ P , E ) are stably outer conjugate. Proof :
The triple ( ρ , P , E ) and its stabilisation ( ⊗ ρ , PU ( L ( G )) ⊗ P , E ) arestably isomorphic and the triples ( ⊗ ρ , PU ( L ( G )) ⊗ P , E ) and ( Ad ◦ λ G ⊗ ρ , PU ( L ( G )) ⊗ P , E ) are exterior equivalent by c i ( u , g , z ) : = λ G ( g ) ⊗ H . (cid:4) By Dyn we denote the set valued functor that sends a base space B to theset of equivalence classes of stably outer conjugate dynamical triples over it,i.e. Dyn ( B ) : = { dynamical triples over B } (cid:14) stable outer conj. .In the same manner as we did for the functor Par we can fix a bundle E → B and defineDyn ( E , B ) : = { dynamical triples over B with fixed E } (cid:14) stable outer conj. over id E .Isomorphic bundles E , E ′ lead to isomorphic sets Dyn ( E , B ) ∼ = Dyn ( E ′ , B ) , andthe bundle automorphisms Aut B ( E ) act on Dyn ( E , B ) by pullback. This yieldsa decomposition Dyn ( B ) ∼ = ∐ [ E ] ( Dyn ( E , B ) /Aut B ( E )) .Our next goal is to find the link between dynamical triples and the co-homology theory we introduce now. Let M n be the abelian sheaf on B de-fined by M n ( U ) : = C ( U , Bor ( G × n , L ∞ ( G / N , U ( )))) , for n =
0, 1, 2, . . . . Let U • = { U i | i ∈ I } be an open cover of B and let g ∈ ˇ Z ( U • , G / N ) . Note that M n is a right G / N -module, for all n =
0, 1, 2, . . . , by shifting the G / N -variable. We28onsider the the double complex... ... ...ˇ C ( U • , M ) d ∗ / / δ g O O ˇ C ( U • , M ) d ∗ / / δ g O O ˇ C ( U • , M ) d ∗ / / δ g O O · · · ˇ C ( U • , M ) d ∗ / / δ g O O ˇ C ( U • , M ) d ∗ / / δ g O O ˇ C ( U • , M ) d ∗ / / δ g O O · · · ˇ C ( U • , M ) d ∗ / / δ g O O ˇ C ( U • , M ) d ∗ / / δ g O O ˇ C ( U • , M ) d ∗ / / δ g O O · · · ,wherein the horizontal arrows d ∗ are induced by the boundary operator d :Bor ( G × n , L ∞ ( G / N , U ( ))) → Bor ( G × n + , L ∞ ( G / N , U ( ))) of group cohomol-ogy by acting point-wise on functions and the vertical arrows δ g are the twistedˇCech coboundary operators. In fact, all of the squares are commutative, hencewe obtain a resulting total complex (cid:0) C • tot ( U • , M • ) , ∂ g (cid:1) , i.e. C p tot ( U • , M • ) : = L p = k + l ˇ C k ( U • , M l ) and ∂ g | C p tot ( U • , M • ) : = δ g − ( − ) p d ∗ define a cochain com-plex. The choice of the sign in ∂ g will be convenient. By H • tot ( U • , M • , g ) we de-note the corresponding cohomology groups, and by passing to the limit overall refinements of the open covering U • we obtain H • tot ( B , M • , g ) : = lim V • H • tot ( V • , M • , g ) .We call this group the total cohomology of B with twist g . In the same manneras explained on page 21 the limit does not depend on the covering U • and isindependent of the choice of the refinement maps.It is also similar to the discussion on twisted ˇCech cohomology that the to-tal cohomology groups are well defined objects for the class [ g ] of a cocycle g up to an isomorphism.The connexion of dynamical triples and total cohomology has its origin inthe local structure of triples as we explain now. Let ( ρ , P , E ) be a dynamicaltriple. Let U • = { U i | i ∈ I } be an atlas for the underlying pair with transitionfunctions g ij , ζ ij and continuous cocycles µ i as in Proposition 2.5. One shouldrealise at this point that when we suppress the non-commutativity of PU ( H ) for a moment the equations δ g ζ .. = , d ( µ i ( u )) = and equation (9) are equiv-alent to ∂ g ( ζ .. , µ . ) = . We lift the transition functions and the cocycles to Borelfunctions. This will define a 2-cocycle for the total cohomology of B ; in detail:Without restriction (see equation (7)) we can assume that the atlas is chosensuch that the transition functions can be lifted continuously to Borel valued See A.2 ζ ij ; they define a twisted ˇCech 2-cocycle δ g ζ = : ψ ... ∈ ˇ C ( U • , M ) .Further we can assume (if necessary we refine the atlas once more) that allcharts are contractible. Therefore we can apply Corollary A.2 to each of the µ i and obtain continuous functions µ i : U i → Bor ( G × G / N , U ( H )) lifting µ i , forall i ∈ I . By Lemma A.2 we can also pass to µ i : U i → Bor ( G , L ∞ ( G / N , U ( H ))) which is important to as L ∞ ( G / N , U ( H )) is a continuous G / N -module. Fromthe cocycle identity d ( µ i ( u )) = we see that d ( µ i ( u )) = : ω i ( u ) defines a groupcohomology 2-cocycle ω . ∈ ˇ C ( U • , M ) . On twofold intersections we can de-fine φ ji ( u )( g , z ) : = µ i ( u )( g , z ) ζ ji ( u )( z ) − µ j ( u )( g , g ji ( u ) + z ) − ζ ji ( u )( z + gN ) which is due to equation (9) U ( ) -valued, i.e. φ .. ∈ ˇ C ( U • , M ) . Now, the threefamilies of functions ψ ... , φ .. and ω . satisfy the algebraic relations δ g ψ = δ g φ = d ∗ ψ d ∗ φ = δ g ω d ∗ ω = ∂ g ( ψ ... , φ .. , ω . ) = ∈ C ( U • , M • ) , (10)i.e. ( ψ ... , φ .. , ω . ) is a total 2-cocycle. Of course, one can verify this by directcomputation, but indeed it is implicitly clear, because, informally , we havedefined ( ψ ... , φ .. , ω . ) : = ∂ g ( ζ .. , µ . ) ∈ C ( U • , M • ) . Proposition 2.8
The assignment ( ρ , P , E ) ( ψ ... , φ .. , ω . ) constructed above de-fines a homomorphism of groups Dyn ( E , B ) → H ( B , M • , g .. ) . Proof :
We must check that the defined total cohomology class is independentof all choices. This is simple to verify for the choice of the atlas, and the choiceof the lifts of the transition functions and cocycles. As stably isomorphic pairshave the same local description, it is also clear that stably isomorphic pairsdefine the same total cohomology class. In detail we give the calculation thatexterior equivalent triples define the same class:Let ρ and ρ ′ be exterior equivalent deckers on ( P , E ) . Let c i : U i × G × G / N → U ( H ) be such that (E0), (E1) and (E2) of Defintion 2.5 are satisfied.If µ i and µ ′ i are the cocycles for the deckers, they both satisfy (9) for the samefamily of transition functions ζ ji . They are related by µ ′ i = µ i ( Ad ◦ c i ) , andif µ i is a lift for µ i , then µ ′ i : = µ i c i defines a lift for µ ′ i . Let ( ψ ... , φ .. , ω . ) be the i.e. up to the non-comutativity of U ( H ) ( ζ ji , µ i ) . Then we have µ ′ i ( u )( h + g , z )= µ i ( u )( h + g , z ) c i ( u , h + g , z )= µ i ( u )( h , z + gN ) µ i ( u )( h , z ) c i ( u , h + g , z ) ω i ( u )( g , h , z ) ( E2 ) = µ i ( u )( h , z + gN ) µ i ( u )( g , z ) µ i ( u )( g , z ) − (cid:0) c i ( u , h , z + gN ) (cid:1) c i ( g , z ) ω i ( u )( g , h , z )= µ i ( u )( h , z + gN ) c i ( u , h , z + gN ) µ i ( u )( g , z ) c i ( g , z ) ω i ( u )( g , h , z )= µ ′ i ( u )( h , z + gN ) µ ′ i ( u )( g , z ) ω i ( u )( g , h , z ) ,so ω ′ i = ω i , and µ ′ i ( u )( g , z )= µ i ( u )( g , z ) c i ( u , g , z )= ζ ji ( u )( z + gN ) − µ j ( u )( g , g ji ( u ) + z ) ζ ji ( u )( z ) c i ( u , g , z ) φ ji ( u )( g , z ) ( E1 ) = ζ ji ( u )( z + gN ) − µ j ( u )( g , g ji ( u ) + z ) c j ( u , g , g ji ( u ) + z ) ζ ji ( u )( z ) φ ji ( u )( g , z )= ζ ji ( u )( z + gN ) − µ ′ j ( u )( g , g ji ( u ) + z ) ζ ji ( u )( z ) φ ji ( u )( g , z ) ,so φ ′ ji = φ ji . (cid:4) Of course, the whole discussion we made so far for ( G , N ) can be done for ( b G , N ⊥ ) . I.e. we can replace G by its dual group b G : = Hom ( G , U ( )) and N bythe annihilator N ⊥ : = { χ | χ | N = } ⊂ b G of N everywhere. This is meaningfulas b G is second countable, N ⊥ is discrete and b G / N ⊥ compact (see A.1). Definition 2.6
Let B be a base space.i ) A dual pair ( b P , b E ) over B with underlying Hilbert space H is a sequence b P → b E → B, wherein b E → B is a b G / N ⊥ -principal fibre bundel and b P → b E a PU ( H ) -principal fibre bundle, such that the latter bundle is already trivial overthe fibres of b E → B.ii ) A dual decker ˆ ρ is an action ˆ ρ : b P × b G → b P that lifts the induced b G actionon b E and ˆ ρ ( , χ ) : b P → b P is a bundle isomorphisms for all χ ∈ b G.iii ) A dual dynamical triple ( ˆ ρ , b P , b E ) over B is a pair ( b P , b E ) over B equipped witha dual decker ˆ ρ It is clear now how we define d Par ( B ) , d Par ( b E , B ) , d Dyn ( B ) , d Dyn ( b E , B ) andhow all statements we have achieved so far translate to dual pairs and triples.31 .6 Topological Triples We introduce topological triples built out of a pair and a dual one. They wereintroduced first in [BRS] under the name T-duality triples in the special case G = R n , N = Z . Our definition won’t be exactly the same as in [BRS] - wecomment on this in section 3.3.There is a canonical U ( ) -principal fibre bundle over G / N × b G / N ⊥ which iscalled Poincar´e bundle. We recall its definition. Let Q : = (cid:0) G / N × b G × U ( ) (cid:1) / N ⊥ , (11)where the action of N ⊥ is defined by ( z , χ , t ) · n ⊥ : = ( z , χ + n ⊥ , t h n ⊥ , z i − ) .Then the obvious map Q → G / N × b G / N ⊥ is a U ( ) -principal fibre bundle.Indeed, U ( ) acts freely and transitive in each fibre by multiplication in thethird component and local sections of Q → G / N × b G / N ⊥ are given by G / N × V a ∋ ( z , ˆ z ) [( z , ˆ s a ( ˆ z ) , 1 )] ∈ Q , where, by Lemma A.1, ˆ s a : V a → b G is a familyof local sections of the N ⊥ -principal bundle b G → b G / N ⊥ , V a ⊂ b G / N ⊥ . On theoverlap G / N × V ab two such sections are related by [( z , ˆ s a ( ˆ z ) , 1 )] = [( z , ˆ s b ( ˆ z ) − n ⊥ ab ( ˆ z ) , 1 )]= [( z , ˆ s b ( ˆ z ) , h n ⊥ ( ˆ z ) , z i )]= [( z , ˆ s b ( ˆ z ) , 1 )] · h n ⊥ ( ˆ z ) , z i ,wherein n ⊥ ab ( ˆ z ) : = − ˆ s a ( ˆ z ) + ˆ s b ( ˆ z ) which defines a family of transition functions n ⊥ ab : V ab → N ⊥ for the bundle b G → b G / N ⊥ . Thus we have found that ν ⊥ ab : G / N × V ab → U ( ) defined by ν ⊥ ab ( z , ˆ z ) : = h n ⊥ ab ( ˆ z ) , z i are transition functionsfor the bundle Q .Dually, there is a second U ( ) -bundle R : = ( G × b G / N ⊥ × U ( )) / N → G / N × b G / N ⊥ which has transition functions ν cd : W cd × b G / N ⊥ → U ( ) de-fined by ν cd ( z , ˆ z ) : = h ˆ z , n cd ( z ) i for an open cover { W a } of G / N and transitionfunctions n cd : W cd → N of G → G / N .We denote the ˇCech classes in ˇ H ( G / N × b G / N ⊥ , U ( )) which these bun-dles define by [ Q ] and [ R ] . Definition 2.7
The class π : = − [ Q ] constucted above is called the Poincar´e class of G / N × b G / N ⊥ . Of course, in this definition we made a choice, but up to a sign there is none.
Lemma 2.2 [ Q ] and [ R ] are inverses of each other, i.e. [ Q ] + [ R ] = ∈ ˇ H ( G / N × b G / N ⊥ , U ( )) . Proof :
We have to show that { ν cd ν ⊥ ab : W cd × V ab → U ( ) } ( a , c ) , ( b , d ) is a ˇCechcoboundary. Let ˆ s a : V a → b G and s c : W c → G be families of local sections suchthat n cd ( z ) = − s c ( z ) + s d ( z ) and n ⊥ ab ( ˆ z ) = − ˆ s a ( ˆ z ) + ˆ s b ( ˆ z ) . We show that { ν cd ν ⊥ ab : W cd × V ab → U ( ) } ( a , c ) , ( b , d ) = δ (cid:0) {h ˆ s a ( .. ) , s c ( ) i : W c × V a → U ( ) } ( a , c ) (cid:1) ,32herin δ is the usual ˇCech coboundary operator. It is h ˆ z , n cd ( z ) i = h ˆ s a ( ˆ z ) , n cd ( z ) i and h n ⊥ ab ( ˆ z ) , z i = h n ⊥ ab ( ˆ z ) , s d ( z ) i , because n cd ( z ) ∈ N and n ⊥ ab ( ˆ z ) ∈ N ⊥ ; thereinthe right hand side is the pairing G × b G → U ( ) . Thus ν cd ( z , ˆ z ) ν ⊥ ab ( z , ˆ z ) = h ˆ s a ( ˆ z ) , n cd ( z ) ih n ⊥ ab ( ˆ z ) , s d ( z ) i = h ˆ s a ( ˆ z ) , − s c ( z ) + s d ( z ) ih− ˆ s a ( ˆ z ) + ˆ s b ( ˆ z ) , s d ( z ) i = h ˆ s b ( ˆ z ) , s d ( z ) ih ˆ s a ( ˆ z ) , s c ( z ) i − which proves the lemma. (cid:4) We now turn to the definition of topological triples. Let P → E → B be apair and let b P → b E → B be a dual pair with same underlying Hilbert space H .We consider the following diagram of Cartesian squares P × B b E $ $ IIIIIIIII | | yyyyyyyyy E × B b P ! ! DDDDDDDDD z z uuuuuuuuu P s " " EEEEEEEEEE E × B b E $ $ JJJJJJJJJJ z z tttttttttt b P | | zzzzzzzzzz E % % KKKKKKKKKKKK b E y y ssssssssssss B . (12)Assume that there is a PU ( H ) -bundle isomorphism κ : E × B b P → P × B b E whichfits into the above diagram (12), i.e. it fixes its base E × B b E . Let us choose a chart U i ⊂ B common for the pair and the dual pair and trivialise (12) locally. Foreach u ∈ U i this induces an automorphism κ i ( u ) of the trivial PU ( H ) -bundleover G / N × b G / N ⊥ , G / N × b G / N ⊥ × PU ( H ) (cid:15) (cid:15) G / N × b G / N ⊥ × PU ( H ) κ i ( u ) o o (cid:15) (cid:15) G / N × b G / N ⊥ G / N × b G / N ⊥ . = o o This automorphism defines a ˇCech class [ κ i ( u )] ∈ ˇ H ( G / N × b G / N ⊥ , U ( )) (cp. Therem A.1). Definition 2.8
We say κ satisfies the Poincar´e condition if for each chart U i andeach u ∈ U i the equality [ κ i ( u )] = π + p ∗ a + p ∗ b holds, for the Poincar´e class π andsome classes a ∈ ˇ H ( G / N , U ( )) and b ∈ ˇ H ( b G / N ⊥ , U ( )) . Here p , p are theprojections from G / N × b G / N ⊥ on the first and second factor. a , b are just of minor importance.They are manifestations of the freedom to choose another atlas as they varyunder the change of the local trivialisations. In fact, one can always modify thelocal trivialisations of the given atlas such that a and b vanish. Definition 2.9 A topological triple (cid:0) κ , ( P , E ) , ( b P , b E ) (cid:1) over B is a pair ( P , E ) and dual pair ( b P , b E ) over B (with same underlying Hilbert space H ) together with acommutative diagram P × B b E $ $ IIIIIIIII } } zzzzzzzzz E × B b P ! ! DDDDDDDDD z z uuuuuuuuu κ o o P " " DDDDDDDDDD E × B b E $ $ JJJJJJJJJJ z z tttttttttt b P | | zzzzzzzzzz E % % JJJJJJJJJJJJ b E y y tttttttttttt B , (13) wherein all squares are Cartesian and κ is an isomorphism that satisfies the Poincar´econdition. We call two topological triples (cid:0) κ , ( P , E ) , ( b P , b E ) (cid:1) and (cid:0) κ ′ , ( P ′ , E ′ ) , ( b P ′ , b E ′ ) (cid:1) (with underlying Hilbert spaces H , H ′ respectively) equivalent if there is amorphisms of pairs ( ϕ , ϑ , θ ) from ( P , E ) to ( P ′ , E ′ ) and a morphism of dualpairs ( ˆ ϕ , ˆ ϑ , ˆ θ ) from ( b P , b E ) to ( b P ′ , b E ′ ) such that the induced diagram P × B b E ϑ × B ˆ θ (cid:15) (cid:15) E × B b P κ o o θ × B ˆ ϑ (cid:15) (cid:15) ϕ ∗ P ′ × B b E ′ E ′ × B ˆ ϕ ∗ b P ′ κ ′ o o (14)is commutative up to homotopy, i.e. the ˇCech class of the bundle automor-phism ( θ × B ˆ ϑ ) − ◦ κ ′− ◦ ( ϑ × B ˆ θ ) ◦ κ in ˇ H ( E × B b P , U ( )) vanishes. The triplesare called stably equivalent if the stabilised triples (cid:0) ⊗ κ , ( P H , E ) , ( b P H , b E ) (cid:1) and (cid:0) ⊗ κ ′ , ( P ′ H , E ′ ) , ( b P ′ H , b E ′ ) (cid:1) are equivalent for some separable Hilbert space H . The meaning of the index H is stabilisation as in equation (4). Stableequivalence will be the right choice of equivalence for us, and we introducethe set valued functor Top which associates to a base space B the set of stableequivalence classes of topological triples, i.e.Top ( B ) : = { topological triples over B } (cid:14) stable equivalence .34f we choose a G / N -bundle E → B we can consider all the topological tripleswith this bundle fixed and those stable equivalences for which the identity over E can be extendend to a morphisms of the underlying pairs. We defineTop ( E , B ) : = { topological triples with fixed E → B } (cid:14) stable equivalence over id E .The automorphisms Aut B ( E ) of E over the identity of B act on Top ( E , B ) bypullback, and there is a correspondence Top ( B ) ∼ = ∐ [ E ] ( Top ( E , B ) /Aut B ( E )) . Remark 2.4
We already stated in Remark 1.1 that our notion of topological triplesin the case of G = R n , N = Z n and the notion of T-duality triples as found in [BRS]do not agree. Nevertheless the two notions lead to the same isomorphism classes, butwe postpone this clarification to section 3.3, Lemma 3.7. By definition, the Poincar´e class π has a geometric interpretation in termsof the Poincar´e bundle (11). For our purposes it will be important that we cangive an analytical description of π . Lemma 2.3
Choose a Borel section σ : G / N → G and an arbitrary section ˆ σ : b G / N ⊥ → b G of the corresponding quotient maps. Theni ) the map κ σ : G / N × b G / N ⊥ → PU ( L ( G / N ) ⊗ H )( z , ˆ z ) Ad (cid:0) h ˆ σ ( ˆ z ) , σ ( − z ) − σ ( ) i | {z } ⊗ H (cid:1) = : κ σ ( z , ˆ z ) ∈ L ∞ ( G / N , U ( )) is continuous and independent of the choice of ˆ σ . Therefore it defines (a bundleisomorphism of the trivial PU ( L ( G / N , H )) -bundle and) a class [ κ σ ] ∈ ˇ H ( G / N × b G / N ⊥ , U ( )) , and this class is independent of the choice of σ ;ii ) [ κ σ ] = π . Proof : i ) Firstly, we observe that b G ∋ χ
7→ h χ , σ ( ) i ∈ U ( L ( G / N )) is (strongly)continuous. In fact, the sequential continuity of this map follows by domi-nated convergence. Therefore the composition ( z , χ ) λ G / N ( z ) ◦ h χ , σ ( ) i ◦ λ G / N ( − z ) ◦ h χ , − σ ( ) i = h χ , σ ( − z ) − σ ( ) i is continous. Here λ G / N is the left regular representation, i.e. λ G / N ( z ) F ( x ) : = F ( − z + x ) , F ∈ L ( G / N ) . Now, if n ⊥ ∈ N ⊥ we obtain h n ⊥ , σ ( − z ) − σ ( ) i = An element f = f ( ) ∈ L ∞ ( G / N , U ( )) is a multiplication operator on the Hilbert space L ( G / N ) , see section A.3. b G is first countable. n ⊥ , σ ( − z ) i ∈ U ( ) ⊂ U ( L ( G / N ) , because the difference σ ( − z ) − ( σ ( − z ) − σ ( )) is in N . Thus ( z , χ ) Ad ( h χ , σ ( − z ) − σ ( ) i ) factors through the quo-tient map G / N × b G → G / N × b G / N ⊥ which establishes κ σ as stated above.To see that the class of κ σ is independent of σ it is sufficient to recognise that κ n , defined by the same formula as κ σ , is unitary implemented for any Borelfunction n : G / N → N . But this is the case, for ( z , χ )
7→ h χ , n ( − z ) − n ( ) i ∈ U ( L ( G / N )) is continuous and factors through the quotient G / N × b G / N ⊥ . ii ) Let b G / N ⊥ ⊃ V a ˆ s a → b G be a family of local sections, so ˆ s b ( ˆ z ) − ˆ s a ( ˆ z ) = : n ⊥ ab ( ˆ z ) ∈ N ⊥ defines a set of transition functions. Let W a : = G / N × V a , then κ a : W a ∋ ( z , ˆ z )
7→ h ˆ s a ( ˆ z ) , σ ( − z ) − σ ( ) i ∈ U ( L ( G / N )) defines locally a con-tinuous unitary lift of κ σ . Therefore, on twofold intersections W ba ∋ ( z , ˆ z ) wehave κ b ( z , ˆ z ) κ a ( z , ˆ z ) − = : κ ab ( z , ˆ z ) ∈ U ( ) , and the class of κ σ is by definitionthe class [ κ .. ] of the cocycle κ .. . We obtain κ b ( z , ˆ z ) κ a ( z , ˆ z ) − = h ˆ s b ( ˆ z ) − ˆ s a ( ˆ z ) , σ ( − z ) − σ ( ) i = h n ⊥ ab ( ˆ z ) , σ ( − z ) − σ ( ) i = h n ⊥ ab ( ˆ z ) , σ ( − z ) i = h n ⊥ ab ( ˆ z ) , z i − ∈ U ( ) ,wherein the last equality identifies N ⊥ with [ G / N . Therefore the class of κ σ coincides with the negative of the class of the bundle Q . (cid:4) Of course, the situation is symmetric and there is an analogous statementinvolving the class of bundle R . In that case, if we replace everything by itsdual counterpart, we deal with the functionˆ κ ˆ σ : G / N × b G / N ⊥ → PU ( L ( b G / N ⊥ ) ⊗ H ) (15) ( z , ˆ z ) Ad ( h ˆ σ ( .. − ˆ z ) − ˆ σ ( .. ) , σ ( z ) i | {z } ⊗ H )= : ˆ κ ˆ σ ( z , ˆ z ) ∈ L ∞ ( b G / N ⊥ , U ( )) with a Borel section ˆ σ : b G / N ⊥ → b G . Lemma 2.4
The class of ˆ κ ˆ σ is the negative of the Poincar´e class, i.e. π = [ κ σ ] = − [ ˆ κ ˆ σ ] . Proof :
We show that ( z , ˆ z ) κ σ ( z , ˆ z ) ⊗ ˆ κ ˆ σ ( z , ˆ z ) is already unitarily imple-36ented. Indeed, κ σ ( z , ˆ z ) ⊗ ˆ κ ˆ σ ( z , ˆ z )= Ad ( h ˆ σ ( ˆ z ) , σ ( − z ) − σ ( ) ih ˆ σ ( .. − ˆ z ) − ˆ σ ( .. ) , σ ( z ) i )= Ad ( h ˆ σ ( ˆ z ) , σ ( − z ) − σ ( ) + σ ( z ) ih ˆ σ ( .. − ˆ z ) − ˆ σ ( .. ) , σ ( z ) i )= Ad ( h− ˆ σ ( .. − ˆ z ) + ˆ σ ( .. ) , σ ( − z ) − σ ( ) + σ ( z ) ih ˆ σ ( .. − ˆ z ) − ˆ σ ( .. ) , σ ( z ) i )= Ad ( h− ˆ σ ( .. − ˆ z ) + ˆ σ ( .. ) , σ ( − z ) − σ ( ) i )= Ad ( λ b G / N ⊥ ( ˆ z ) λ G / N ( z ) h− ˆ σ ( .. ) , σ ( ) i ) λ b G / N ⊥ ( − ˆ z ) h ˆ σ ( .. ) , σ ( ) i ) λ G / N ( − z ) λ b G / N ⊥ ( ˆ z ) h− ˆ σ ( .. ) , − σ ( ) i ) λ b G / N ⊥ ( − ˆ z ) h ˆ σ ( .. ) , − σ ( ) i ) ∈ PU ( L ( G / N , H ) ⊗ L ( b G / N ⊥ , H )) .The argument of Ad is a continuous, unitary expression, since the left regularrepresentations λ b G / N ⊥ and λ G / N are (strongly) continuous. (cid:4) T-Duality
In the last sections we have introduced our main objects (dynamical and topo-logical triples). In the following sections we single out specific subclasses ofthose and show that they are related to each other. In addition, we show thattheir relations are precisely those which are obtained from the associated C ∗ -dynamical picture. Most part of this C ∗ -algebraic structure been observed incontext of continuous trace algebras alone in [RR, Thm 2.2, Cor. 2.5], [OR, Cor.2.1] and more recent in [MR, Thm. 3.1] with applications to T-duality.However, we establish a different approach by use of the local structure ofthe underlying objects. In particular, we do not need any assumption aboutlocal compactness of the underlying spaces. If we assume our base spaces B tobe locally compact, then the bundles E , b E over B will be locally compact andspectra of the involved continuous trace algebras; but the proof we present isindependent of such an assumption. Moreover, it has the advantage of beingexplicit enough to point out the connexion between the C ∗ -algebraic approachand the topological approach to T-duality. Notation:
In several proofs of the following sections we have to check thevalidity of local identities. All of these are straight forward computations ingeneral, but to keep the formulas readable we drop the base variable u ∈ B .E.g. an identity like ζ ji ( u )( z + gN ) µ i ( u )( g , z ) = µ j ( u )( g , g ji ( u ) + z ) ζ ji ( u )( z ) becomes ζ ji ( z + gN ) µ i ( g , z ) = µ j ( g , g ji + z ) ζ ji ( z ) .We indicate this with the label u (cid:13) before any of these computations. We start with the definition of an important subclass of dynamical triples.
Definition 3.1
A dynamical triple is called dualisable if its associated total coho-mology class (Proposition 2.8) is of the form [ ψ ... , φ .. , ω . = d ∗ ν . ] , i.e. the pair permitsa sufficiently refined atlas such that ω . is in the image of the boundary operator d ∗ . Let us denote the set of dualisable dynamical triples over B by Dyn † ( B ) ,respectively Dyn † ( E , B ) for fixed E . Similarly, d Dyn † ( B ) and d Dyn † ( E , B ) arethe sets of dualisable dual dynamical triples. We are going to present a con-struction that associates to a dualisable dynamical triple x a dual dualisabledynamical triple b x . This construction will be an honest map on the level ofequivalence classes, and there it turns out to be involutive, i.e. [ bb x ] = [ x ] .Let ( ρ , P , E ) be a dualisable dynamical triple over B . Choose a sufficiently re-fined atlas { U i | i ∈ I } in the sense that the transition functions ζ ij and cocycles38 i lift continuously to ζ ij and µ i . Then let ( ψ ... , φ .. , ω . ) be the associated 2-cocycle. Since the triple is dualisable, we can assume that ω . is in the image of d ∗ , let ω . : = d ∗ ν . . We can modify µ i by multiplying with ν − i . Therefore we canassume without restriction that d ( µ i ( u )) = and ω i =
1, for all i ∈ I , u ∈ U i . µ i is then unique up to a function U i → Z ( G , L ∞ ( G / N , U ( ))) .We are going to define a set of transition functionsˆ a ij : = ˆ g ij × ˆ ζ ij : U ji → b G / N ⊥ ⋉ Map ( b G / N ⊥ , PU ( L ( G / N ) ⊗ H )) for a dual pair. The vanishing of ω . implies that φ ij ( u ) ∈ Z ( G , L ∞ ( G / N ; U ( ))) ,for all i , j , u . This tells us u (cid:13) φ ij ( g , hN + ) φ ij ( h , ) = φ ij ( g + h , )= φ ij ( h + g , )= φ ij ( h , gN + ) φ ij ( g , ) ∈ L ∞ ( G / N , U ( )) which implies for g = n ∈ N and all hN ∈ G / N that φ kl ( u )( n , hN + ) = φ kl ( u )( n , ) holds, hence φ kl ( u )( n , ) ∈ L ∞ ( G / N , U ( )) is a constant. Further, n φ kl ( u )( n , ) is a homomorphism and continuous as N is discrete. Thusthere exists ˆ g ij : U ji → b G / N ⊥ ( ∼ = b N ) such that h ˆ g ij ( u ) , n i = ( φ ij ( u )( n , z )) − (16)for all u ∈ U ji , n ∈ N and (almost) all z ∈ G / N . Proposition 3.1 { ˆ g ij : U ji → b G / N ⊥ | i , j ∈ I } is a ˇC ech 1-cocycle. Proof :
We have δ g φ = d ∗ ψ , explicitely this reads for g ∈ G , u ∈ U ijk φ jk ( u )( g , ) φ ik ( u )( g , ) − φ ij ( u )( g , g jk ( u ) + ) = ψ ijk ( u )( + gN ) ψ ijk ( u )( ) − .For g = n ∈ N the right-hand side vanishes, hence the ˇCech cocycle equationfollows. (cid:4) In the next lemma we state some properties of the lifted cocycles µ i . It willbe a useful technical tool later. Lemma 3.1
The maps ( i ) U i × G / N ∋ ( u , z ) µ i ( u )( . , z ) | N ∈ U ( L ( N ) ⊗ H ) , ( ii ) U i × G / N ∋ ( u , z ) Ad ( µ i ( u )( . , z )) ∈ PU ( L ( G ) ⊗ H ) , ( iii ) U i × G / N ∋ ( u , z ) Ad ( µ i ( u )( − σ ( ) , z )) ∈ PU ( L ( G / N ) ⊗ H ) , ( iv ) U ji × G / N ∋ ( u , z ) Ad ( φ ij ( u )( . , z )) ∈ PU ( L ( G )) ,39 v ) U ji × G / N ∋ ( u , z ) Ad ( φ ij ( u )( − σ ( ) , z )) ∈ PU ( L ( G / N )) are continuous; and for all u ∈ U ji , z ∈ G / N the formula Ad ( ζ ji ( u )( − ) φ ji ( u )( − σ ( ) , 0 ) − )= Ad ( µ j ( u )( − σ ( + z ) , g ji ( u ) + z )) ζ ji ( u )( z ) Ad ( µ i ( u )( − σ ( + z ) , z ) − ) Ad ( h ˆ σ ( ˆ g ji ( u )) , σ ( + z ) − σ ( ) i ) ∈ PU ( L ( G / N , H )) (17) holds. Proof : ( i ) Let n ∈ N , g ∈ G and z ∈ G / N , then µ i ( u )( n , z + gN ) µ i ( u )( g , z ) = µ i ( u )( n + g , z )= µ i ( u )( g , z ) µ i ( u )( n , z ) .This implies that µ i ( u )( n , z + gN ) = µ i ( u )( g , z ) (cid:0) µ i ( u )( n , z ) (cid:1) ∈ U ( H ) . Thereinthe right hand side is continuous in g , hence the left hand side is continuous in z ′ : = gN . As the left hand side is symmetric in z and z ′ it is continuos in z . ( ii ) Let ( u α , z α ) → ( u , z ) be a converging net. Let x α : = z α − z and choose g α → ∈ G such that g α N = x α . Such g α exist – take a local section of thequotient (Lemma A.1). ThenAd ( µ i ( u α )( . , z α )) = Ad ( µ i ( u α )( . , z + x α ))= Ad ( µ i ( u α )( . + g α , z )) Ad ( µ i ( u α )( g α , z ) − )= Ad ( µ i ( u α )( . + g α , z )) µ i ( u α )( g α , z ) − The second factor converges to µ i ( u )( z ) = . For the first factor, note that ( u , g ) Ad ( µ i ( u )( . + g , z )) = Ad ( λ G ( g ) µ i ( u )( . , z ) λ G ( − g )) ∈ PU ( L ( G , H )) is continuous. This because Bor ( G × G / N , U ( H )) ∋ µ i ( u ) µ i ( u )( . , z ) ∈ L ∞ ( G , U ( H )) is continuous. ( iii ) We shall show that ν ( . ) ν ( σ ( )) is a continuous map from the multi-plication operators L ∞ ( G , U ( H )) to L ∞ ( G / N , U ( H )) . Indeed, let f ∈ L ( G / N ) and let χ be the characteristic function of σ ( G / N ) ⊂ G , so ( g f σ ( g ) : = f ( gN ) χ ( g )) ∈ L ( G ) . Let ν n ( . ) → ν ( . ) ∈ L ∞ ( G , U ( H )) be a convergingsequence. Then k ν n ( σ ( )) f ( ) − ν ( σ ( )) f ( ) k = Z G / N | ν n ( σ ( z )) f ( z ) − ν ( σ ( z )) f ( z ) | dz = Z G | ν n ( g ) f σ ( g ) − ν ( g ) f σ ( g ) | dg →
0, for n → ∞ . ( iv ) , ( v ) The fourth and fifth statement follow directly from ( i ) and ( ii ) andfrom the definition of φ ji on page 30. 40quation (17) follows by threefold application of the definition of φ ji . Welet n ( x , y ) : = σ ( x + y ) − σ ( x ) − σ ( y ) ∈ N , then u (cid:13) Ad (cid:16) ζ ji ( − ) φ ji ( − σ ( ) , 0 ) − (cid:17) = Ad (cid:16) µ j ( − σ ( ) , g ji ) ζ ji ( ) µ i ( − σ ( ) , 0 ) − (cid:17) = Ad (cid:16) µ j ( − σ ( ) , g ji ) µ j ( σ ( z ) , g ji ) − ζ ji ( z ) µ i ( σ ( z ) , 0 ) µ i ( − σ ( ) , 0 ) − (cid:17) = Ad (cid:16) µ j ( − σ ( ) − σ ( z ) , g ji + z ) ζ ji ( z ) µ i ( − σ ( ) − σ ( z ) , z ) − (cid:17) = Ad (cid:16) µ j ( − σ ( + z ) , g ji + z ) µ j ( n ( , z ) , g ji + z ) ζ ji ( z ) µ i ( n ( , z ) , z ) − µ i ( − σ ( + z ) , z ) − (cid:17) = Ad (cid:16) µ j ( − σ ( + z ) , g ji + z ) ζ ji ( z ) φ ji ( n ( , z ) , z ) − µ i ( − σ ( + z ) , z ) − (cid:17) .Since Ad ( φ ji ( u )( n ( , z ) , z ) − ) = Ad ( h ˆ g ji ( u ) , n ( , z ) i ) = Ad ( h ˆ σ ( g ji ( u )) , σ ( + z ) − σ ( ) i ) , the assertion is proven. (cid:4) We now turn to the definition of the projective unitary transition functionsˆ ζ ji for the dual pair, but before we remark on the local definition we make. Remark 3.1
The ad hoc definition of the transition functions of the dual pair by for-mula (18) below may seem very unsatisfactory, because one cannot even guess the ori-gin of this formula. In Theorem 3.8 we will see that the crossed product G × ρ Γ ( E , F ) (see appendix A.4) of the associated C ∗ -algebra of sections Γ ( E , F ) , F : = P × PU ( H ) K ( H ) , can be explicitly computed by a fibre-wise, modified Fourier transform. Thebehaviour of this transformation under the change of charts will lead finally to formula(18) (and shows that the crossed product is isomorphic to the associated C ∗ -algebra ofsections of the dual pair).However, we are in the pleasant situation that we can avoid the C ∗ -algebraic appa-ratus at this point and can formulate the theory in bundle theoretic terms only. By Lemma 3.1, ( u , z ) Ad ( φ ji ( u )( − σ ( ) , z )) is continuous hence its re-striction to U ji × { } is: u Ad ( φ ji ( u )( − σ ( ) , 0 )) . Then we letˆ ζ ji ( u )( ˆ z ) : = Ad (cid:16) ( κ σ ( − g ji ( u ) , ˆ g ji ( u ) + ˆ z ) ⊗ H ) ( λ G / N ( − g ji ( u ) ⊗ H ) ζ ji ( u )( − ) ( φ ji ( u )( − σ ( ) , 0 ) − ⊗ H ) (cid:17) . (18)41herein λ G / N is the left regular representation on L ( G / N ) , and κ σ is takenfrom Lemma 2.3. Indeed, this defines a continuous mapˆ ζ ji : U ji → Map ( b G / N ⊥ , PU ( L ( G / N ) ⊗ H )) ,but its definition involved several choices, namely, the atlas, the liftings ζ ij , µ i and the section σ . Theorem 3.1
The family ˆ a .. = { ˆ a ij = g ji × ζ ji | i , j ∈ I } is a ˇC ech cocycle and itsclass [ ˆ a .. ] is independent of the choices involved. Proof :
We have to show that δ ˆ g ˆ ζ = . We insert (18) and obtain u (cid:13) ˆ ζ ji ( ˆ z ) ˆ ζ ki ( ˆ z ) − ˆ ζ kj ( ˆ g ji + ˆ z )= Ad (cid:16) ( h ˆ σ ( ˆ g ji + ˆ z ) , σ ( + g ji ) − σ ( ) i ⊗ )( λ G / N ( − g ji ) ⊗ ) ζ ji ( − ) ( φ ji ( − σ ( ) , 0 ) − ⊗ )( φ ki ( − σ ( ) , 0 ) ⊗ ) ζ ki ( − ) − ( λ G / N ( g ki ) ⊗ )( h ˆ σ ( ˆ g ki + ˆ z ) , σ ( + g ki ) − σ ( ) i − ⊗ )( h ˆ σ ( ˆ g kj + ˆ g ji + ˆ z ) , σ ( + g kj ) − σ ( ) i ⊗ )( λ G / N ( − g kj ) ⊗ ) ζ kj ( − )( φ kj ( − σ ( ) , 0 ) − ⊗ ) (cid:17) = Ad (cid:16) ( h ˆ σ ( ˆ g ji + ˆ z ) − ˆ σ ( ˆ g ki + ˆ z ) , σ ( + g ji ) − σ ( ) i ⊗ )( ψ kji ( − − g ji ) ⊗ )( φ ji ( − σ ( + g ji ) , 0 ) − ⊗ )( φ ki ( − σ ( + g ji ) , 0 ) ⊗ )( φ kj ( − σ ( ) , 0 ) − ⊗ ) (cid:17) .The argument of Ad has simplified to a multiplication operator, so it remainsto show that it is a constant in U ( ) . By use of δ g φ = d ∗ ψ and d ∗ φ = = Ad (cid:16) ( h ˆ σ ( ˆ g ji + ˆ z ) − ˆ σ ( ˆ g ki + ˆ z ) , σ ( + g ji ) − σ ( ) i ⊗ ) ψ kji ( ) ( φ kj ( − σ ( + g ji ) , g ji ) ⊗ )( φ kj ( − σ ( ) , 0 ) − ⊗ ) (cid:17) = Ad (cid:16) ( h ˆ σ ( ˆ g ji + ˆ z ) − ˆ σ ( ˆ g ki + ˆ z ) , σ ( + g ji ) − σ ( ) i ⊗ ) ψ kji ( ) ( φ kj ( − σ ( ) + σ ( + g ji ) , 0 ) − ⊗ ) (cid:17) = Ad (cid:16) ψ kji ( ) ( φ kj ( − σ ( ) + σ ( + g ji ) , 0 ) − ⊗ ) h ˆ σ ( ˆ g ji + ˆ z ) − ˆ σ ( ˆ g ki + ˆ z ) , σ ( g ji ) i ( h ˆ σ ( ˆ g ji + ˆ z ) − ˆ σ ( ˆ g ki + ˆ z ) , σ ( + g ji ) − σ ( ) − σ ( g ji ) | {z } i ⊗ ) (cid:17) , ∈ N g ji . . . = Ad (cid:16) ψ kji ( ) ( φ kj ( − σ ( ) + σ ( + g ji ) , 0 ) − ⊗ ) h ˆ σ ( ˆ g ji + ˆ z ) − ˆ σ ( ˆ g ki + ˆ z ) , σ ( g ji ) i ( h− ˆ g kj , σ ( + g ji ) − σ ( ) − σ ( g ji ) i ⊗ ) (cid:17) = Ad (cid:16) ψ kji ( ) φ kj ( σ ( g ji ) , 0 ) − h ˆ σ ( ˆ g ji + ˆ z ) − ˆ σ ( ˆ g ki + ˆ z ) , σ ( g ji ) i (cid:17) = L ( G / N ) ⊗ H . (19)This is the desired result.To check that all the choices involved have no effect on the class of thiscocycle is straight forward, and we skip the tedious computation here. (cid:4) Let us define a pair for ˆ g ij , ˆ ζ ij explicitly. Let b E : = ∐ i ( U i × b G / N ⊥ ) (cid:14) ∼ with relation ( i , u , ˆ z ) ∼ ( j , u , ˆ g ji ( u ) + ˆ z ) and b P : = ∐ i ( U i × b G / N ⊥ × PU ( L ( G / N ) ⊗ H )) (cid:14) ∼ with relation ( i , u , ˆ z , U ) ∼ ( j , u , ˆ g ji ( u ) + ˆ z , ˆ ζ ji ( u )( ˆ z ) U ) . ( b P , b E ) is a dual pair over B with underlying Hilbert space b H : = L ( G / N ) ⊗ H . Then the statement ofthe theorem is that we have constructed a mapDyn † ( B ) → d Par ( B ) , [( ρ , P , E )] [( b P , b E )] for one easily checks that outer conjugate triples define isomorphic duals. Weare going to improve this statement in the next theorem.Let us define a family of projective unitary 1-cocycles by the following sim-ple formula. Let χ ∈ b G , ˆ z ∈ b G / N ⊥ , u ∈ U i , then we defineˆ µ i ( u )( χ , ˆ z ) : = Ad ( h χ , − σ ( ) i ⊗ H ) ∈ PU ( L ( G / N ) ⊗ H ) . (20)Therein σ : G / N → G is the same Borel section which we have used todefine ˆ ζ ji . By dominated convergence , it is clear that χ
7→ h χ , − σ ( ) i ∈ L ∞ ( G / N , U ( )) is continuous, so ˆ µ i is. Theorem 3.2
The family { ˆ µ i | i ∈ I } defines a dual decker ˆ ρ on the pair ( b P , b E ) , andwe obtain a bijection from the set of dualisable dynamical triples to the set of dualisabledual dynamical triples, Dyn † ( B ) → d Dyn † ( B ) . [ ρ , P , E ] [ ˆ ρ , b P , b E ] b G is first countable. n formulas (16),(18) and (20) we can replace everything by its dual counterpart, i.e.we interchange the rˆole of triples and dual triples, then we obtain a map d Dyn † ( B ) → Dyn † ( B ) . Moreover, these two maps are natural and inverse to each other, so we have an equiva-lence of functors
Dyn † ∼ = d Dyn † . Proof :
To see that the cocycles ˆ µ i define a decker we have to verify thatˆ µ i ( u )( χ , ˆ z ) = ˆ ζ ji ( u )( ˆ z + χ N ⊥ ) − ˆ µ j ( u )( χ , ˆ g ji ( u ) + ˆ z ) ˆ ζ ji ( u )( ˆ z ) .We just compute the right-hand side u (cid:13) ˆ ζ ji ( ˆ z + χ N ⊥ ) − ˆ µ j ( χ , ˆ g ji + ˆ z ) ˆ ζ ji ( ˆ z )= Ad (cid:16) ζ ji ( − ) − ( φ ji ( − σ ( ) , 0 ) ⊗ ) ( λ G / N ( g ji ) ⊗ )( h ˆ σ ( ˆ z + χ N ⊥ + ˆ g ji ) , σ ( ) − σ ( + g ji ) i ⊗ )( h χ , − σ ( ) i ⊗ )( h ˆ σ ( ˆ z + ˆ g ji ) , − σ ( ) + σ ( + g ji ) i ⊗ )( λ G / N ( − g ji ) ⊗ ) ζ ji ( − ) ( φ ji ( − σ ( ) , 0 ) − ⊗ ) (cid:17) = Ad (cid:16) ( h ˆ σ ( ˆ z + χ N ⊥ + ˆ g ji ) , σ ( − g ji ) − σ ( ) i ⊗ )( h χ , − σ ( − g ji ) i ⊗ )( h ˆ σ ( ˆ z + ˆ g ji ) , − σ ( − g ji ) + σ ( ) i ⊗ ) (cid:17) = Ad (cid:16) ( h χ , − σ ( ) ih ˆ σ ( ˆ z + ˆ g ji + χ N ⊥ ) − χ − ˆ σ ( ˆ z + ˆ g ji ) , σ ( − g ji ) − σ ( ) i ⊗ ) (cid:17) = Ad (cid:16) ( h χ , − σ ( ) i ⊗ ) h ˆ σ ( ˆ z + ˆ g ji + χ N ⊥ ) − χ − ˆ σ ( ˆ z + ˆ g ji ) , − σ ( g ji ) i | {z } (cid:17) = ˆ µ i ( χ , ˆ z ) . = : ( ˆ φ ji ( χ , ˆ z )) − ≡ mod U ( ) This establishes ˆ ρ . A careful look at this calculation shows that ˆ φ .. indeed isthe second term of the total dual cocycle ( ˆ ψ ... , ˆ φ .. , 1 ) defined by our constructeddual dynamical triple. It is easy to see that another choice of the section σ altersˆ µ i and ˆ ζ ji precisely in such a way that they define an isomorphic pair. So wehave established a map Dyn † ( B ) → d Dyn † ( B ) and, by replacing everything byits dual, a map in opposite direction.To prove the remaining assertions of the theorem, we shall apply our con-struction twice. We will find that the double dual ( ˆˆ ρ , bb P , bb E ) is isomorphic to ( Ad ◦ λ G ⊗ ρ , PU ( L ( G )) ⊗ P , E ) . In particular, we already see from the defini-tion of ˆ φ ji that the double dual bundle bb E has cocycle ˆˆ g ji ( u ) : = ˆ φ ji ( u )( , 0 ) | N ⊥ = ji ( u ) , so there is θ : bb E ∼ = E . We compute the double dual cocycleˆˆ ζ ji : U ij → Map ( G / N , PU ( L ( b G / N ⊥ ) ⊗ L ( G / N ) ⊗ H )) .By definition, we have by (18) and (15) u (cid:13) ˆˆ ζ ji ( z )= Ad (cid:16) ( ˆ κ ˆ σ ( g ji + z , − ˆ g ji ) ⊗ ⊗ )( λ b G / N ⊥ ( − ˆ g ji ) ⊗ ⊗ ) ˆ ζ ji ( − .. )( ˆ φ ji ( − ˆ σ ( .. ) , 0 ) − ⊗ ⊗ ) (cid:17) = Ad (cid:16) ( ˆ κ ˆ σ ( g ji + z , − ˆ g ji ) ⊗ ⊗ )( λ b G / N ⊥ ( − ˆ g ji ) ⊗ ⊗ )( κ σ ( − g ji , ˆ g ji − .. ) ⊗ )( ⊗ λ G / N ( − g ji ) ⊗ )( ⊗ ζ ji ( − ))( ⊗ φ ji ( − σ ( ) , 0 ) − ⊗ )( ˆ φ ji ( − ˆ σ ( .. ) , 0 ) − ⊗ ⊗ ) (cid:17) .With equation (17) and the definitions of κ σ , φ ji , . . . this reads. . . = Ad (cid:16) ( h ˆ σ ( .. + ˆ g ji ) − ˆ σ ( .. ) , σ ( g ji + z ) i ⊗ ⊗ )( λ b G / N ⊥ ( − ˆ g ji ) ⊗ ⊗ )( h ˆ σ ( ˆ g ji − .. ) , σ ( + g ji ) − σ ( ) i ⊗ )( ⊗ λ G / N ( − g ji ) ⊗ )( ⊗ µ j ( − σ ( + z ) , g ji + z )) (cid:17) ( ⊗ ⊗ ζ ji ( z )) Ad (cid:16) ( ⊗ µ i ( − σ ( + z ) , z ))( ⊗ h ˆ σ ( ˆ g ji ) , σ ( + z ) − σ ( ) i ⊗ )( h ˆ σ ( ˆ g ji − .. ) + ˆ σ ( .. ) − ˆ σ ( ˆ g ji , − σ ( g ji + z ) + σ ( z ) i ⊗ ⊗ ) (cid:17) ,and after some intermediate steps we findˆˆ ζ ji ( u )( z ) = η j ( u )( g ji ( u ) + z ) − Ad (cid:16) λ b G / N ⊥ ( − ˆ g ji ( u )) (cid:17) ⊗ ⊗ ζ ji ( u )( z ) η i ( u )( z ) ,wherein η i ( u )( z ) : = Ad (cid:16) ( h ˆ σ ( .. ) + ˆ σ ( − .. ) , σ ( z ) ih ˆ σ ( − .. ) , σ ( − z ) − σ ( ) i ⊗ ⊗ )( ⊗ µ i ( u )( − σ ( ) , z ) − )( ⊗ λ G / N ( z ) ⊗ ) (cid:17) .This already proves part of the second half of the theorem, for we see that thedouble dual is isomorphic to a pair with transition functions ( u , z ) Ad ( λ b G / N ⊥ ( − ˆ g ji ( u ))) ⊗ ⊗ ζ ji ( u )( z ) .For general reasons such a pair is isomorphic to a pair with transition function ζ ji , for ( u , z ) λ b G / N ⊥ ( − ˆ g ji ( u )) ⊗ ⊗ ∈ U ( L ( b G / N ⊥ ) ⊗ L ( G / N ) ⊗ H )
45s a unitary cocycle. However, we can give a concrete isomorphism. Let usdenote by F : L ( b G / N ⊥ ) → L ( N ) the Fourier transform, then we obtain amultiplication operator F ◦ λ b G / N ⊥ ( − ˆ g ji ( u )) ◦ F − = h ˆ g ji , . i − = φ ji ( u )( . , 0 ) | N ∈ U ( L ( N )) .Thus, by definition of φ ji we haveAd ( λ b G / N ⊥ ( − ˆ g ji ( u ))) ⊗ ⊗ ζ ji ( u )( z )= Ad (cid:16) ( F − ⊗ ⊗ ) ◦ µ j ( u )( . , g ji ( u ) + z ) − | N ◦ ( F ⊗ ⊗ ) (cid:17) ( ⊗ ⊗ ζ ji ( u )( z )) Ad (cid:16) ( F − ⊗ ⊗ ) ◦ µ i ( u )( . , z ) | N ◦ ( F ⊗ ⊗ ) (cid:17) ,and we let ϑ i ( u )( z ) : = Ad (cid:16) ( F − ⊗ ⊗ ) ◦ µ i ( u )( . , z ) | N ◦ ( F ⊗ ⊗ ) (cid:17) η i ( u )( z ) .Due to Lemma 3.1 ( i ) ϑ i is continuous. We obtainˆˆ ζ ji ( u )( z ) = ϑ j ( u )( g ji ( u ) + z ) − (cid:0) ⊗ ⊗ ζ ji ( u )( z ) (cid:1) ϑ i ( u )( z ) ,so ϑ i define local isomorphisms between bb P and PU ( L ( b G / N ⊥ ) ⊗ L ( G / N )) ⊗ P which fit together to a global isomorphism ( , ϑ , θ ) : ( bb P , bb E ) → ( PU ( L ( b G / N ⊥ ) ⊗ L ( G / N )) ⊗ P , E ) .We have to compute the behaviour of the double dual decker under this iso-morphism, i.e. we have to compute µ ′ i ( u )( g , z ) : = ϑ i ( u )( z + gN ) ˆˆ µ i ( u )( g , z )) ϑ i ( u )( z ) − ,for ˆˆ µ i ( u )( g , z ) = Ad ( h ˆ σ ( .. ) , − g i ⊗ ⊗ ) . This yields u (cid:13) µ ′ i ( g , z ) = Ad (cid:16) ( F − ⊗ ⊗ ) ◦ µ i ( . , z + gN ) | N ◦ ( F ⊗ ⊗ )( ⊗ λ G / N ( gN ) ⊗ )( h ˆ σ ( .. ) , − g − σ ( ) + σ ( + gN ) i ⊗ )( ⊗ µ i ( σ ( + gN ) − σ ( ) , z ))( F − ⊗ ⊗ ) ◦ µ i ( . , z ) − | N ◦ ( F ⊗ ⊗ ) (cid:17) = Ad (cid:16) ( F − ⊗ ⊗ ) ◦ µ i ( . , z + gN ) | N ( ⊗ λ G / N ( gN ) ⊗ )( λ N ( g + σ ( ) − σ ( + gN )) ⊗ )( ⊗ µ i ( σ ( + gN ) − σ ( ) , z )) µ i ( . , z ) − | N ◦ ( F ⊗ ⊗ ) (cid:17) = Ad (cid:16) ( F − ⊗ ⊗ ) ◦ ( ⊗ λ G / N ( gN ) ⊗ )( λ N ( g + σ ( ) − σ ( + gN )) ⊗ ) ◦ ( F ⊗ ⊗ )( ⊗ ⊗ µ i ( g , z )) (cid:17) . 46he last equality is just the cocycle condition for µ i . We can make a furthermanipulation by use of the following isomorphism S : L ( N ) ⊗ L ( G / N ) ∼ = L ( N × G / N ) → L ( G ) .We let ( S ( f ))( g ) : = f ( g − σ ( gN ) , gN ) . Its inverse is given by ( S − ( f ))( n , z ) : = f ( n + σ ( z )) , and it is immediate to verify that S − ◦ λ G ( g ) ◦ S = λ G / N ( gN ) λ N ( g + σ ( ) − σ ( + gN )) .This implies for the cocycle that µ ′ i ( u )( g , z ) = Ad (cid:16) ( F − ⊗ ⊗ ) ◦ ( S ⊗ ) ◦ ( λ G ( g ) ⊗ µ i ( u )( g , z )) ◦ ( S ⊗ ) ◦ ( F ⊗ ⊗ ) (cid:17) To summarise, we have shown that there is an isomorphism of dynamicaltriples (cid:0) ( S ⊗ ) ◦ ( F ⊗ ⊗ ) , ϑ , θ (cid:1) : ( ˆˆ ρ , bb P , bb E ) → (( Ad ◦ λ G ) ⊗ ρ , PU ( L ( G )) ⊗ P , E ) .So, as we discussed in Example 2.2, the double dual triple is outer conjugate tothe triple we started with.We finally comment on the naturality of the defined maps. Let f : B ′ → B be a map of bases spaces. Then we have a diagramDyn † ( B ) / / f ∗ (cid:15) (cid:15) d Dyn † ( B ) f ∗ (cid:15) (cid:15) Dyn † ( B ′ ) / / d Dyn † ( B ′ ) .On the local level pullback with f ∗ is the purely formal substitution of u ∈ B by f ( u ′ ) for u ′ ∈ B ′ in all formulas, and it follows that the diagram commutes.This proves the theorem. (cid:4) So far we have found that dualisable dynamical triples and dualisable dual dy-namical triples have the same isomorphism classes. We demonstrate that thetheory developed so far is intimately connected to topological T-duality.Let ( ρ , P , E ) be dualisable and b P , b E as above. Let P top : = PU ( L ( G / N )) ⊗ P . Theorem 3.3
The two pairs ( P top , E ) and ( b P , b E ) with underlying Hilbert space b H = L ( G / N ) ⊗ H span a topological triple ( κ top , ( P top , E ) , ( b P , b E )) , and we obtaina map τ ( B ) : Dyn † ( B ) → Top ( B ) .47 oreover, this map is natural, i.e. it defines a natural transformation of functors τ : Dyn † → Top.
Proof :
We have to show that there exists (a natural choice of) an isomorphism κ top : E × B b P → P top × B b E which satisfies the Poincar´e condition. We are goingto define κ top by local isomorphisms of the locally trivialised pairs. Let κ top i : U i → Map ( G / N × b G / N ⊥ , PU ( L ( G / N ) ⊗ H )) be given by the formula κ top i ( u )( z , ˆ z ) : = Ad (cid:16) ( κ σ ( z , ˆ z ) ⊗ ) µ i ( u )( − σ ( ) , z ) − ( λ G / N ( z ) ⊗ ) | {z } (cid:17) = : κ top i ( u )( z , ˆ z ) (21) κ top i is continuous, for the first factor is exactly κ σ of Lemma 2.3 which definesthe Poincar´e class, thus it is continuous. The left regular representation λ G / N is continuous and by Lemma 3.1 ( iii ) ( u , z ) Ad ( µ i ( u )( − σ ( ) , z )) is contin-uous. The functions ( u , z , ˆ z ) ⊗ ζ ji ( u )( z ) and ( u , z , ˆ z ) ⊗ ˆ ζ ji ( u )( ˆ z ) aretransition functions for the bundles P dyn × B b E and E × B b P , and the functions κ top i will define a global isomorphism if and only if κ top j ( u , g ji ( u ) + z , ˆ g ji ( u ) + ˆ z ) ˆ ζ ji ( u )( ˆ z ) κ top i ( u , z , ˆ z ) − = ⊗ ζ ji ( u )( z ) ,for ( u , z , ˆ z ) ∈ U ij × G / N × b G / N ⊥ . By construction, the Poincar´e condition willbe satisfied automatically.We are able to do this calculation on the projective unitary level directly, butlater we need the same calculation on the unitary level, so we prepare ourselvesfirst with a choice of lifts ˆ ζ ji .Recall that u Ad ( φ ji ( u )( − σ ( ) , 0 )) is continuous. If we pass once moreto a refined atlas which again we denote by { U i } , then there exist continuouslifts φ ji : U ji → L ∞ ( G / N , U ( )) such that Ad ( φ ji ( u )) = Ad ( φ ji ( u )( − σ ( ) , 0 )) ; by those we define lifted transi-tion functions functions ˆ ζ ji : U ji → L ∞ ( b G / N ⊥ , L ∞ ( G / N , U ( H ))) byˆ ζ ji ( u )( ˆ z ) : = ( κ σ ( − g ji ( u ) , ˆ g ji ( u ) + ˆ z ) ⊗ H ) ( λ G / N ( − g ji ( u )) ⊗ H ) ζ ji ( u )( − ) ( φ ji ( u ) − ⊗ H ) , (22)for u ∈ U ij , ˆ z ∈ b G / N ⊥ . 48ow, we calculate u (cid:13) κ top j ( g ji + z , ˆ g ji + ˆ z ) ˆ ζ ji ( ˆ z ) κ top i ( z , ˆ z ) − = ( h ˆ σ ( ˆ g ji + ˆ z ) , − σ ( ) + σ ( − g ji − z ) i ⊗ ) µ j ( − σ ( ) , g ji + z ) − ( λ G / N ( g ji + z ) ⊗ )( h ˆ σ ( ˆ g ji + ˆ z ) , − σ ( ) + σ ( + g ji ) i ⊗ ) ( λ G / N ( − g ji ) ⊗ ) ζ ji ( − ) ( φ − ji ⊗ )( λ G / N ( − z ) ⊗ ) µ i ( − σ ( ) , z ) ( h ˆ σ ( ˆ z ) , σ ( ) − σ ( − z ) i ⊗ )= ( h ˆ σ ( ˆ g ji + ˆ z ) − ˆ σ ( ˆ z ) , − σ ( ) + σ ( − z ) i ⊗ ) µ j ( − σ ( ) , g ji + z ) − ζ ji ( z − ) ( φ ji ( − z ) − ⊗ ) µ i ( − σ ( ) , z )= ( h ˆ σ ( ˆ g ji + ˆ z ) − ˆ σ ( ˆ z ) , − σ ( ) + σ ( − z ) + σ ( z ) i ⊗ ) h ˆ σ ( ˆ g ji + ˆ z ) − ˆ σ ( ˆ z ) , − σ ( z ) i µ j ( − σ ( ))( u , g ji + z ) − ζ ji ( z − ) ( φ ji ( − z ) − ⊗ ) µ i ( − σ ( ) , z )= ( h ˆ g ji , − σ ( ) + σ ( − z ) + σ ( z ) i ⊗ ) h ˆ σ ( ˆ g ji + ˆ z ) − ˆ σ ( ˆ z ) , − σ ( z ) i µ j ( − σ ( ) , g ji + z ) − ζ ji ( z − ) ( φ ji ( − z ) − ⊗ ) µ i ( − σ ( ) , z ) .The variable ˆ z only occurs in h ˆ σ ( ˆ g ji + ˆ z ) − ˆ σ ( ˆ z ) , − σ ( z ) i . To the remaining ex-pressinon we can apply equation (17) of Lemma 3.1 which is valid on the uni-tary level up to a U ( ) -valued perturbation α ′ ji : U ji → L ∞ ( G / N , U ( )) . Weobtain. . . = h ˆ σ ( ˆ g ji ( u ) + ˆ z ) − ˆ σ ( ˆ z ) , − σ ( z ) i α ′ ji ( u )( z ) − | {z } ( ⊗ ζ ji ( u )( z )) . (23) = : α ji ( u )( z , ˆ z ) − ∈ U ( ) This establishes the existence of κ top .It remains to check that the constructed equivalence class of the topolog-ical triple only depends on the equivalence class of the dynamical triple, butthis not difficult to establish. We just mention that if we start with an exteriorequivalent decker which differs locally by c i : U i × G × G / N → U ( H ) fromthe cocycle µ i , then κ top differs by the null-homotopic bundle automorphismwhich is locally of the form c i ( u , − σ ( + z ) , z ) .Finally, we remark that τ is natural. Indeed, let f : B ′ → B be a map ofbases spaces. Then there is a commutative diagramDyn † ( B ) τ ( B ) / / f ∗ (cid:15) (cid:15) Top ( B ) f ∗ (cid:15) (cid:15) Dyn † ( B ′ ) τ ( B ′ ) / / Top ( B ′ ) ,because in our local construction pullback with f is just pullback of the under-lying locally defined functions by f which corresponds to the formal substitu-tion of u ∈ B by f ( u ′ ) for u ′ ∈ B ′ in all formulas. (cid:4) τ ( B ) : Dyn † ( B ) → Top ( B ) . In partic-ular, we wish to construct from the data of a topological triple P × B b E $ $ IIIIIIIII } } zzzzzzzzz E × B b P z z uuuuuuuuu ! ! DDDDDDDDD κ o o P " " DDDDDDDDDD E × B b E $ $ JJJJJJJJJJ z z tttttttttt b P | | zzzzzzzzzz E % % KKKKKKKKKKKK b E y y ssssssssssss B a decker ρ on (a pair stably isomorphic to) ( P , E ) . The space E × B b P clearly hasa G / N -action, and by use of κ we can define a G / N -action on P × B b E . The lead-ing idea is that the Poincar´e condition enables us to push forward this actiondown to a G -action on P . We will see that this fails in general as an obstructionagainst a decker occurs.Let ( κ , ( P , E ) , ( b P , b E )) be a topological triple over B , and let g ij , ζ ij and ˆ g ij , ˆ ζ ij be transition functions for the pairs ( P , E ) and ( b P , b E ) . Without restriction welet b H = L ( G / N ) ⊗ H be the underlying Hilbert space. The existence of κ im-plies the existence of locally defined κ i : U i → Map ( G / N × b G / N ⊥ , PU ( b H )) such that κ j ( u )( g ji ( u ) + z , ˆ g ji ( u ) + ˆ z ) ˆ ζ ji ( u )( ˆ z ) κ i ( u )( z , ˆ z ) − = ζ ji ( u )( z ) .Since κ satisfies the Poincar´e condition each κ i can be written in the form κ i ( u )( z , ˆ z ) = κ ai ( u )( z ) κ σ ( z , ˆ z ) Ad ( v i ( u , z , ˆ z )) κ bi ( u )( ˆ z ) ,wherein κ σ is from Lemma 2.3 and κ ai , κ bi are some continuous projective uni-tary functions and v i : U i × G / N × b G / N ⊥ → U ( b H ) is a continuous unitaryfunction. We introduce some short hands to get rid of κ ai and κ bi . We let ζ ′ ji ( u )( z ) : = κ aj ( u )( g ji ( u ) + z ) − ζ ji ( u )( z ) κ ai ( u )( z ) and analogously ˆ ζ ′ ji ( u )( ˆ z ) : = κ bj ( u )( g ji ( u ) + z ) ˆ ζ ji ( u )( z ) κ bi ( u )( z ) − . Thus we have u (cid:13) ζ ′ ji ( z ) = κ σ ( g ji + z , ˆ g ji + ˆ z ) Ad ( v j ( g ji + z , ˆ g ji + ˆ z )) ˆ ζ ′ ji ( ˆ z ) Ad ( v i ( z , ˆ z )) − κ σ ( z , ˆ z ) − . (24)Let us choose, if necessary after a refinement of the given atlas, lifts ζ ′ ji : U ij → L ∞ ( G / N , U ( b H )) and ˆ ζ ′ ji : U ij → L ∞ ( b G / N ⊥ , U ( b H )) of the transition functions50Lemma A.8). Then we obtain u (cid:13) ζ ′ ji ( z ) = ( κ σ ( g ji + z , ˆ g ji + ˆ z ) ⊗ ) v j ( u , g ji + z , ˆ g ji + ˆ z ) ˆ ζ ′ ji ( ˆ z ) v i ( z , ˆ z ) − ( κ σ ( z , ˆ z ) − ⊗ ) α ji ( z , ˆ z ) , (25)for some continuous α ji : U ij → L ∞ ( G / N × b G / N ⊥ , U ( )) and κ σ ( z , ˆ z ) : = h ˆ σ ( ˆ z ) , σ ( − z ) − σ ( ) i ∈ L ∞ ( G / N , U ( )) ⊂ U ( b H ) ; σ and ˆ σ are both chosento be Borel. By direct computation, it follows that ( δ g × ˆ g α .. ) kji ( u )( z , ˆ z ) = ψ kji ( u )( z ) ˆ ψ kji ( u )( ˆ z ) − , (26)for the twisted ˇ C ech cocycles ψ ... · : = δ g ζ ′ .. and ˆ ψ ... · : = δ ˆ g ˆ ζ ′ .. . It is ourtask now to use the Poincar´e condition of the isomorphism κ to deduce a moreconcrete formula for α ji from which we can extract the existence of a decker. Itwill be sufficient to investigate the structure of A ji ( u , z ) : = Ad (cid:0) α ji ( u )( z , .. ) (cid:1) ∈ P L ∞ ( b G / N ⊥ , U ( )) .We let β ji ( u , z ) : = Ad ( h ˆ σ ( .. + ˆ g ji ( u )) − ˆ σ ( .. ) , σ ( g ji ( u ) + z ) i ) ∈ P L ∞ ( b G / N ⊥ , U ( )) Note for the next lemma that β ji ( u , z ) equals ˆ κ ˆ σ ( g ji ( u ) + z , ˆ g ji ( u )) − up to con-jugation by the unitary implemented Ad ( λ b G / N ⊥ ( ˆ g ji ( u ))) , so the ˇCech classes [ β ji ] and [( u , z ) ˆ κ ˆ σ ( g ji ( u ) + z , ˆ g ji ( u )) − ] in ˇ H ( U ji × G / N , U ( )) defined bythese functions agree. (Cp. Lemma 2.4)The Hilbert space H occurs in the following to force that certain operatorswhich are defined on different tensor factors of H ⊗ L ( b G / N ⊥ ) ⊗ b H commute. Lemma 3.2 A ji is continuous, and there are continuous functions w ji : U ji → Map ( G / N , L ∞ ( b G / N ⊥ , U ab ( H )))) ⊂ Map ( G / N , U ( H ⊗ L ( b G / N ⊥ ))) and γ ji : U ji → PU ab ( H ) such that H ⊗ A ji ( u , z ) = H ⊗ β ji ( u , z ) Ad ( w ji ( u )( z )) ( γ ji ( u ) ⊗ L ( b G / N ⊥ ) ) . U ab ( H ) is a commutative, contractible subgroup of U ( H ) such that U ( ) · ⊂ U ab ( H ) . Seesection A.3, page 92. roof : Equation (25) implies ⊗ ζ ′ ji ( u )( z ) = Ad (cid:0) L ( b G / N ⊥ ) ⊗ ζ ji ( u )( z ) (cid:1) = Ad (cid:16) ( κ σ ( g ji ( u ) + z , ˆ g ji ( u ) + .. ) ⊗ H ) v j ( u , g ji ( u ) + z , ˆ g ji ( u ) + .. ) ˆ ζ ′ ji ( u )( .. ) v i ( u , z , .. )) − ( κ σ ( z , .. ) − ⊗ H ) (cid:17) ( A ji ( u )( z ) ⊗ c H ) ∈ PU ( L ( b G / N ⊥ ) ⊗ b H )) .The terms inside the bracket are continuous unitary functions on U ji × G / N .Thus we have equality of the ˇCech classes [ ζ ′ ji ] = [ ⊗ ζ ′ ji ] = [ A ji ⊗ ] = [ A ji ] ∈ ˇ H ( U ji × G / N , U ( )) . We compute the ˇCech class of ⊗ ζ ′ ji by equation (24)for ˆ z =
0. Then κ σ ( z , 0 ) = , hence has trivial ˇCech class. The map ( u , z ) ˆ ζ ′ ji ( u , 0 ) has ˇCech class [( u , z ) ˆ ζ ′ ji ( u , 0 )] . By Remark A.2, let γ ji : U ji → PU ab ( H ) be such that [( u , z ) ˆ ζ ′ ji ( u , 0 )] = [( u , z ) γ ji ( u )] = pr ∗ U ji [ γ ji ] . Then [ ⊗ ζ ′ ji ] = [( u , z ) κ σ ( g ji ( u ) + z , ˆ g ji ( u ))] + [ u ˆ ζ ′ ji ( u , 0 )]= (cid:0) ( g ji ◦ pr U ji + pr G / N ) × ( ˆ g ji ◦ pr U ji ) (cid:1) ∗ π + pr ∗ U ji [ γ ji ]= [ β ji ] + pr ∗ U ji [ γ ji ] .This implies that A ji ⊗ H equals β ji ⊗ γ ji up to Ad of a continuous unitaryfunction w ji : U ji × G / N → U ( L ( b G / N ⊥ , H )) . w ji takes values in the sub-group L ∞ ( b G / N ⊥ , U ab ( b H )) only, since A ji , β ji and γ ji are maps into the sub-group P L ∞ ( b G / N ⊥ , U ab ( b H )) ⊂ PU ( L ( b G / N ⊥ , b H )) .Finally, we just interchange the order of the Hilbert spaces in the tensorproduct L ( b G / N ⊥ ) ⊗ H to H ⊗ L ( b G / N ⊥ ) . This will be convenient later. (cid:4) Remark 3.2
By passing to a refined atlas, we can always achieve that γ ji = , forwe can apply Lemma A.8 to the functions U ji ∋ u ˆ ζ ′ ji ( u )( ) . In that case we havew ji ( u )( z )( ˆ z ) ∈ U ( ) . It is important to note that although w ji depends on the choice of γ ji theterm ( d ∗ w .. ) ji ( u ) = d ( w ji ( u )) does not; here d is the (first) boundary operatorof group cohomologyMap ( G / N , L ∞ ( b G / N ⊥ , U ab ( H ))) d −→ Map ( G , Map ( G / N , L ∞ ( b G / N ⊥ , U ab ( H )))) w (cid:0) ( h , z ) w ( z + hN )( .. ) w ( z )( .. ) − (cid:1) The sheaf of continuous functions into the range (or source) of d is a G / N × b G / N ⊥ -module in the obvious way. So the following statement is meaningful for thetwisted ˇCech boundary operator δ g .. × ˆ g .. . Lemma 3.3 Ad (cid:16) ( δ g .. × ˆ g .. d ∗ w .. ) kji ( u )( h , z )( .. ) (cid:17) = , for all u ∈ U kji , z ∈ G / N , h ∈ G . 52 roof : Since d ∗ and δ g .. × ˆ g .. commute, we have to show that the expressionAd (( δ g .. × ˆ g .. w .. ) kji ( u )( z )( .. )) is independent of z . Because Ad (( δ g × ˆ g α .. ) kji ( u )( z , .. )) = Ad ( ˆ ψ kji ( u )( .. ) − ) and γ ji ( u ) are independent of z we have to compute δ g .. × ˆ g .. β .. .This yields u (cid:13) ( δ g .. × ˆ g .. β .. ) kji ( z )= Ad (cid:0) h ˆ σ ( .. + ˆ g ji ) − ˆ σ ( .. ) , − σ ( g ji + z ) ih ˆ σ ( .. + ˆ g ki ) − ˆ σ ( .. ) , σ ( g ki + z ) ih ˆ σ ( .. + ˆ g ji + ˆ g kj ) − ˆ σ ( .. + ˆ g ji ) , − σ ( g ki + z ) i (cid:1) = Ad ( h ˆ σ ( .. + ˆ g ji ) − ˆ σ ( .. ) , σ ( g ki ) − σ ( g ji ) i ) ,and the lemma is proven. (cid:4) It follows that ϕ kji ( u )( h , z ) : = ( δ g .. × ˆ g .. d ∗ w .. ) kji ( u )( h , z )( .. ) ∈ U ( ) (27)satisfies δ g ϕ = d ∗ ϕ ji =
1, i.e. it defines a twisted ˇCech 2-class [ ϕ ... ] ∈ ˇ H ( B , Z ( G , Map ( G / N , U ( ))) , g .. ) . (28) Lemma 3.4
The construction of the class [ ϕ ... ] defines a natural map Top ( E , B ) → ˇ H ( B , Z ( G , Map ( G / N , U ( ))) , g .. ) . Proof :
We must show that the constructed class only depends on the isomor-phism class of the triple. Let us check all choices in reversed order of theirappearance.Firstly, the choice of w ji is only unique up to a continuous scalar function U ji × G / N → U ( ) , so ϕ ... = δ g × ˆ g d ∗ w .. changes by a boundary.Secondly, the choice of γ ji , is only unique up to a unitary implementedfunction U ji → U ab ( H ) , so w ji may change by this function. But this functionis independent of z ∈ G / N , so d ∗ w ji does not change.Thirdly, if we choose different lifts of the transition functions ζ ji , this wouldchange α ji by a function U ji → L ∞ ( G / N , U ( )) , so A ji does not change.Fourthly, if we choose different lifts of the transition functions ˆ ζ ji , then α ji is changed by a function U ji → L ∞ ( b G / N ⊥ , U ( )) , so A ji , thus w ji , changes bya function independent of z ∈ G / N , so d ∗ w ji does not change.Fifthly, if we choose another atlas for our construction, we can take a com-mon refinement and the normalisation procedure ζ ji ζ ′ ji , ˆ ζ ji ˆ ζ ′ ji , leads tothe same equations as above, except we restricted us to the refined atlas, so theclass of ϕ ... is not changed.Sixthly, if we start with a topological triple which is isomorphic to ( κ , ( P , E ) , ( b P , b E ) ,then, locally, the isomorphisms of the underlying pairs have the same effectas a change of the atlas which does not change the class of ϕ ... . However,53e must take care of the homotopy commutativity of diagram (14). So if κ ′ is homotopic to κ , then they differ locally by a continuous unitary function v ′ i : U i × G / N × b G / N ⊥ → U ( b H ) , and equation (25) becomes v ′ j ( u , g ji ( u ) + z , ˆ g ji ( u ) + ˆ z ) − ζ ′ ji ( u )( z ) v ′ i ( u , z , ˆ z )= ( κ σ ( g ji ( u ) + z , ˆ g ji ( u ) + ˆ z ) ⊗ ) v j ( u , g ji ( u ) + z , ˆ g ji ( u ) + ˆ z ) ˆ ζ ′ ji ( u )( ˆ z ) v i ( u , z , ˆ z ) − ( κ σ ( z , ˆ z ) − ⊗ ) α ′ ji ( u )( z , ˆ z ) .We must investigate how α ′ ji is related to α ji . The family v ′ i defines the bundleautomorphism κ ′ ◦ κ − on P × B b E so ζ ′ ji ( u )( z ) (cid:0) v ′ i ( u , g ji ( u ) + z , ˆ g ji ( u ) + ˆ z ) (cid:1) = v ′ j ( u , z , ˆ z ) α ′′ ji ( u )( z , ˆ z ) , (29)for a scalar α ′′ ji : U i × G / N × b G / N ⊥ → U ( ) . It follows from the three equa-tions for α ji , α ′ ji and α ′′ ji that α ′ ji = α ji α ′′ ji . So w ji changes by α ′′ ji . But α ′′ .. is acocycle δ g × ˆ g α ′′ .. = δ g × ˆ g w .. and ϕ ... do not change.Finally, we just remark that the defined map is natural with respect to pull-back. I.e. if f : B ′ → B is a map of base spaces, then there is a commutativediagram Top ( E , B ) / / f ∗ (cid:15) (cid:15) ˇ H ( B , Z ( G , Map ( G / N , U ( ))) , g .. ) f ∗ (cid:15) (cid:15) Top ( f ∗ E , B ′ ) / / ˇ H ( B ′ , Z ( G , Map ( G / N , U ( ))) , f ∗ g .. ) .This proves the lemma. (cid:4) The quotient map G → G / N induces a map on the twisted ˇCech groupsˇ H ( B , Z ( G / N , Map ( G / N , U ( ))) , g .. ) (cid:15) (cid:15) ˇ H ( B , Z ( G , Map ( G / N , U ( ))) , g .. ) ,and the map of Lemma 3.4 has an obvious factorisationTop ( E , B ) Lemma + + WWWWWWWWWWWWWWWWWWWWWWW / / ˇ H ( B , Z ( G / N , Map ( G / N , U ( ))) , g .. ) (cid:15) (cid:15) ˇ H ( B , Z ( G , Map ( G / N , U ( ))) , g .. ) ,54ince the definition (27) of ϕ ... implies ϕ kji ( u )( h + n , z ) = ϕ kji ( u )( h , z ) , (30)for all n ∈ N . Definition 3.2
A topological T-duality triple over B is called almost strict if its ˇC ech class [ ϕ ... ] defined by Lemma 3.4 vanishes. The triple is called strict if its ˇC echclass defined by the horizontal map in the diagram above vanishes already. We denote by Top as ( B ) respectively Top s ( B ) the set of all almost strict re-spectively strict topological triples over B ; so we have obvious inclusionsTop s ( B ) ⊂ Top as ( B ) ⊂ Top ( B ) . Remark 3.3
For the class of a topological triple [( κ , ( P , E ) , ( b P , b E ))] ∈ Top ( B ) itsclass in Top ( E , B ) is only well-defined up to the action of Aut B ( E ) on Top ( E , B ) .However, the the vanishing of the obstruction class in Definition 3.2 is independent ofthe possible choices, so Top as ( B ) and Top s ( B ) are well-defined. We will see that strict and almost strict play a major r ˆole in our theory. Thefollowing two lemmata give a first feeling.
Lemma 3.5
The image of the map τ ( B ) from dualisable dynamical to topologicaltriples is contained in the set of strict topological triples, im ( τ ( B )) ⊂ Top s ( B ) . Proof :
In equation (23) we already computed α ji for a topological triple whichis constructed out of a dynamical one. (In (23) we did not normalise ζ ji ζ ′ ji , but this does not change α ji .) The result is α ji ( u )( z , ˆ z ) = h ˆ σ ( ˆ g ji ( u ) + ˆ z ) − ˆ σ ( ˆ z ) , σ ( z ) i α ′ ji ( u )( z ) . So in this case we have A ji ( u )( z ) = Ad ( h ˆ σ ( ˆ g ji ( u ) + .. ) − ˆ σ ( .. ) , σ ( z ) i )= Ad ( h ˆ σ ( ˆ g ji ( u ) + .. ) − ˆ σ ( .. ) , σ ( g ji ( u ) + z ) − σ ( g ji ( u ) i )= β ji ( u , z ) Ad ( h ˆ σ ( ˆ g ji ( u ) + .. ) − ˆ σ ( .. ) , − σ ( g ji ( u ) i ) | {z } ,independent of z ∈ G / N i.e. we can choose w ji such that w ji ( u )( z ) does not depend on z . Of course, thechoice of w ji is only determined up to a scalar U ji × G / N → U ( ) , but in anycase χ ji · : = d ∗ w ji defines a function U ji → Z ( G , Map ( G / N , U ( ))) , so byconstruction ϕ kji = ( δ g χ .. ) kji is a boundary. (cid:4) The next (technical) lemma will be the crucial point in the construction of adecker from the data of a topological triple.55 emma 3.6
Assume ( κ , ( P , E ) , ( b P , b E )) is almost strict. Then we can find a (suf-ficiently refined) atlas { U i | i ∈ I } such that for w ji from above there exists a familym i : U i → Z ( G , Map ( G / N , L ∞ ( b G / N ⊥ , U ab ( H )))) such that Ad ( d ∗ w ji ( u )( h , z )( .. )) = Ad (( δ g .. × ˆ g .. m . ) ji ( u )( h , z )( .. )) (31) ∈ P L ∞ ( b G / N ⊥ , U ab ( H )) Proof :
The proof uses a standard Zorn’s lemma argument as it can be foundin [Di].First we note that the space Z : = Z ( G , Map ( G / N , L ∞ ( b G / N ⊥ , U ab ( H )))) is contractible. In fact, if H : [
0, 1 ] × U ab ( H ) → U ab ( H ) is a contraction witheach H ( t , ) a group homomorphism, then the push-forward H : [
0, 1 ] × Z → Z preserves the cocycle relation and is a contraction.We can assume that the atlas is sufficiently refined such that, firstly, ϕ ... from(27) is a boundary, i.e. there exist χ ji : U ji → Z ( G , Map ( G / N , U ( ))) suchthat ( δ g χ ) kji = ϕ kji , and secondly, as B is a paracompact Hausdorff space, eventhe closed cover S i ∈ I U i = B is locally finite and that w ji , χ ji are well-definedon U ji , for all j , i ∈ I . We let M : = n ( J , m . ) | J ⊂ I , for all j ∈ J : m j : U j → Z such thatfor all i , j ∈ J and all u ∈ U ji , z ∈ G / N we have m i ( u )( h , z )( .. ) = d ∗ w ji ( u )( h )( z )( .. ) − χ ji ( u )( h , z ) m j ( u )( h , g ji ( u ) + z )( ˆ g ji ( u ) + .. ) o (32)For each i ∈ I , ( { i } , { } ) ∈ M , because we can assume that w ii = , χ ii = M is non-empty. We define a partial order on M by ( J , m . ) ≤ ( J ′ , m . ) ifand only if J ⊂ J ′ ⊂ I and m j = m ′ j , for all j ∈ J . For each chain in M the union of the index and cocycle sets is an upper bound, hence by Zorn’slemma there exists a maximal element ( J , m . ) . Assume J = I , so there is some a ∈ I \ J . Let R : = S j ∈ J ( U j ∩ U a ) ⊂ U a . For u ∈ R we define ˜ m a ( u )( h , z )( .. ) : = d ∗ w ja ( u )( h , z )( .. ) − m j ( u )( h , g ja ( u ) + z )( ˆ g ja ( u ) + .. ) χ ja ( u )( h , z ) , if u ∈ U j . Dueto δ g χ = δ g × ˆ g d ∗ w , this definition is independent of j ∈ J . We end up with adiagram R ˜ m a / / ∩ (cid:15) (cid:15) ZU a m a ? ? .Since our cover is locally finite, R is closed, but Z is contractible, therefore anextension m a exists [DD, Lem. 4]. This contradicts the maximality of ( J , m . ) , so J = I . Finally, equation (31) holds, since χ ji ( u )( h , z ) ∈ U ( ) . (cid:4) The constructed family { m i } is the last ingredient to write down an explicitformula for a decker ρ dyn on (a stabilisation of) the pair ( P , E ) . It is not hard to56uess that the cocycles m i will be an essential part of the cocycles µ dyn i whichwe will define to implement the decker ρ dyn locally. But unfortunately theconstructed family { m i } is by no means unique, and this non-uniqueness isthe origin of the following discussion. As a matter of fact this discussion willsimplify drastically when we consider the special case of G = R n with lattice N = Z n in the next section below. However, now we continue with the discus-sion of almost strict triples from above and work out the general framework.Let ϕ ... be the twisted ˇCech cocycle from equation (27). The triple under consid-eration is assumed to be almost strict, so (after refining the atlas U • ) we have ϕ kji = ( δ g χ .. ) kji for a chain χ .. ∈ ˇ C ( U • , Z ( G , Map ( G / N , U ( )) , g .. ) as inthe proof above. Obviously, this chain χ .. is only well defined up to a cocycle χ ∈ ˇ Z ( U • , Z ( G , Map ( G / N , U ( )) , g .. ) .A choice of χ .. determines the family { m i } in equation (32) not completelybut up to a family n i : U i → Z ( G , Map ( G / N , L ∞ ( b G / N ⊥ , U ab ( H )))) whichsatisfies ( δ g × ˆ g n . ) ji = . We already mentioned that m i will be part of the co-cycles µ dyn i . Then it will turn out that the two families { m i } and { m i n i } de-fine exterior equivalent deckers. However, this is not the case when make weanother choice of of χ .. , say χ .. χ for χ as above. Then { m i } must be re-placed by { m i m i } for m i being such that ( δ g × ˆ g m ) ji = χ ji · , and we will findthat the corresponding deckers are exterior equivalent if and only if the class [ χ ] ∈ ˇ H ( B , Z ( G / N , Map ( G / N , U ( ))) , g .. ) vanishes. In other words, for eachclass [ χ ] we obtain a different class of dynamical triples. It is then the obviousquestion, whether the different dynamical triples still have something in com-mon. Or if x is the class of almost strict topological triple and x dyn is the classof one of the possible dynamical triples indicated, is there any relation between x and τ ( B )( x dyn ) ? Can we describe the difference, in particuar, when do theyequal? A partial answer of this question is already given in Lemma 3.5 whichshows that x and τ ( B )( x dyn ) only can equal if x is strict.57onsider the short exact sequences0 (cid:15) (cid:15) (cid:15) (cid:15) N ⊥ (cid:15) (cid:15) / / Z ( G / N , Map ( G / N , U ( ))) (cid:15) (cid:15) b G (cid:15) (cid:15) ⊂ / / Z ( G , Map ( G / N , U ( ))) restr. (cid:15) (cid:15) b G / N ⊥ (cid:15) (cid:15) ∼ = / / b N (cid:15) (cid:15) Q G / N : = ˇ H ( B , Z ( G / N , Map ( G / N , U ( ))) , g .. ) q (cid:15) (cid:15) ˇ H ( B , b G ) (cid:15) (cid:15) / / Q G : = ˇ H ( B , Z ( G , Map ( G / N , U ( ))) , g .. ) (cid:15) (cid:15) ˇ H ( B , b G / N ⊥ ) (cid:15) (cid:15) ∼ = / / ˇ H ( B , b N , g .. ) ˇ H ( B , N ⊥ ) . (33)Due to equation (30) restriction χ ji ( u ) | N defines a class [ χ .. | N ] ∈ ˇ H ( B , b N , g .. ) and by diagram (33) a class in ˇ H ( B , b G / N ⊥ ) , i.e the class of a b G / N ⊥ -principalfibre bundle b E χ .. → B . Exactness of the columns in diagram (33) implies thatthis class is only well-defined up to the quotient group Q G / q ( Q G / N ) , becausethe class varies with the choice of χ .. . The bundle b E χ .. will be connected tothe description of τ ( B )( x dyn ) . Namely, if x = [( κ , ( P , E ) , ( b P , b E ))] and x dyn =[( ρ dyn , P dyn , E )] ( P dyn will be a stabilisation of P with a certain Hilbert space),then the result will be τ ( B )( x dyn ) = [( κ top , ( P dyntop , E ) , ( d P dyn , b E ⊕ b E χ .. ))] .If the topological triple is strict, then by definition we can choose χ .. such that χ ji ( u ) | N =
1, so the class [ χ .. | N ] vanishes and the bundle b E χ .. is trivialisable. In58hat case we give a description of d P dyn in the proof of the theorem below. If thetopological triple is in the image of τ ( B ) , then we even know more as the nexttheorem makes precise. Therein P ( M ) denotes the power set of a set M , andwe denote the image of τ ( B ) by Top im ( B ) , soTop im ( B ) ⊂ Top s ( B ) ⊂ Top as ( B ) ⊂ Top ( B ) . Theorem 3.4
The map τ ( B ) is injective, and there are three maps δ as ( B ) : Top as ( B ) → P ( Dyn † ( B )) δ s ( B ) : Top s ( B ) → P ( Dyn † ( B )) δ im ( B ) : Top im ( B ) → Dyn † ( B ) with the following properties:(a) For each x ∈ Top as ( B ) the set δ as ( B )( x ) ⊂ Dyn † ( B ) is a Q G -torsor.(b) If x ∈ Top s ( B ) , then δ s ( B )( x ) ⊂ δ as ( B )( x ) is a q ( Q G / N ) -subtorsor, and foreach x dyn ∈ δ s ( B )( x ) the (class of the) b G / N ⊥ -bundle [ E dyn of the (class of the)dual pair of τ ( B )( x dyn ) is (the class of) the b G / N ⊥ -bundle b E of x.(c) If x is in the image of τ ( B ) , then δ im ( B )( x ) ∈ δ s ( B )( x ) , and δ im ( B ) is theinverse of τ ( B ) .Moreover, δ ? ( B ) is natural in the sense that it extends to a natural transformation offunctors δ ? : Top ? → P ◦ Dyn † , ? = as, s, im. Proof :
The proof consists of several steps which are rather technical, so wefirst give an overview of the proof:In Step 1 we construct from the data of an almost strict topological triple adecker on the underlying pair (after stabilisation). The local construction willdepend on many choices and it is the statement of Step 2 that almost all ofthese choices do not interfere with the equivalence class of triple defined. Anexception is the choice of χ .. (above) which will cause the Q G -torsor and the q ( Q G / N ) -subtorsor structure in ( a ) , ( b ) . Then in Step 3 we compute the compo-sition δ im ( B ) ◦ τ ( B ) and find that this is the identity on Dyn † ( B ) , hence τ ( B ) is injective and δ im ( B ) is surjective. In Step 4 we compute the reverse compo-sition τ ( B ) ◦ δ s ( B ) , which will lead us to the result stated in ( b ) . In particularthis calculation shows that τ ( B ) ◦ δ im ( B ) is the identity on the image of τ ( B ) .As both compositions δ im ( B ) ◦ τ ( B ) and τ ( B ) ◦ δ im are the identity maps thestatement of ( c ) is clear then. In Step 5 we finally comment on the naturality ofthe maps.In the hole of the proof we maintain the notation introduced above.59 tep 1: For an almost strict topological triple ( κ , ( P , E ) , ( b P , b E )) we constructa dualisable dynamical triple ( ρ , P dyn , E dyn ) .The idea is simple. All we have to do is to write down an explicit formulafor a family of cocycles which satisfies equation (9) for the transition functionsof the pair ( P , E ) .Assume the topological triple has underlying Hilbert space b H = L ( G / N ) ⊗ H . Let ζ ji , ζ ′ ji , v i , w ji , ϕ kji , . . . be as above. We can assume that the atlas U • issufficiently refined such that ϕ ... is a boundary. So we choose χ .. as above suchthat ϕ ... = δ g χ .. , then let m i be defined by (32) in the last lemma.Let us consider U ji ∋ u H ⊗ λ b G / N ⊥ ( ˆ g ji ( u )) ∈ U ( H ⊗ L ( b G / N ⊥ )) ,this defines a unitary implemented ˇCech cocycle, therefore it is a boundary,i.e. there are l i : U i → U ( H ⊗ L ( b G / N ⊥ )) such that l j ( u ) − l i ( u ) = H ⊗ λ b G / N ⊥ ( ˆ g ji ( u )) . Note that this need not be true for λ b G / N ⊥ ( ˆ g ji ( u )) alone as b G / N ⊥ may be finite, L ( b G / N ⊥ ) finite dimensional and U ( L ( b G / N ⊥ )) not contractible.We conclude that there exists a family λ i : U i → Map ( G / N , PU ( H ⊗ L ( b G / N ⊥ ) ⊗ L ( G / N ) ⊗ H )) such that this family permits lifts l i : U i → Map ( G / N , U ( H ⊗ L ( b G / N ⊥ ) ⊗ L ( G / N ) ⊗ H )) such that l j ( u )( g ji ( u ) + z ) − ( ⊗ ⊗ ζ ji ( u )( z )) l i ( u )( z )= ⊗ λ b G / N ⊥ ( ˆ g ji ( u )) ⊗ ζ ji ( u )( z ) . (34)For example we may take λ i ( u )( z ) = Ad ( l i ( u ) ⊗ ⊗ ) , but it will be impor-tant that we allow ourselves to have the freedom of a more flexible form of λ i . One should also note that equation (34) is stated for unitary and not forprojective unitary operators.We define a family (of local isomorphisms ) ϑ i : U i → Map ( G / N , PU ( H ⊗ L ( b G / N ⊥ ) ⊗ L ( G / N ) ⊗ H )) by ϑ i ( u )( z ) : = λ i ( u )( z ) ( H ⊗ L ( b G / N ⊥ ) ⊗ κ ai ( u , z )) Ad (cid:16) ( H ⊗ κ σ ( z , .. ) ⊗ H ) ( H ⊗ v i ( u , z , .. )) (cid:17) which we use to define another family (of cocycles) µ dyn i : U i → Z ( G , Map ( G / N , PU ( H ⊗ L ( b G / N ⊥ ) ⊗ L ( G / N ) ⊗ H )) by µ dyn i ( u )( h , z ) : = ϑ i ( u )( z + hN ) Ad (cid:16) ( m i ( u )( h , z ) ⊗ L ( G / N ) ⊗ H )( H ⊗ h ˆ σ ( .. ) , h i ⊗ L ( G / N ) ⊗ H ) (cid:17) ϑ i ( u )( z ) − . (35)60he stabilised pair ( P dyn , E dyn ) : = ( PU ( H ⊗ L ( b G / N ⊥ )) ⊗ P , E ) has transitionfunctions g ji , ζ dyn ji : = H ⊗ L ( b G / N ⊥ ) ⊗ ζ ji , and we claim that on ( P dyn , E dyn ) the family { µ dyn i } i ∈ I defines a dualisable decker ρ dyn . To verify this we have tocheck that µ dyn j ( u )( h , g ji ( u ) + z ) − ζ dyn ji ( u )( z + hN ) µ dyn i ( u )( h , z ) = ζ dyn ji ( u )( z ) .Although lengthy, this is a straight forward calculation. Indeed, we have u (cid:13) µ dyn j ( h , g ji + z ) − ζ dyn ji ( z + hN ) µ dyn i ( h , z )= ϑ j ( g ji + z ) Ad (( m j ( h , g ji + z )( .. ) − ⊗ ⊗ )( ⊗ h ˆ σ ( .. ) , − h i ⊗ ⊗ )) ϑ j ( g ji + z + hN ) − ( ⊗ ⊗ ζ ji ( z + hN )) ϑ i ( z + hN ) Ad (( m i ( h , z )( .. ) ⊗ ⊗ )( ⊗ h ˆ σ ( .. ) , h i ⊗ ⊗ )) ϑ i ( z ) − = ϑ j ( g ji + z ) Ad (( m j ( h , g ji + z )( .. ) − ⊗ ⊗ )( ⊗ h ˆ σ ( .. ) , − h i ⊗ ⊗ )) Ad (cid:16) ( ⊗ v j ( g ji + z + hN , .. ) − )( ⊗ κ σ ( g ji + z + hN , .. ) − ⊗ ) (cid:17) ( ⊗ Ad ( λ b G / N ⊥ ( ˆ g ji )) ⊗ ζ ′ ji ( z + hN )) Ad (cid:16) ( ⊗ κ σ ( z + hN , .. ) ⊗ )( ⊗ v i ( z + hN , .. )) (cid:17) Ad (( m i ( h , z )( .. ) ⊗ ⊗ )( ⊗ h ˆ σ ( .. ) , h i ⊗ ⊗ )) ϑ i ( z ) − = ϑ j ( g ji + z ) Ad (( m j ( h , g ji + z )( .. ) − ⊗ ⊗ )( ⊗ h ˆ σ ( .. ) , − h i ⊗ ⊗ )) Ad ( ⊗ λ b G / N ⊥ ( g ji ) ⊗ ⊗ ) ⊗ h Ad (cid:16) v j ( g ji + z + hN , ˆ g ji + .. ) − ( κ σ ( z + hN , ˆ g ji + .. ) − ⊗ ) (cid:17) ζ ′ ji ( z + hN )) Ad (cid:16) ( κ σ ( z + hN , .. ) ⊗ ) v i ( z + hN , .. ) (cid:17)i Ad (( m i ( h , z )( .. ) ⊗ ⊗ )( ⊗ h ˆ σ ( .. ) , h i ⊗ ⊗ )) ϑ i ( z ) − .The expression inside the squared brackets can be rewritten by equation (25).This reads. . . = ϑ j ( g ji + z ) Ad (( m j ( h , g ji + z )( .. ) − ⊗ ⊗ )( ⊗ h ˆ σ ( .. ) , − h i ⊗ ⊗ )) Ad ( ⊗ λ b G / N ⊥ ( g ji ) ⊗ ⊗ ) ⊗ h Ad ( ˆ ζ ′ ji ( .. )) ( A ji ( z + hN ) ⊗ ⊗ ) i Ad (( m i ( h , z )( .. ) ⊗ ⊗ )( ⊗ h ˆ σ ( .. ) , h i ⊗ ⊗ )) ϑ i ( z ) − .61f we insert furthermore the results of the previous lemmata, we obtain. . . = ϑ j ( z ) Ad (( m j ( h , g ji + z )( .. ) − ⊗ ⊗ )( ⊗ h ˆ σ ( .. ) , − h i ⊗ ⊗ )) Ad ( ⊗ λ b G / N ⊥ ( g ji ) ⊗ ⊗ ) (36)Ad ( ⊗ ˆ ζ ′ ji ( .. )) ( ⊗ β ji ( z + hN ) ⊗ ⊗ )( Ad ( w ji ( z + hN )( .. )) ⊗ ⊗ )( γ ji ⊗ ⊗ ) Ad (( m i ( h , z )( .. ) ⊗ ⊗ )( ⊗ h ˆ σ ( .. ) , h i ⊗ ⊗ )) ϑ i ( z ) − = ϑ j ( z ) Ad ( ⊗ λ b G / N ⊥ ( g ji ) ⊗ ⊗ ) Ad (cid:16) ( ⊗ ˆ ζ ′ ji ( .. ))( ⊗ h ˆ σ ( .. + ˆ g ji ) − ˆ σ ( .. ) , σ ( g ji + z + hN ) − h i ⊗ ⊗ )( w ji ( z )( .. ) ⊗ ⊗ ) (cid:17) ( γ ji ⊗ ⊗ ) ϑ i ( z ) − . (37)As σ ( g ji + z + hN ) − h and σ ( g ji + z ) differ by some element in N , we obtainthe identity h ˆ σ ( .. + ˆ g ji ) − ˆ σ ( .. ) , σ ( g ji + z + hN ) − h i = h ˆ σ ( .. + ˆ g ji ) − ˆ σ ( .. ) , σ ( g ji + z + hN ) − h − σ ( g ji + z ) ih ˆ σ ( .. + ˆ g ji ) − ˆ σ ( .. ) , σ ( g ji + z ) i = h ˆ g ji ( u ) , σ ( g ji + z + hN ) − h − σ ( g ji + z ) i (38) h ˆ σ ( .. + ˆ g ji ) − ˆ σ ( .. ) , σ ( g ji + z ) i .Therein the first factor is a scalar and the second defines β ji . So the main calcu-lation continues. . . = ϑ j ( z ) Ad ( ⊗ λ b G / N ⊥ ( g ji ) ⊗ ⊗ ) Ad (( ⊗ ˆ ζ ′ ji ( .. ))) (39) ( ⊗ β ji ( z ) ⊗ ⊗ )( Ad ( w ji ( z )( .. )) ⊗ ⊗ )( γ ji ⊗ ⊗ ) ϑ i ( z ) − ,and again by equation (25). . . = ϑ j ( z ) Ad ( ⊗ λ b G / N ⊥ ( g ji ) ⊗ ⊗ ) ⊗ h Ad (cid:16) v j ( g ji + z , ˆ g ji + .. ) − ( κ σ ( z , ˆ g ji + .. ) − ⊗ ) (cid:17) ( ⊗ ζ ′ ji ( z )) Ad (cid:16) ( κ σ ( z , .. ) ⊗ ) v i ( z , .. ) (cid:17)i ϑ i ( z ) − (40) = ϑ j ( z ) ⊗ Ad (cid:16) v j ( g ji + z , .. ) − ( κ σ ( z , .. ) − ⊗ ) (cid:17) Ad ( ⊗ λ b G / N ⊥ ( g ji )) ⊗ ζ ′ ji ( z ) ⊗ Ad (cid:16) ( κ σ ( z , .. ) ⊗ ) v i ( z , .. ) (cid:17) ϑ i ( z ) − = ⊗ ⊗ ζ ji ( z )= ζ dyn ji ( z ) .This shows that the µ dyn i s define a decker ρ dyn on ( P dyn , E dyn ) . By construction ρ dyn is dualisable, because ( h , z ) ( m i ( u )( h , z )( .. ) ⊗ ⊗ )( ⊗ h ˆ σ ( .. ) , h i ⊗ ⊗ ) is a continuous and unitary implemented 1-cocycle. Step 2:
The construction of Step 1 defines a map δ as ( B ) : Top as ( B ) → P ( Dyn † ( B )) and δ as ( B )( x ) is a Q G -torsor, for each x ∈ Top as ( B ) . If x is strict then there isa distinguished q ( Q G / N ) -subtorsor X dyn , and if x is in the image of τ ( B ) wesingle out a specific element x ∈ X dyn . We then just define δ s ( B )( x ) : = X dyn and δ im ( B )( x ) : = x in the respective cases.We have to show that all choices involved do not change the class of the dy-namical triple ( ρ dyn , P dyn , E dyn ) . That the choice of the atlas has no effect on theclass of the constructed dynamical triple is rather obvious. It is less obvious forthe choices of λ i (or l i ), m i and the homotopy class of κ , i.e. the choice of anisomorphic topological triple. We convince ourselves that three other choicesof these define exterior equivalent deckers.Firstly, if we choose another λ i , say λ ′ i = Ad ( l ′ i ⊗ ⊗ ) , then by equation(34) the family ν i ( u )( z ) : = λ i ( u )( z ) λ ′− i ( u )( z ) defines an automorphism of P dyn , and the ˇCech class of this automorphism [ ν . ] ∈ ˇ H ( B , U ( )) vanishes.Thus we are precisely in the situation of Example 2.1.Secondly, equation (32) shows that m i is unique up to functions n i : U i → Z ( G , Map ( G / N , L ∞ ( b G / N ⊥ , U ab ( H )))) such that δ g × ˆ g n . = . We changethe atlas of the the constructed dynamical triple such that we have transi-tion functions ζ ji ( u )( z ) : = ϑ j ( u )( g ji ( u ) + z ) − ζ ji ( u )( z ) ϑ i ( u )( z ) and cocycles µ i ( u )( h , z ) : = Ad (( m i ( u )( h , z ) ⊗ L ( G / N ) ⊗ H )( H ⊗ h ˆ σ ( .. ) , h i ⊗ L ( G / N ) ⊗ H )) . Assume m i is changed by n i . Then because of the commutativity ofU ab ( H ) we have n i ( u )( h + g , z ) = µ i ( u )( g , z ) − ( n i ( u )( h , z + gN )) n i ( u )( g , z ) ;and because of the λ i -terms in ϑ i we have that ζ ji ( u )( z )( n i ( u )( h , z )( .. )) = n i ( u )( h , z )( .. − ˆ g ji ( u )) = n j ( u )( h , g ji ( u ) + z )( .. ) . So c i : = n i defines an exte-rior equivalence (Definition 2.5).Thirdly, if κ ′ is homotopic to κ , the ˇCech class of the bundle automorphism κ ′ ◦ κ − vanishes, and the change of µ dyn i caused by this automorphism is againcovered by Example 2.1.Let us now discuss the choice of χ .. . We always have the freedom to changeit by a cocycle χ ∈ ˇ Z ( U • , Z ( G , Map ( G / N , U ( ))) , g .. ) . The consequenceis a change of m i by m i : U i → Z ( G , Map ( G / N , L ∞ ( b G / N ⊥ , U ab ( H )))) suchthat δ g × ˆ g m = χ . It is clear that if χ = δ g χ is a boundary, then the deckers ρ dyn and ρ dyn1 which correspond to m i and m i m i are exterior equivalent, forwe can define n i : = χ i m i , so δ g × ˆ g n . = which leads us to the case we alreadydiscussed above. If the class [ χ ] does not vanish, then it follows that the cor-responding deckers are not exterior equivalent and the corresponding triplesare not stably outer conjugate.Thus it follows that for each [ χ ] ∈ Q G we get a different dynamical triple,and we define δ as ( B )( x ) ⊂ Dyn † ( B ) to be the set of these dynamical triples.63t is obvious that Q G acts freely and transitively on δ as ( B )( x ) , i.e. it is a Q G -torsor. If the triple x is strict, then we can choose χ .. such that χ ji ( u ) | N =
1, andthis property is preserved by the action of q ( Q G / N ) ⊂ Q G . So we singled out aspecific q ( Q G / N ) -subtorsor X dyn ⊂ δ as ( B )( x ) , and we define δ s ( B )( x ) : = X dyn .If x is in the image of τ ( B ) , then we saw in in the proof of Lemma 3.5 that w ji satisfies d ∗ w ji ( u )( z ) ∈ U ( ) · , so it is meaningful to define χ ji : = d ∗ w ji . Thenwe let x ∈ X dyn be the element which corresponds to this particular choice of χ .. , and we put δ im ( B )( x ) : = x . Step 3: δ im ( B ) ◦ τ ( B ) = id Dyn † ( B ) , so τ ( B ) is injective and δ ( B ) is surjective.The formal calculation is similar to to what we did in the of the proof of Theo-rem 3.2.Let ( ρ , P , E ) be a dualisable dynamical triple having transition functions g ji , ζ ji and cocycles µ i . Recall the definition of τ ( B ) in particular of the topologi-cal triple ( κ top , ( P top , E ) , ( b P , b E )) out of which we must compute ( ρ dyn , P topdyn , E ) . P topdyn = PU ( H ⊗ L ( b G / N ⊥ ) ⊗ L ( G / N )) ⊗ P is stably isomorphic to P and we shall compute µ dyn i . We first give one possible, explicit formula for λ i ( u )( z ) . Let ˆ F : L ( b G / N ⊥ ) → L ( N ) be the inverse Fourier transform, thenˆ F ◦ λ b G / N ⊥ ( ˆ g ji ( u )) ◦ ˆ F − = h ˆ g ji ( u ) , . i − ∈ L ∞ ( N , U ( )) ,and the definition of ˆ g ji impliesAd ( λ b G / N ⊥ ( ˆ g ji ( u ))) ⊗ ζ top ji ( u )( z )= Ad (cid:16) ( ˆ F − ⊗ ⊗ ) ◦ µ j ( u )( . , g ji ( u ) + z ) − | N ◦ ( ˆ F ⊗ ⊗ ) (cid:17) ⊗ ζ top ji ( u )( z ) Ad (cid:16) ( ˆ F − ⊗ ⊗ ) ◦ µ i ( u )( . , z ) | N ◦ ( ˆ F ⊗ ⊗ ) (cid:17) .Therefore we have l i ( u )( z ) = (( ⊗ ˆ F − ⊗ ⊗ ) ◦ ( ⊗ µ i ( u )( . , z ) | N ) ◦ ( ⊗ ˆ F ⊗ ⊗ ) ∈ U ( H ⊗ L ( b G / N ⊥ ) ⊗ L ( G / N ) ⊗ H ) which is continuous by Lemma 3.1 ( i ) , hence λ i ( u )( z ) = Ad ( l i ( u )( z )) . Fromthe local definition of κ top in equation (21) we read off that κ i ( u )( z , ˆ z ) = κ ai ( u , z ) κ σ ( z , ˆ z ) Ad ( v i ( u , z , ˆ z )) for κ ai ( u , z ) : = Ad ( µ i ( u )( − σ ( ) , z ) − ) and v i ( u , z , ˆ z ) : = λ G / N ( z ) ⊗ . In theproof of Lemma 3.5 we already observed that in the case of a topological tripleconstructed out of dynamical one we have d ∗ w ji ( u )( z ) ∈ U ( ) · , and by thedefinition of δ im ( B ) we choose χ .. : = d ∗ w .. . Therefore we can choose m i = . As a consequence the first H -slot of the tensor product H ⊗ L ( b G / N ⊥ ) ⊗ ( G / N ) ⊗ H will contain the identity operator only. We have all ingredientstogether to compute µ dyn i u (cid:13) . µ dyn i ( h , z ) = ϑ i ( z + hN )( ⊗ h ˆ σ ( .. ) , h i ⊗ ⊗ ) ϑ i ( z ) − = Ad (cid:16) ( ⊗ ˆ F − ⊗ ⊗ ) ◦ ⊗ µ i ( . , z + hN ) | N ◦ ( ⊗ ˆ F ⊗ ⊗ )( ⊗ ⊗ µ i ( − σ ( ) , z + hN ) − )( ⊗ κ σ ( z + hN , .. ))( ⊗ ⊗ λ G / N ( z + hN ) ⊗ )))( ⊗ h ˆ σ ( .. ) , h i ⊗ ⊗ )( ⊗ ⊗ λ G / N ( − z ) ⊗ )( ⊗ κ σ ( z + hN , .. ) − ))( ⊗ ⊗ µ i ( − σ ( ) , z ))( ⊗ ˆ F − ⊗ ⊗ ) ◦ ⊗ ( µ i ( . , z ) | N ) − ◦ ( ⊗ ˆ F ⊗ ⊗ ) (cid:17) .Since ( ˆ F ⊗ ) ◦ h ˆ σ ( .. ) , h − σ ( + hN ) + σ ( ) i ◦ ( ˆ F − ⊗ ) = λ N ( h − σ ( + hN ) + σ ( )) ∈ U ( L ( b G / N ⊥ ) ⊗ L ( G / N )) and by the cocycle identity for µ i this trans-forms to. . . = Ad (cid:16) ( ⊗ ˆ F − ⊗ ⊗ ) ◦ ( ⊗ µ i ( σ ( ) + . , z + hN − ))( ⊗ λ G / N ( hN ) ⊗ ⊗ )( ⊗ λ N ( h − σ ( + hN ) + σ ( ))) ⊗ )( ⊗ ( µ i ( σ ( ) + . , z − )) − ) ◦ ( ⊗ ˆ F ⊗ ⊗ ) (cid:17) = Ad (cid:16) ( ⊗ ˆ F − ⊗ ⊗ ) ◦ ( ⊗ λ G / N ( hN ) ⊗ ⊗ )( ⊗ λ N ( h − σ ( + hN ) + σ ( ))) ⊗ )( ⊗ µ i ( σ ( + hN ) + . + h − σ ( + hN ) + σ ( ) , z − ))( ⊗ ( µ i ( − σ ( ) − . , z ))) ◦ ( ⊗ ˆ F ⊗ ⊗ ) (cid:17) = Ad (cid:16) ( ⊗ ˆ F − ⊗ ⊗ ) ◦ ( ⊗ λ G / N ( hN ) ⊗ ⊗ )( ⊗ λ N ( h − σ ( + hN ) + σ ( ))) ⊗ )( ⊗ ⊗ ⊗ ( µ i ( h , z ))) ◦ ( ⊗ ˆ F ⊗ ⊗ ) (cid:17) .Let S : L ( N ) ⊗ L ( G / N ) → L ( G ) be the isomorphism introduced on page 47.There we discussed its behaviour with respect to the left regular representationon L ( G ) . This leads us finally to µ dyn i ( u )( h , z ) = Ad (cid:16) ( ⊗ ˆ F − ⊗ ⊗ ) ◦ ( ⊗ S − ⊗ ) ◦ ( ⊗ λ G ( h ) ⊗ µ i ( u )( h , z ))) ◦ ( ⊗ S ⊗ ) ◦ ( ⊗ ˆ F ⊗ ⊗ ) (cid:17) .65hus we have shown that ( ρ , P , E ) and ( ρ dyn , P topdyn , E ) are outer conjugate. Step 4:
We compute τ ( B )( δ s )( B )( x )) which will complete the statement of ( b ) .In particular this calculation shows that τ ( B ) ◦ δ im ( B ) = id im ( τ ( B )) .Let ( κ , ( P , E ) , ( b P , b E )) be a topological triple with underlying Hilbert space b H = L ( G / N ) ⊗ H . For the first steps we only need the assumption that the tripleis almost strict. So let ( ρ dyn , P dyn , E dyn ) be the dynamical triple which we con-structed in Step 1. As we saw E dyn is nothing but E itself. From this dynamicaltriple we are going to construct the topological triple ( κ top , ( P dyntop , E dyn ) , ( d P dyn , [ E dyn )) (according to the construction of τ ( B ) ) which we then have to compare with ( κ , ( P , E ) , ( b P , b E )) .To compute the dual pair ( d P dyn , [ E dyn ) we with determination of φ dyn ji thesecond term of the total cocycle ( ψ dyn... , φ dyn.. , 1 ) of the constructed dynamicaltriple ( ρ dyn , P dyn , E dyn ) . By equation (16), the restriction of φ dyn ji to N definesthe (the cocycle of) bundle [ E dyn .The definition of the cocycles µ dyn i of the decker ρ dyn contains λ i . We makeuse of the possibility to choose λ i ( u )( z ) = Ad ( l i ( u ) ⊗ ⊗ ) as we explainedin Step 1. In Step 2 we saw that this choice does not have any effect on the levelof equivalence classes. Assume that the atlas is sufficiently refined and consistsof contractible charts such that the following continuous lifts exist. Lifts of thetransition functions ζ dyn ji = ⊗ ⊗ ζ ji and lifts κ ai : U i → L ∞ ( G / N , U ( b H )) of κ a which we use to define unitary lifts µ dyn i of µ dyn i according to (35) in theobvious way, i.e. we drop the Ad. Let δ ji : U ji → L ∞ ( G / N , U ( )) be such that α ji ( u )( z , ˆ z ) = δ ji ( u )( z ) h ˆ σ ( ˆ g ji ( u ) + ˆ z ) − ˆ σ ( ˆ z ) , σ ( g ji ( u ) + z i w ji ( u )( z )( ˆ z ) γ ji ( u ) ,for a lift γ ji : U ji → U ( H ) of γ ji ; therein all notation is as above. Then bydefinition φ dyn ji ( u )( h , z )= µ dyn i ( u )( h , z ) ζ dyn ji ( u )( z ) − µ dyn j ( u )( h , g ji ( u ) + z ) − ζ dyn ji ( u )( z + hN ) ,and if we repeat the calculation of Step 1 on the unitary level, there are fourequalities which must be modified by U ( ) -valued functions. Namely, equa-tion (36) by δ ( u )( z + hN ) , equation (37) by χ ji from equation (32), equation (39)by the scalar term of (38) and equation (40) by δ ( u )( z ) − . We finally find φ dyn ji ( u )( h , z ) = h ˆ g ji ( u ) , σ ( g ji ( u ) + z + hN ) − h − σ ( g ji ( u ) + z ) i χ ji ( u )( h , z ) ( d δ ji ( u ))( h , z ) .66t follows that [ E dyn ∼ = b E if and only if the topological triple we started with isstrict and we have chosen χ .. such that χ ji ( u ) | N =
1. Indeed, the cocycles ofthese bundles are ˆ g dyn ji ( u ) : = ( φ dyn ji ( u )( . , z ) | N ) − = ˆ g ji ( u ) χ ji ( u )( − . , z ) | N .This proves ( b ) .To complete the computation of the dual we have to compute d ζ dyn ji : U ji → Map ( b G / N ⊥ , PU ( L ( G / N ) ⊗ H ⊗ L ( b G / N ⊥ ) ⊗ L ( G / N ) ⊗ H )) .by equation (18). The reader should not be confused by the two different L ( G / N ) factors in the tensor product. The first is due to the stabilisation inthe definition of the dual (18), and the second is due to the Hilbert space westarted with which is L ( G / N ) ⊗ H . We use the symbols ⌣ and to distinguishmultiplication operators on the two Hilbert spaces; ⌣ for the first factor, due tothe definition of the dual and for the second factor as we did all the time. Thedual transition functions are given by u (cid:13) d ζ dyn ji ( ˆ z ) = Ad (cid:16) ( κ σ ( − g ji , ˆ g dyn ji + ˆ z ) ⊗ ⊗ ⊗ ⊗ )( λ G / N ( − g ji ) ⊗ ⊗ ⊗ ⊗ )( ζ dyn ji ( − ⌣ ))( φ dyn ji ( − σ ( ⌣ ) , 0 ) − ⊗ ⊗ ⊗ ⊗ ) (cid:17) = Ad (cid:16) ( κ σ ( − g ji , ˆ g dyn ji + ˆ z ) ⊗ ⊗ ⊗ ⊗ )( λ G / N ( − g ji ) ⊗ ⊗ ⊗ ⊗ )( ⊗ l j ⊗ ⊗ )[ ⊗ λ b G / N ⊥ ( ˆ g ji ) ⊗ ( ζ ji ( − ⌣ )]( ⊗ l i − ⊗ ⊗ )( φ dyn ji ( − σ ( ⌣ ) , 0 ) − ⊗ ⊗ ⊗ ⊗ ) (cid:17) .By equation (25) this is u (cid:13) . . . = Ad (cid:16) ( ⊗ l j ⊗ ⊗ )( κ σ ( − g ji , ˆ g dyn ji + ˆ z ) ⊗ ⊗ ⊗ ⊗ )( λ G / N ( − g ji ) ⊗ ⊗ ⊗ ⊗ )( ⊗ ⊗ κ aj ( g ji − ⌣ ))( ⊗ ⊗ κ σ ( g ji − ⌣ , ˆ g ji + ˆ z ) ⊗ )( ⊗ ⊗ v j ( g ji − ⌣ , ˆ g ji + ˆ z ))( ⊗ ⊗ κ bj ( ˆ g ji + ˆ z ))( ⊗ ⊗ λ b G / N ⊥ ( ˆ g ji ) ⊗ ˆ ζ ji ( ˆ z ))( α ji ( − ⌣ , ˆ z ) ⊗ ⊗ ⊗ ⊗ )( ⊗ ⊗ κ bi ( ˆ z ) − )( ⊗ ⊗ v i ( − ⌣ , ˆ z ) − )( ⊗ ⊗ κ σ ( − ⌣ , ˆ z ) − ⊗ )( ⊗ ⊗ κ ai ( − ⌣ ) − )( φ dyn ji ( − σ ( ⌣ ) , 0 ) − ⊗ ⊗ ⊗ ⊗ )( ⊗ l i − ⊗ ⊗ ) (cid:17) , 67nd if we insert the formulas for κ σ , α ji and φ dyn ji from above, we end up with u (cid:13) . . . = Ad (cid:16) ( ⊗ l j ⊗ ⊗ )( h ˆ σ ( ˆ g dyn ji + ˆ z ) , σ ( ⌣ + g ji ) − σ ( ⌣ ) i ⊗ ⊗ ⊗ ⊗ )( λ G / N ( − g ji ) ⊗ ⊗ ⊗ ⊗ )( ⊗ ⊗ κ aj ( g ji − ⌣ ))( ⊗ ⊗ h ˆ σ ( ˆ g ji + ˆ z ) , σ ( − g ji + ⌣ ) − σ ( ) i ⊗ )( ⊗ ⊗ v j ( g ji − ⌣ , ˆ g ji + ˆ z ))( ⊗ ⊗ κ bj ( ˆ g ji + ˆ z ))( ⊗ ⊗ λ b G / N ⊥ ( ˆ g ji ) ⊗ ˆ ζ ji ( ˆ z ))( δ ji ( − ⌣ ) ⊗ ⊗ ⊗ ⊗ )( h ˆ σ ( ˆ g ji + ˆ z ) − ˆ σ ( ˆ z ) , σ ( g ji − ⌣ ) i ⊗ ⊗ ⊗ ⊗ )( w ji ( − ⌣ )( ˆ z ) ⊗ ⊗ ⊗ )( ⊗ γ ji ⊗ ⊗ ⊗ )( ⊗ ⊗ κ bi ( ˆ z ) − )( ⊗ ⊗ v i ( − ⌣ , ˆ z ) − )( ⊗ ⊗ h ˆ σ ( ˆ z ) , − σ ( + ⌣ ) + σ ( ) i ⊗ )( ⊗ ⊗ κ ai ( − ⌣ ) − )([ h ˆ g ji , σ ( g ji − ⌣ ) + σ ( ⌣ ) − σ ( g ji ) i χ ji ( − σ ( ⌣ ) , 0 )( d δ ji )( − σ ( ⌣ ) , 0 ) ] − ⊗ ⊗ ⊗ ⊗ )( ⊗ l i − ⊗ ⊗ ) (cid:17) .It follows from the definition of γ ji that Ad ( w ji ( u )( )( ˆ z ) − ) = γ ji ( u ) , so af-ter some further manipulation with all the h . . . i -expressions this finally trans-forms to u (cid:13) . . . = Ad (cid:16) ( ⊗ l j ⊗ ⊗ )( ⊗ ⊗ κ aj ( − ⌣ ))( ⊗ ⊗ h ˆ σ ( ˆ g ji + ˆ z ) , − σ ( ⌣ ) + σ ( + ⌣ ) − σ ( ) i ⊗ )( ⊗ ⊗ v j ( − ⌣ , ˆ g ji + ˆ z ))( ⊗ ⊗ κ bj ( ˆ g ji + ˆ z )) λ G / N ( − g ji ) ⊗ ⊗ λ b G / N ⊥ ( ˆ g ji ) ⊗ ˆ ζ ji ( ˆ z ))( w ji ( − ⌣ )( ˆ z ) w ji ( )( ˆ z ) − χ ji ( − σ ( ⌣ ) , 0 ) − ⊗ ⊗ ⊗ )( ⊗ ⊗ κ bi ( ˆ z ) − )( ⊗ ⊗ v i ( − ⌣ , ˆ z ) − )( ⊗ ⊗ h ˆ σ ( ˆ z ) , σ ( ⌣ ) − σ ( + ⌣ ) + σ ( ) i ⊗ )( ⊗ ⊗ κ ai ( − ⌣ ) − )( ⊗ l i − ⊗ ⊗ )( h ˆ σ ( ˆ g dyn ji + ˆ z ) − ˆ σ ( ˆ g ji + ˆ z ) , − σ ( ⌣ − g ji ) + σ ( ⌣ ) i ⊗ ⊗ ⊗ ⊗ ) (cid:17) .So far we did not use that the topological triple we started with is strict and68hat we want to compute the composition τ ( B ) ◦ δ s ( B ) . We use this now, so the G -slot of χ ji factors through G / N and in particular ˆ g dyn ji = ˆ g ji , so we find. . . = η ′ j ( u , ˆ g ji ( u ) + ˆ z ) ( ξ ji ( u )( ˆ z ) ⊗ ˆ ζ ji ( u )( ˆ z )) η ′ i ( u )( z ) − wherein we have used the short hands ξ ji ( u )( ˆ z ) : = Ad (cid:16) [( λ G / N ( − g ji ( u )) ⊗ ) w ji ( u )( − ⌣ )( ˆ z ) w ji ( u )( )( ˆ z ) − χ ji ( u )( − σ ( ⌣ ) , 0 ) − ] ⊗ λ b G / N ⊥ ( ˆ g ji ( u )) (cid:17) ∈ U ( L ( G / N ) ⊗ H ⊗ L ( b G / N ⊥ )) and η ′ i ( u )( z ) : = Ad (cid:16) ( ⊗ l i ( u ) ⊗ ⊗ )( ⊗ ⊗ κ ai ( u )( − ⌣ ))( ⊗ ⊗ h ˆ σ ( ˆ z ) , − σ ( ⌣ ) + σ ( + ⌣ ) − σ ( ) i ⊗ )( ⊗ ⊗ v i ( u , − ⌣ , ˆ z ))( ⊗ ⊗ κ bi ( u )( ˆ z )) (cid:17) ∈ PU ( L ( G / N ) ⊗ H ⊗ L ( b G / N ⊥ ) ⊗ L ( G / N ) ⊗ H ) .At this point we can read off that the bundle defined by ξ ji is trivialisable if χ − d ∗ w ji = which is the datum we must consider when we compute τ ( B ) ◦ δ im ( B ) (cp. Step 2). In this case let x i : U i → Map ( b G / N ⊥ , U ( L ( G / N ) ⊗ H ⊗ L ( b G / N ⊥ ))) be such that Ad ( x j ( u )( ˆ g ji ( u ) + ˆ z ) x i ( u )( ˆ z ) − ) = ξ ji ( u )( ˆ z ) . Thenwe have shown that η i ( u )( ˆ z ) : = η ′ i ( u )( ˆ z ) Ad ( x i ( u )( ˆ z ) ⊗ ⊗ ) defines a familyof local isomorphisms which fit together to a global isomorphism of principalbundles η : b P s → d P dyn . The subscript s denotes stabilisation with respect tothe Hilbert space L ( G / N ) ⊗ H ⊗ L ( b G / N ⊥ ) . We claim that the topologicaltriple ( κ , ( P , E ) , ( b P , b E )) and the constructed triple ( κ top , ( P dyntop , E ) , ( d P dyn , b E )) are equivalent; κ top is defined by equation (21) out of µ dyn i . This will prove theidentity τ ( B ) ◦ δ ( B ) = id im ( τ ( B ) . We show that the diagram P s × B b E = (cid:15) (cid:15) E × B b P s ⊗ ⊗ ⊗ κ o o id × B η (cid:15) (cid:15) P dyntop × B b E E × B d P dyn κ top o o commutes up to homotpy. Locally, this means that there exist continuous maps V i : U i × G / N × b G / N ⊥ → PU ( L ( G / N ) ⊗ H ⊗ L ( b G / N ⊥ ) ⊗ L ( G / N ) ⊗ H ) κ top i ( u )( z , ˆ z ) η i ( u )( ˆ z ) = ( ⊗ ⊗ ⊗ κ i ( u )( z , ˆ z )) Ad ( V i ( u , z , ˆ z )) . Toprove this we will use that the projective unitary group is homotopy comuta-tive in the sense of Corollary A.1. We have u (cid:13) κ top i ( z , ˆ z ) η i ( ˆ z )= Ad (cid:16) ( h ˆ σ ( ˆ z ) , σ ( ⌣ − z ) − σ ( ⌣ ) i ⊗ ⊗ ⊗ ⊗ )( µ dyn i ( − σ ( ⌣ ) , z )) − ( λ G / N ( z ) ⊗ ⊗ ⊗ ⊗ ) (cid:17) η i ( ˆ z )= Ad (cid:16) ( h ˆ σ ( ˆ z ) , σ ( ⌣ − z ) − σ ( ⌣ ) i ⊗ ⊗ ⊗ ⊗ )( ⊗ l i ⊗ ⊗ )( ⊗ ⊗ ⊗ κ ai ( z ))( ⊗ ⊗ h ˆ σ ( .. ) , σ ( − z ) − σ ( ) i ⊗ )( ⊗ ⊗ v i ( z , .. ))( ⊗ h ˆ σ ( .. ) , σ ( ⌣ ) i ⊗ ⊗ )( m i ( − σ ( ⌣ ) , z )( .. ) − ⊗ ⊗ )( ⊗ v i ( z − ⌣ , .. ) − )( ⊗ h ˆ σ ( .. ) , − σ ( − z + ⌣ ) + σ ( ) i ⊗ )( ⊗ ⊗ ⊗ κ ai ( z − ⌣ ) − )( ⊗ l i − ⊗ ⊗ )( λ G / N ( z ) ⊗ ⊗ ⊗ ⊗ )( ⊗ l i ⊗ ⊗ )( ⊗ ⊗ κ ai ( − ⌣ ))( ⊗ ⊗ h ˆ σ ( ˆ z ) , − σ ( ⌣ ) + σ ( + ⌣ ) − σ ( ) i ⊗ )( ⊗ ⊗ v i ( − ⌣ , ˆ z ))( ⊗ ⊗ κ bi ( ˆ z ))( x i ( ˆ z ) ⊗ ⊗ ) (cid:17) .We see that the terms κ ai ( u )( z ) and κ bi ( u )( z ) occur. All terms with l i , m i , v i , x i are continuous unitary maps The only bracket h . . . i terms not continuous asunitary maps are the two terms in the first and third last line. If we collectthem, we shall better pay attention to λ G / N ( z ) which acts in the ⌣ -variable.Then these two terms equal h ˆ σ ( ˆ z ) , σ ( + ⌣ − z ) − σ ( ) − σ ( ⌣ ) i = h ˆ σ ( ˆ z ) , σ ( − z ) − σ ( ) ih ˆ σ ( ˆ z ) , − σ ( − z ) + σ ( + ⌣ − z ) − σ ( ⌣ ) i = ( ⊗ κ σ ( z , ˆ z )) h ˆ z , − σ ( − z ) + σ ( + ⌣ − z ) − σ ( ⌣ ) i∈ U ( L ( G / N ) ⊗ L ( G / N )) κ top i ( u )( z , ˆ z ) η i ( u )( ˆ z ) = . . . = ( ⊗ ⊗ ⊗ κ ai ( u )( z ))( ⊗ ⊗ ⊗ κ σ ( z , ˆ z ))( ⊗ ⊗ ⊗ κ bi ( u )( z )) Ad ( V ′ i ( u , z , ˆ z ))= ( ⊗ ⊗ ⊗ κ i ( u )( z , ˆ z ) Ad ( V i ( u , z , ˆ z )) ,for suitable continuous unitary maps V ′ i , V i , and we are done. Step 5:
Naturality.In the same way as we already stated for τ , we have a commutative diagramTop ? ( B ) f ∗ (cid:15) (cid:15) δ ? ( B ) / / P ( Dyn † ( B )) f ∗ (cid:15) (cid:15) Top ? ( B ′ ) δ ? ( B ′ ) / / P ( Dyn † ( B ′ )) , ? = as, s, im,for each continuous map between base spaces f : B ′ → B (cid:4) Definition 3.3
Let P → E → B be a pair over the base space B. We say that this pairhas an extension to a topological (resp. almost strict topological, strict topological,dynamical or dualisable dynamical) triple if the class [( P , E )] ∈ Par ( B ) is in the imageof the forgetful map from topological (resp. almost strict topological, strict topological,dynamical or dualisable dynamical) triples over B to pairs over B. Corollary 3.1
Let ( P , E ) be a pair over B. Then the following are equivalent: ( i ) ( P , E ) has an extension to a strict topological triple; ( ii ) ( P , E ) has an extension to an almost strict topological triple; ( iii ) ( P , E ) has an extension to a dualisable dynamical triple. Proof :
This is an immediate consequence of the previous theorem. (cid:4) G = R n and N = Z n So far we kept our analysis completely general in the sense that we did notspecify the groups G , N . We now turn to the important case of G = R n withlattice N = Z n , n =
1, 2, 3, . . . . In the whole of this section we use the nota-tion T n : = R n / Z n for the torus and ˆ T n : = c R n / Z n ⊥ for the dual torus. Thereshould be no confusion to decide between ˆ T n and the dual group c T n ∼ = Z n .The first thing we should check is that in case of G = R n , N = Z n our def-inition of topological triples agrees with the one introduced in [BRS].71he definition of T-duality triples in [BRS, Def. 2.8] differs in two pointsfrom what we stated in Definiton 2.9. The first point is that they use the lan-guage of twists [BRS, A.1] instead of PU ( H ) -principal fibre bundles to modelT-duality diagrams. But the category of PU ( H ) -principal bundles with ho-motopy classes of bundle isomorphisms as morphisms is a model of twists,and our notion of stable equivalence of topological triples leads to the sameequivalence classes as twists modulo isomorphism. This, because the notionof equivalence we use for topological triples requires the commutativity of di-agram (14) only up to homotopy. To explain the second point let us considerthe filtration of H ( E , Z ) associated to the Leray-Serre spectral sequence0 ⊂ F H ( E , Z ) ⊂ F H ( E , Z ) ⊂ F H ( E , Z ) ⊂ H ( E , Z ) .By definition, an element h ∈ H ( E , Z ) is in the subgroup F k + H ( E , Z ) if forany k -dimensional CW-complex C and any map f : C → B h is in the kernelof the induced map f ∗ : H ( E , Z ) → H ( C × B E , Z ) . In our definition of atopological triple ( κ , ( P , E ) , ( b P , b E )) we require that the class [ P ] ∈ H ( E , Z ) of the bundle P → E lies in the subgroup F H ( E , Z ) which is equivalent tothe requirement of the triviality of P over the fibres of E → B (see Definition2.2); analogously for b P . In [BRS] the definition of T-duality triples requiresthat the class [ P ] is even in the second step of the filtration [ P ] ∈ F H ( E , Z ) ;analogously for b P . The following lemma states that in the case of G = R n and N = Z n these two conditions are equivalent. Lemma 3.7
Let G = R n and N = Z n , and let ( κ , ( P , E ) , ( b P , b E )) be a topologicaltriple over B, then [ P ] ∈ F H ( E , Z ) and [ b P ] ∈ F H ( b E , Z ) . Proof :
Let C be any 1-dimensional CW-complex and f : C → B any contin-uous map. We have to show that the class [ P ] is in the kernel of the inducedmap f ∗ : H ( E , Z ) → H ( C × B E , Z ) . As C is 1-dimensional, H ( C , Z ) = ( κ , ( P , E ) , ( b P , b E )) along f , it becomes ( κ C , ( P C , C × B E ∼ = C × T n ) , ( b P C , C × B b E ∼ = C × ˆ T n )) and the centre part of the corresponding diagram degenerates to the projec-tions C × T n × ˆ T n ' ' NNNNNNNNNNN x x ppppppppppp C × T n C × ˆ T n .72e extend this diagram to C × T n w w ppppppppppp (cid:15) (cid:15) " " FFFFFFFFF C × T n × ˆ T n & & NNNNNNNNNNN x x ppppppppppp C | | yyyyyyyyy C × T n C × ˆ T n ,with the obvious inclusions and projection, so the diagonal composition is theidentity. When we apply the the cohomology functor H ( . , Z ) to this diagramthe vertical arrow becomes zero as it factors over H ( C , Z ) =
0, but as f ∗ [ P ] =[ P C ] ∈ H ( C × T n , Z ) is mapped by the identity from the left lower to theright upper group and as [ P C ] and [ b P C ] equal when pulled back to H ( C × T n × ˆ T n , Z ) , we conclude it is zero, since its image in the right upper groupcoincides with the image of f ∗ [ b P ] = [ b P C ] under the vertical arrow.This proves that f ∗ [ P ] = ∈ H ( C × B E , Z ) and the same argument showsthat the corresponding statement is true for [ b P ] . (cid:4) So we observe that our functor Top : { base spaces } → { sets } is the samefunctor which is introduced in [BRS, Def. 2.11] under the name Triple n , andbelow we are going to use a central result of [BRS] about this functor, namelythat the functor Top = Triple n is representable by a space R n (see Lemma 3.9below).We now want to discuss Theorem 3.4 in the case of G = R n , N = Z n . In thiscase the question which of the topological triples are almost strict has a triv-ial answer and also the torsor structure of Theorem 3.4 becomes trivial. Thecriterion we make use of is the following lemma. Lemma 3.8
Let Z be a topological abelian group which is contractible as topologicalspace and is equipped with a continuous (right) G / N-action. Then ˇ H k ( B , Z , [ g .. ]) = k =
1, 2.
Proof :
Similar to the proof of Lemma 3.6, the proof makes use of the standardZorn’s lemma argument.k=1: Let { U i } i ∈ I be an open cover of B , and let ϕ .. ∈ ˇ Z ( U • , Z ) be a twisted1-cocycle. We shall construct functions χ i : U i → Z such that ( δ g χ ) .. = ϕ .. . B is paracompact, hence without restriction we can assume that even the closed73over { U i } is locally finite and all ϕ ij are defined on the whole of U ij . Let K : = n ( J , χ . ) | J ⊂ I , for all j ∈ J : χ j : U j → Z ,such that for all j , k ∈ J and u ∈ U jk : χ j ( u ) χ k ( u ) − · g kj ( u ) = ϕ kj ( u ) o K is non-empty, since ( { i } , { ϕ ii } ) ∈ L , for each i ∈ I . We define a partial orderon K such that every chain has an upper bound. We let ( J , χ . ) ≤ ( J ′ , χ ′ . ) if andonly if J ⊂ J ′ ⊂ I and χ j = χ ′ j , for all j ∈ J . By Zorn’s lemma, let ( J , χ . ) denotea maximal element of K .Assume J = I , so there is some a ∈ I \ J . Let R : = S j ∈ J ( U j ∩ U a ) ⊂ U a .For u ∈ R we define ˜ χ a ( u ) : = ϕ ja ( u ) χ j ( u ) · g ja ( u ) if u ∈ U j . This definition isindependent of j ∈ J . We end up with a diagram R ˜ χ a / / ∩ (cid:15) (cid:15) ZU a χ a ? ? .Since our cover is locally finite, R is closed, but Z is contractible, therefore anextension χ a exists [DD, Lem. 4]. This contradicts the maximality of ( J , χ . ) , so J = I .k=2: Let ϕ ... ∈ ˇ Z ( U • , Z ) be a twisted 2-cocycle. We shall construct functions χ ij : U ji → Z such that ( δ g χ ) ... = ϕ ... . Again by paracompactness of B we canassume that even the closed cover { U i } is locally finite and all ϕ ijk are definedon the whole of U ijk . Let L : = n ( J , χ .. ) | J ⊂ I , for all i , j ∈ J : χ ij : U ij → Z ,such that for all i , j , k ∈ J and u ∈ U ijk : χ ji ( u ) χ ki ( u ) − χ kj ( u ) · g ji ( u ) = ϕ kji ( u ) o The set L is non-empty, since for each i ∈ I ( { i } , { ϕ iii } ) ∈ L . We define a partialorder on L such that every chain has an upper bound. We let ( J , χ .. ) ≤ ( J ′ , χ ′ .. ) if and only if J ⊂ J ′ ⊂ I and χ ij = χ ′ ij , for all i , j ∈ J . By Zorn’s lemma, let ( J , χ .. ) denote a maximal element of L . Assume a ∈ I \ J . Then let M a : = n ( K , ψ . a ) | K ⊂ J , for all k ∈ K : ψ ka : U ka → Z ,such that for all k , l ∈ K and for u ∈ U lka ψ ka ( u ) ψ la ( u ) − χ kl ( u ) · g la ( u ) = ϕ lka ( u ) o M a is non-empty as for each j ∈ J we find ( { j } , { } ) ∈ M a . Thisbecause we always have 1 · · χ jj ( u ) · g ja ( u ) = ϕ jjj ( u ) · g ja ( u ) = ϕ jja ( u ) . In thesame manner as before, let ( K , ψ . a ) ≤ ( K ′ , ψ ′ . a ) if and only if K ⊂ K ′ ⊂ I and ψ ka = ψ ′ ka , for all k ∈ K . ≤ is a partial order on M a and every chain has anupper bound. Let ( K , ψ . a ) be a maximal element. Assume b ∈ J \ K . Let S : = S k ∈ K ( U k ∩ U b ∩ U a ) ⊂ U ba . For u ∈ R we define ˜ ψ ba ( u ) : = ϕ kba ( u ) χ kb ( u ) · g ba ( u ) − ψ ka ( u ) , if u ∈ U kba , k ∈ K . By a one line calculation we find that thisdefinition is independent of k ∈ K . So we have a diagram S ˜ ψ ba / / ∩ (cid:15) (cid:15) ZU ba ψ ba > > . S is closed, since our cover is locally finite. Thus, since Z is contractible, thereis an extension ψ ba [DD, Lem. 4]. This contradicts the maximality of ( K , ψ . a ) ,so K = J . We define ψ aa : = ϕ aaa and then ψ aj ( u ) : = ϕ aja ( u ) · g aj ( u ) ψ aa ( u ) · g aj ( u ) − ψ ja ( u ) · g aj ( u ) , for j ∈ J . We let J ′ : = { a } ∪ J and extend χ .. to J ′ by χ ′ ij : = χ ij , if i , j ∈ J , ψ ia , if j = a , i ∈ J , ψ aa , if i = j = a , ψ aj , if i = a , j ∈ J .It is straight forward to check that ( J ′ , χ ′ .. ) ∈ L , and as clearly ( J , χ .. ) ≤ ( J ′ , χ ′ .. ) we have a contradiction. Hence J = I , and the lemma is proven. (cid:4) Theorem 3.5
Let G = R n , N = Z n . Then every topological triple is almost strict,i.e. in this case Top as ( B ) = Top ( B ) . Moreover, the group Q R defined in diagram (33)vanishes, hence the three natural transformations of Theorem 3.4 reduce to one singletransformation δ : Top → Dyn † . Proof :
By the lemma above, it is sufficient to show that Z : = Z ( R n , Map ( T n , U ( ))) is contractible. Then it follows that ˇ H ( B , Z , g .. ) = { } and every topologicaltriple is almost strict. Because Q R = ˇ H ( B , Z , g .. ) = { } , the torsor struc-tures in Theorem 3.4 are trivial and we obtain a natural map δ ( B ) : Top ( B ) → Dyn † ( B ) .We now show that Z is contractible. For a cocycle α ∈ Z we have α ( )( z ) = α ( + )( z ) = α ( )( z ) α ( )( z ) for all z ∈ T n , so α ( )( z ) = α ( g ) : T n → U ( ) is null homotopic as R n is path connected. Thus Z = Z ( R n , Map ( T n , U ( ))) = Z ( R n , Map ( T n , U ( ))) ,75or Map ( T n , U ( )) : = { f ≃ const. } . But for each α : R n → Map ( T n , U ( )) there exists a lift α in Map ( T n , R n ) (cid:15) (cid:15) R n α / / α llllllllllllllll Map ( T n , U ( )) ,and because of d α =
1, we have d α = m ∈ Z ( R n , Map ( T n , Z )) ∼ = Z . Then α ′ ( g )( z ) : = α ( g )( z ) − m defines a unique element α ′ ∈ Z ( R n , Map ( T n , R )) such that α ′ ( g )( z ) Z = α ( g )( z ) . In fact, the mapping α α ′ is a homoeomor-phism Z ( R n , Map ( T n , U ( ))) → Z ( R n , Map ( T n , R )) ,and the latter space is easily seen to be contractible by h ( t , α ′ )( g )( z ) : = t α ′ ( g )( z ) , t ∈ [
0, 1 ] . This proves the theorem. (cid:4) Corollary 3.2
Let G = R n , N = Z n . Then a pair has an extension to a toplogicalT-duality triple if and only if it has an extension to a dualisable dynamical triple. Proof :
This follows from Corollary 3.1 and Theorem 3.5. (cid:4)
So far we have seen that in the case of G = R n , N = Z n Top im ( B ) ⊂ Top s ( B ) ⊂ Top as ( B ) ! = Top ( B ) , (41)and the composition of the natural transformations τ and δ is the identity trans-formation δ ◦ τ = id : Dyn † → Dyn † . The opposite composition J : = τ ◦ δ :Top → Top is an idempotent J ◦ J = J . Unfortunately, inside our local theorywe won’t be able to answer the question whether J is the identity on Top, i.e. allinclusions in (41) are equalities and δ and τ are both equivalences of functorsinverse to each other. Nevertheless we can give this appealing answer whenwe restrict the functors Top and Dyn † to the subcategory CW ⊂ { bases spaces } of CW-complexes (or, more general, of base spaces with the homotopy type ofa CW-complex). The point is that we can use a main result of [BRS] about therepresentability of the functor Top | CW = Triple n . However, the statementtherein is not completely precise. We restate it for convenience. Lemma 3.9 ([BRS, Thm. 7.24])
The functor
Top | CW : CW → { sets } is repre-sentable by a space R n ∈ CW , i.e. there is an equivalence of functors Ψ : [ . , R n ] ∼ = Top | CW ( . ) , where the squared brackets denote the homotopy classes of continuous maps.The definition of Ψ is via pullback of a certain topological triple x univ ∈ Top ( R n ) (the universal triple) over R n ; for B ∈ CW, [ f ] ∈ [ B , R n ] it is Ψ ( B )([ f ]) : = f ∗ x univ .76urther, it is shown in [BRS, Sec. 4] that the space R n has an homotopyaction of the so-called T-duality group O ( n , n , Z ) which is the group of 2 n × n -matrices that fix the form Z n ∋ ( a , . . . , a n , b , . . . , b n ) ∑ a i b i ∈ Z .So each element of O ( n , n , Z ) defines a homotopy class of maps R n → R n . Inparticular, the element n n n n ! ∈ O ( n , n , Z ) defines (the homotopy class of) a function T : R n → R n . T is constructedsuch that T ◦ T = id R n and such that the pullback T ∗ : Top ( R n ) → Top ( R n ) exchanges the underlying torus bundles . To be precise, the construction of T is such that if [( κ , ( P , E ) , ( b P , b E ))] ∈ Top ( R n ) is a topological triple, then T ∗ maps this triple to a triple [( κ ′ , ( P ′ , E ′ ) , ( b P ′ , b E ′ ))] , such that E ′ ∼ = b E and b E ′ ∼ = E .Note that b P × B E $ $ IIIIIIIII } } zzzzzzzzz b E × B P z z uuuuuuuuu ! ! DDDDDDDDD κ − o o b P " " DDDDDDDDDD b E × B E $ $ JJJJJJJJJJ z z tttttttttt P | | zzzzzzzzzz b E % % JJJJJJJJJJJ E y y ttttttttttt R n is not a topological triples as the inverse of κ does not satisfy the Poincar´econdition, but also note that b P × B E % % JJJJJJJJJ { { xxxxxxxxx b E × B P z z ttttttttt FFFFFFFFF ( κ − ) o o b P GGGGGGGGGG b E × B E % % JJJJJJJJJJJ y y ttttttttttt P { { b E % % KKKKKKKKKKKK E y y ssssssssssss R n This is meaningful, as the case of the groups G = R n , N = Z n is self-dual, i.e. G ∼ = b G , N ∼ = N ⊥ .
77s a topological triple, wherein the superscript bundles and isomorphism.We claim that T ∗ [( κ , ( P , E ) , ( b P , b E ))] = [(( κ − ) , ( b P , b E ) , ( P , b E ))] . (42)In fact, by [BRS, Prop. 7.4] the set of topological triples over R n with fixed torusbundles E ′ , b E ′ is a torsor over H ( R n , Z ) and by [BRS, Lem. 3.3] H ( R n , Z ) = R n which has underlying torusbundles E ′ ∼ = b E and b E ′ ∼ = E , hence equation (42) is valid.The properties of the map T which is called universal T-duality enables usto prove the statement indicated above: Theorem 3.6
Let G = R n and N = Z n . Then we have an equivalence of functors δ | CW : Top | CW ∼ = Dyn † | CW : τ | CW . In particular
Top im ( B ) = Top s ( B ) = Top as ( B ) = Top ( B ) , for all B ∈ CW . Proof :
It suffices to show that J ( B ) = id, for all B ∈ CW. J is a natural trans-formation, so Top ( R n ) J ( R n ) / / T ∗ (cid:15) (cid:15) Top ( R n ) T ∗ (cid:15) (cid:15) Top ( R n ) J ( R n ) / / Top ( R n ) is a commutative diagram. But by construction J ( B ) does not change the un-derlying pairs of the corresponding topological triples, for any B , so by thecommutativity of the diagram J ( R n ) at least does not change the underlyingpairs and underlying dual pairs of the corresponding triples. That J ( R n ) is theidentity, i.e. it does not change the equivalence class of the isomorphisms κ , fol-lows from the trivial H ( R n , Z ) -torsor structure of the set of topological tripleswith fixed torus bundles. Hence J ( R n ) = id.Now, let B ∈ CW and x ∈ Top ( B ) be any topological triple. By the universalproperty of x univ , there is f : B → R n such that x = f ∗ x univ and, by naturalityof J , J ( B )( x ) = J ( B )( f ∗ x univ ) = f ∗ ( J ( R n )( x univ )) = f ∗ ( x univ ) = x .So J ( B ) = id, for all B ∈ CW. (cid:4) Complex conjugation may be defined by taking H = l N and then defining the complexconjugate bundles and isomorphism by complex conjugation of the local transition functions andlocal isomorphisms.
78e end this section with a remark on homotopic deckers. Its content isbased on the fact that the functor Top | CW ∼ = [ . , R n ] is homotopy invariant. Remark 3.4
Let ( P , E ) be a pair over B ∈ CW , then ( P × [
0, 1 ] , E × [
0, 1 ]) is apair over B × [
0, 1 ] . Let ( ρ , P × [
0, 1 ] , E × [
0, 1 ]) be a dualisable dynamical triple –in other words ρ is a homotopy of dualisable deckers ρ and ρ on ( P , E ) – then theclasses [( ρ , P , E )] and [( ρ , P , E )] coincide. Proof :
As the the inclusions i t : B ֒ → B × [
0, 1 ] , t =
0, 1, are homotopic, wehave an equality τ ( B )([( ρ , P , E )]) = i ∗ τ ( B × [
0, 1 ])([( ρ , P × [
0, 1 ] , E × [
0, 1 ])])= i ∗ τ ( B × [
0, 1 ])([( ρ , P × [
0, 1 ] , E × [
0, 1 ])])= τ ( B )([( ρ , P , E )]) ,and the claim follows from δ ( B ) ◦ τ ( B ) = id. (cid:4) C ∗ -Dynamical Systems Let ( ρ , P , E ) be a dualisable dynamical triple and let us denote by F : = P × PU ( H ) K ( H ) the associated C ∗ -bundle. The decker ρ induces a G -action on F by [ x , K ] · g : = [ ρ ( x , g ) , K ] x ∈ P , K ∈ K ( H ) . This action defines another action α ρ of G on the C ∗ -algebra of sections Γ ( E , F ) such that ( Γ ( E , F ) , G , α ρ ) becomes a C ∗ -dynamical system. This action is given by ( α ρ g s )( e ) : = s ( e · gN ) · ( − g ) , e ∈ E , g ∈ G , s ∈ Γ ( E , F ) .In the same manner we obtain a dual C ∗ -dynamical system ( Γ ( b E , b F ) , b G , α ˆ ρ ) forthe associated C ∗ -bundle b F : = b P × PU ( L ( G / N ) ⊗ H ) K ( L ( G / N ) ⊗ H ) of the dualtriple ( ˆ ρ , b E , b P ) of ( ρ , E , P ) . The essence of this section is that we can establish anisomorphism of C ∗ -dynamical systems from the crossed product of the firstto the dual C ∗ -dynamical system (Thm. 3.8) (cid:0) G × α ρ Γ ( E , F ) , b G , b α ρ (cid:1) ∼ = (cid:0) Γ ( b E , b F ) , b G , α ˆ ρ (cid:1) .We are going to calculate the crossed product under a series of isomorphismswhich again will be a local calculation. We start with the description of thesimplest case, namely B being a point.In the situation of the trivial pair over the point B = {∗} the sections can beidentified with the continuous functions Γ ( triv. pair ) ∼ = C ( G / N , K ( H )) , See the section on crossed products on page 92 for notation. s : G / N → G / N × K ( H ) is uniquely given by a function f such that s ( z ) = ( z , f ( z )) . Then the action of g ∈ G on such a function f isobtained from ( α ρ g s )( z ) f = s ( z + gN ) · ( − g )= ( z + gN , f ( z + gN ) · ( − g )= ( z , µ ( − g , z + gN )( f ( z + gN )))= ( z , ( µ ( g , z )) − ( f ( z + gN )))= : ( z , ( α µ g f )( z )) (43)for the 1-cocycle µ determined by ρ .For the C ∗ -algebra of the dual trivial pair over the point we have Γ ( dual triv. pair ) ∼ = C ( b G / N ⊥ , K ( L ( N ⊥ , H ))) .It will be convenient at some point to deal with the Hilbert space L ( [ G / N , H ) rather than with L ( G / N ) ⊗ H . So we have to transform the cocycle ˆ µ ofthe dual decker ˆ ρ from equation (20) by Fourier transform F : L ( G / N ) → L ( [ G / N ) , ( F ∗ ˆ µ )( χ , ˆ z ) : = ( F ⊗ H ) ◦ ˆ µ ( χ , ˆ z ) ◦ ( F − ⊗ H ) ,and as in eq. (43) this gives us an action α F ∗ ˆ µ : b G → Aut ( C ( b G / N ⊥ , K ( L ( N ⊥ , H )))) by ( α F ∗ ˆ µχ f )( ˆ z ) : = ( F ∗ ˆ µ )( χ )( ˆ z ) − ( f ( ˆ z + χ N ⊥ )) , for χ ∈ b G , ˆ z ∈ b G / N ⊥ .The next lemma is a simple link between F ∗ ˆ µ and the left regular represen-tation λ [ G / N on L ( [ G / N ) . λ [ G / N ∈ Hom ( [ G / N , U ( L ( [ G / N ))) is a continuoushomomorphism. Lemma 3.10
There exists a continuous extension Λ ∈ Hom ( b G , U ( L ( [ G / N ))) such that [ G / N λ [ G / N / / ∩ (cid:15) (cid:15) U ( L ( [ G / N )) b G Λ rrrrrrrrrrr commutes. Proof :
Let β ∈ [ G / N . The Fourier transform F turns λ [ G / N ( β ) into the multipli-cation operator h− β , i = F − ◦ λ ⊥ ( β ) ◦ F ∈ U ( L ( G / N )) . Let σ : G / N → G be our chosen Borel section. We define an extension of β
7→ h− β , i by b G ∋ χ
7→ h χ , − σ ( ) i . This is a strongly continuous homomorphism, and it fol-lows that Λ ( χ ) : = F ◦ h χ , − σ ( ) i ◦ F − gives us the desired homomorphism Λ : b G → U ( L ( [ G / N )) . (cid:4) F ∗ ˆ µ , we have ( F ∗ ˆ µ )( χ , ˆ z ) = Ad ( Λ ( χ ) ⊗ H ) which is the adequateformula for the next theorem. Therein α α ⊥ denotes the natural isomor-phism [ G / N ∼ = N ⊥ ⊂ b G which is defined by h α ⊥ , g i : = h α , gN i . Theorem 3.7
Assume the considered dynamical triple over B = {∗} is dualisable.Then there is an isomorphism of C ∗ -dynamical systems (cid:16) G × α µ C ( G / N , K ( H )) , b G , c α µ (cid:17) ∼ = −→ (cid:16) C ( b G / N ⊥ , K ( L ( [ G / N , H ))) , b G , α F ∗ ˆ µ (cid:17) . Proof :
First note that naturally G × α µ C ( G / N , K ( H )) ∼ = C c ( G × G / N , K ( H )) k . k ⊂ L ( L ( G × G / N , H )) ,wherein f ∈ C c ( G × G / N , K ( H )) acts on F ∈ L ( G × G / N , H ) by ( f × F )( g , z ) = Z G ( µ ( − g )( z )) − ( f ( h , z − gN )) F ( g − h , z ) dh .We assume the dynamical triple to be dualisable, so we can lift µ to a unitary(Borel) cocycle µ : G × G / N → U ( H ) . Note that for K ∈ K ( H )) µ ( − g , z ) µ ( − g , z ) − ( K ) = K µ ( − g , z ) : H → H .We now define a unitary isomorphism u : L ( G × G / N , H ) → L ( b G , L ( [ G / N , H )) by the composition u : = shift ◦ ( Fourier trans. ) ◦ mult µ ( , ) , explicitely ( uF )( χ )( α ) = \ ( µ ( , ) F )( χ + α ⊥ , − α )= Z G × G / N h χ + α ⊥ , g i h− α , z i µ ( − g , z ) F ( g , z ) d ( g , z ) ,for χ ∈ b G , α ∈ [ G / N . The next step is to calculate u ( f × ) u − . This is straight-forward, but to keep the calculation readable we first introduce some shorthands: µ F ( g , z ) : = µ ( − g , z ) F ( g , z ) , f µ ( g , z ) : = f ( g , z ) µ ( g , z ) − and ˆ f the81ourier transform in the second, the G / N variable only. Now u ( f × F )( χ )( α )= Z G × G / N Z G h χ + α ⊥ , g i h− α , z i f ( h , z − gN ) µ ( − g , z ) F ( g − h , z ) dh d ( g , z )= Z G × G / N Z G h χ + α ⊥ , g i h− α , z i f µ ( h , z − gN ) µ F ( g − h , z ) dh d ( g , z )= Z G × G Z [ G / N h χ + α ⊥ , g i b f µ ( h , β ) h β , gN i c µ F ( g − h , ( − α ) − β ) d β d ( h , g )= Z [ G / N Z G b f µ ( h , β ) h χ + α ⊥ + β ⊥ , h i c µ F ( χ + α ⊥ + β ⊥ , ( − α ) − β ) dh d β = Z [ G / N b f µ ( χ + α ⊥ + β ⊥ , β ) c µ F ( χ + α ⊥ + β ⊥ , ( − α ) − β ) d β = Z [ G / N b f µ ( χ + γ ⊥ , γ − α ) | {z } c µ F ( χ + γ ⊥ , − γ ) | {z } d γ , (44) = : f µ ( χ )( α , γ ) = uF ( χ )( γ ) and we think of f µ as a continuous family of Hilbert-Schmidt operators f µ : b G → K ( L ( [ G / N , H )) , i.e. we do not decide in notation between the operator f µ ( χ ) and its integral kernel. From the definition of the kernel f µ we obtain f µ ( χ + β ⊥ )( α , γ ) = f µ ( χ )( α + β , γ + β ) , α , γ , β ∈ [ G / N , so the operator f µ ( χ ) satisfies the identity f µ ( χ + β ⊥ ) = ( λ [ G / N ( β ) − ⊗ ) f µ ( χ )( λ [ G / N ( β ) ⊗ ) ∈ K ( L ( [ G / N , H )) ,for the left regular representation λ [ G / N . We now use the chosen extension Λ ∈ Hom ( b G , PU ( L ( [ G / N , H ))) from Lemma 3.10 to define T µ : C c ( G × G / N , K ( H )) → C ( b G / N ⊥ , K ( L ( [ G / N , H ))) by ( T µ f )( χ N ⊥ ) : = ( Λ ( χ ) ⊗ ) f µ ( χ ) ( Λ ( χ ) − ⊗ )= Ad ( Λ ( χ ) ⊗ )) f µ ( χ ) . (45)It is now a lengthy but straight forward calculation to check that, firstly, T µ commutes with the ∗ -operation, i.e. T µ ( f × )( ˆ z ) = ( T µ f )( ˆ z ) ∗ , for ˆ z ∈ b G / N ⊥ and f × ( g , z ) = µ ( g )( z ) − ( f ( − g , z + gN ) ∗ ) . Secondly, T µ preserves the product, i.e. T µ ( f × f )( ˆ z ) = T µ f ( ˆ z ) T µ f ( ˆ z ) . Thirdly, T µ preserves the norm, i.e. k f × k = sup ˆ z ∈ b G / N ⊥ k T µ f ( ˆ z ) k .Moreover T µ has dense image, so it extends uniquely to a C ∗ -algebra iso-morphism T µ : G × α µ C ( G / N , K ( H )) → C ( b G / N ⊥ , K ( L ( [ G / N , H ))) .82t remains to show that T µ ( c α µχ f ) = α F ∗ ˆ µχ ( T µ f ) :By definition c α µχ is just the multiplication with the character χ , so theFourier transform gives us simply a shift in the argument by χ . We get T µ ( c α µχ f )( χ ′ N ⊥ ) = Ad ( Λ ( χ ′ ) ⊗ ) f µ ( χ ′ + χ )= Ad ( Λ ( χ ) − ⊗ ) Ad ( Λ ( χ ′ + χ ) ⊗ )( f µ ( χ ′ + χ ))= ( F ∗ ˆ µ )( χ )( χ ′ N ⊥ ) − T µ f ( χ ′ N ⊥ + χ N ⊥ )= α F ∗ ˆ µχ ( T µ f )( χ ′ N ⊥ ) . (46) (cid:4) The discussion in the previous theorem will serve as a description of thelocal situation of the general case. Let ( ρ , P , E ) be a dualisable dynamical triple,and let ( ˆ ρ , b P , b E ) be its dual. We defined the associated C ∗ -bundles F , b F above.Recall the definition of ˆ g ji out of φ ji (p. 39). A priori there need not exist acontinuous lift ϕ ji in the diagram b G (cid:15) (cid:15) U ji ϕ ji ooooooooooooooo ˆ g ji / / b G / N ⊥ ,but by Lemma A.8 we assume without restriction that our atlas is sufficientlyrefined such that ϕ ji exists. We define ϕ ′ ji ( u )( gN ) : = φ ji ( u )( g , 0 ) h ϕ ji ( u ) , g i − .Although the function u ϕ ′ ji ( u ) ∈ L ∞ ( G / N , U ( )) need not to be continu-ous, the function u Ad ( ϕ ′ ji ( u )) ∈ P L ∞ ( G / N , U ( )) is continuous by Lemma3.1, and we have the identity φ nm ( u )( g , hN ) = φ nm ( u )( g + h , 0 ) φ nm ( u )( h , 0 ) − = ϕ nm ( u )( g + h ) ϕ ′ nm ( u )( gN + hN ) ϕ nm ( u )( h ) − ϕ ′ nm ( u )( hN ) − = ϕ nm ( u )( g ) d ( ϕ ′ nm ( u ))( g , hN ) . Theorem 3.8
There is an isomorphism of C ∗ -dynamical systems (cid:16) G × α ρ Γ ( E , F ) , b G , b α ρ (cid:17) ∼ = −→ (cid:16) Γ ( b E , b F ) , b G , α ˆ ρ (cid:17) . Proof :
We generalise the proof of Theorem 3.7. The sections s ∈ Γ ( E , F ) are inone to one correspondence with families of functions s i ∈ C ( U i × G / N , K ( H )) , i ∈ I , which satisfy s i ( u , z ) = ζ ji ( u )( z ) − ( s j ( u , g ji ( u ) + z )) , u ∈ U ij , z ∈ G / N . (47)83t is G × α φ Γ ( E , F ) = C c ( G , Γ ( E , F )) k . k , and for each chart U i of the pair ( P , E ) we have induced restriction maps G × α ρ Γ ( E , F ) → G × α µ i C ( U i × G / N , K ( H )) ∪ ∪ C c ( G , Γ ( E , F )) → C ( U i , C c ( G × G / N , K ( H ))) . f f i The triple ( ρ , P , E ) is dualisable, so we can lift (if necessary after a refinement ofthe atlas) the cocycles µ i to unitary Borel cocycles µ i : U i → Z ( G , L ∞ ( G / N , U ( H ))) .For each such µ i we define an operator T i : C ( U i , C c ( G × G / N , K ( H ))) → C ( U i × b G / N ⊥ , K ( L ( [ G / N , H ))) as in equation (45) by T i f i ( u , χ N ⊥ ) : = ( T µ i ( u ) f i ( u ))( χ N ⊥ ) = Ad ( Λ ( χ )) f i ( u ) µ i ( u ) ( χ ) ,wherein we used the notation of equation (44). In view of equation (47) we areinterested in the relation between T i f i and T j f j on overlaps U ij . So let u ∈ U ji ,then f i ( u ) µ i ( u ) ( χ )( α , γ ) ( ) = Z G × G / N f i ( u )( g , z ) µ i ( u )( g , z ) − h χ + γ ⊥ , g i h γ − α , z i d ( g , z ) ( ) and def. of φ ji = Z G × G / N ζ ji ( u )( z ) ∗ f j ( u )( g , g ji ( u ) + z ) µ j ( u )( g , g ji ( u ) + z ) − ζ ji ( u )( gN + z ) φ ji ( u )( g , z ) h χ + γ ⊥ , g i h γ − α , z i d ( g , z )= Z G × G / N ζ ϕ ji ( u )( z ) ∗ f j ( u ) µ j ( u ) ( g , g ji ( u ) + z ) ζ ϕ ji ( u )( gN + z ) h ϕ ji ( u ) + χ + γ ⊥ , g i h γ − α , z i d ( g , z )= : ♣ .The notation f j ( u ) µ j ( u ) is as in the previous theorem, and ζ ϕ ji ( u )( z ) : = ζ ji ( u )( z ) ϕ ′ ji ( u )( z ) .Since G / N is compact and ζ ji ( u )( z ) ∈ U ( H ) it follows that ζ ϕ ji ( u )( ) v ∈ L ( G / N , H ) , for each v ∈ H . So it possesses a Fourier decomposition ζ ϕ ji ( u )( gN + z ) = Z [ G / N \ ζ ϕ ji ( u )( δ ) h− δ , z i h− δ ⊥ , g i d δ ζ ϕ ji ( u )( z ) ∗ = Z [ G / N \ ζ ϕ ji ( u )( ε ) ∗ h ε , z i d ε ,for some \ ζ ϕ ji ( u )( ) : [ G / N → L ( H ) . The calculation continues inserting thisinto the previous equation, ♣ = Z [ G / N Z [ G / N \ ζ ϕ ji ( u )( ε ) ∗ \ f j ( u ) µ j ( u ) ( ϕ ji ( u ) + χ + γ ⊥ − δ ⊥ , γ − δ − α + ε ) h γ − δ − α + ε , − g ji ( u ) i \ ζ ϕ ij ( u )( δ ) d ε d δ = Z [ G / N Z [ G / N \ ζ ϕ ji ( u )( ε ) ∗ f j ( u ) µ j ( u ) ( ϕ ji ( u ) + χ )( α − ε , γ − δ ) h γ − δ − α + ε , − g ji ( u ) i \ ζ ϕ ij ( u )( δ ) d ε d δ = Z [ G / N Z [ G / N \ ζ ϕ ji ( u )( α − ε ′ ) ∗ h− ε ′ , − g ji ( u ) i f j ( u ) µ j ( u ) ( ϕ ji ( u ) + χ )( ε ′ , δ ′ ) h δ ′ , − g ji ( u ) i \ ζ ϕ ij ( u )( γ − δ ′ ) d ε ′ d δ ′ .Now, bearing in mind that f i ( u ) µ i ( u ) ( χ )( α , γ ) is the kernel of the integral oper-ator f i ( u ) µ i ( u ) ( χ ) ∈ K ( L ( [ G / N , H )) , the last expression becomes f i ( u ) µ i ( u ) ( χ ) = η ji ( u ) ∗ f j ( u ) µ i ( u ) ( ϕ ji ( u ) + χ ) η ji ( u )= Ad ( η ji ( u ) ∗ )( f j ( u ) µ j ( u ) ( ϕ ji ( u ) + χ )) ∈ K ( L ( [ G / N , H )) ,wherein η ji ( u ) ∈ U ( L ( N ⊥ , H )) is the composition of the two operators L ( [ G / N , H ) → L ( [ G / N , H ) (48) F \ ζ ϕ ji ( u )( − ) ∗ F : β Z [ G / N \ ζ ϕ ji ( u )( β − α ) F ( α ⊥ ) d α and L ( [ G / N , H ) → L ( [ G / N , H ) (49) F (cid:16) β
7→ h β , − g ji ( u ) i F ( β ) (cid:17) .Note that both of these operators are in fact unitary, so η ji ( u ) is. The calculation85one so far can now give us the relation we are looking for T i f i ( u , χ N ⊥ ) = Ad ( Λ ( χ ) ⊗ ) f i ( u ) µ i ( u ) ( χ )= Ad ( Λ ( χ ) ⊗ ) Ad ( η ji ( u ) − ) f j ( u ) µ j ( u ) ( ϕ ji ( u ) + χ )= Ad (cid:0) ( Λ ( χ ) ⊗ ) η ji ( u ) − ( Λ ( ϕ ji ( u ) + χ ) − ⊗ ) (cid:1)| {z } = : ( F ∗ ˆ ζ ji )( u )( χ N ⊥ ) − Ad ( Λ ( ϕ ji ( u ) + χ ) ⊗ ) f j ( u ) µ j ( u ) ( ϕ ji ( u ) + χ )= ( F ∗ ˆ ζ ji )( u )( χ N ⊥ ) − T j f j ( u , ϕ ji ( u ) N ⊥ | {z } + χ N ⊥ ) . (50) = ˆ g ji ( u ) ∈ b G / N ⊥ ( F ∗ ˆ ζ ji )( u )( χ N ⊥ ) is in fact well defined for χ N ⊥ ∈ b G / N ⊥ , since Λ ( χ + γ ⊥ ) = Λ ( χ ) Λ ( γ ⊥ ) = Λ ( χ ) λ [ G / N ( γ ) , and the left regular representation operator λ [ G / N ( γ ) commutes with the convolution operator (48) and commutes with the multi-plication operator (49) up to h γ , − g ji ( u ) i , a U ( ) -valued multiple of the iden-tity. Hence the commutator of Ad ( Λ ( γ ⊥ ) ⊗ ) and Ad ( η ji ( u )) vanishes inPU ( L ( [ G / N , H )) .In view of equation (47), equation (50) shows that the family { T i f i } i ∈ I de-fines a section in a K ( L ( [ G / N , H )) -bundle over E with transition functions F ∗ ˆ ζ ji . Up to Fourier transform, this bundle is nothing but b F itself, for we find ( F ∗ ˆ ζ ji )( u )( ˆ z )= Ad (cid:16) ( F ⊗ ) ◦ ( h ϕ ji ( u ) + ˆ σ ( ˆ z ) , − σ ( ) i ⊗ ) ◦ ( F − ⊗ ) ◦ η ji ( u )( ˆ z ) ◦ ( F ⊗ ) ◦ ( h ˆ σ ( ˆ z ) , σ ( ) i ⊗ ) ◦ ( F − ⊗ ) (cid:17) = Ad (cid:16) ( F ⊗ ) ◦ ( h ϕ ji ( u ) + ˆ σ ( ˆ z ) , − σ ( ) i ⊗ )( λ G / N ( − g ji ( u )) ⊗ ) ζ ji ( u )( − ) ( ϕ ′ ji ( u )( − ) ⊗ )( h ˆ σ ( ˆ z ) , σ ( ) i ⊗ ) ◦ ( F − ⊗ ) (cid:17) = Ad (cid:16) ( F ⊗ ) ◦ ( h ϕ ji ( u ) + ˆ σ ( ˆ z ) , σ ( + g ji ( u ) − σ ( ) i ⊗ )( λ G / N ( − g ji ( u )) ⊗ ) ζ ji ( u )( − ) ( h ϕ ji ( u ) , − σ ( ) i ϕ ′ ji ( u )( − ) ⊗ ) ◦ ( F − ⊗ ) (cid:17) = Ad (cid:16) ( F ⊗ ) ◦ ( h ˆ σ ( ˆ g ji ( u ) + ˆ z ) , − σ ( ) i ⊗ )( λ G / N ( − g ji ( u )) ⊗ ) ζ ji ( u )( − ) ( φ ji ( u )( − σ ( ) , 0 ) ⊗ ) ◦ ( F − ⊗ ) (cid:17) = ( F ⊗ ) ◦ ˆ ζ ji ( u )( ˆ z ) ◦ ( F − ⊗ ) .86o the the family { T i f i } defines a section T f ∈ Γ ( b E , b F ) , and we have con-structed a map T : C c ( G , Γ ( E , F )) → Γ ( b E , b F ) which extends to an isomorphismof C ∗ -algebras T : G × α ρ Γ ( E , F )) → Γ ( b E , b F ) Then the relation T ( b α ρχ f ) = α ˆ ρχ ( T f ) is established by local calculation in thesame manner as over the point in equation (46). Thus T is in fact an isomor-phism of C ∗ -dynamical systems. (cid:4) Some Notation and Basic Lemmata
A.1 Groups
Let G be a Hausdorff locally compact abelian group and N some discrete, co-compact subgroup, i.e. G / N is compact. Lemma A.1 ( i ) The quotient map G → G / N has local sections. ( ii ) The quotient map G → G / N has a Borel section.
Proof : ( i ) N ⊂ G is discrete, i.e. there exists an open neighbourhood U of0 ∈ G such that U ∩ N = { } . Let + : G × G → G be the addition. + iscontinuous so + − ( U ) is an open neighbourhood of (
0, 0 ) ∈ G × G . So thereis an open neighbourhood V ⊂ G of 0 ∈ G such that V × V ⊂ + − ( U ) . Let W : = V ∩ ( − V ) . Then W is an open neighbourhood of 0 ∈ G , and for all x ∈ W and n ∈ N \{ } the sum x + n / ∈ W , for x ∈ W implies − x ∈ W and in case x + n ∈ W we would find ( x + n ) + ( − x ) = n ∈ U – a contradiction. There-fore W maps injectively to G / N , and as the quotient map is open it defines ahomoeomorphism from W to its image W / N . This defines a local section from W / N to G , and using addition in G / N we can move W / N all over G / N to geta local section in the neighbourhood of each point in G / N . ( ii ) This follows from compactness of G / N . For let s m : U m → G , m =
1, . . . , n be a family of local sections such that S nm = U m = G / N , then σ ( z ) : = s ( z ) , if z ∈ U , s ( z ) , if z ∈ U \ U ,... ... s n ( z ) , if z ∈ U n \ S n − m = U m defines a Borel section. (cid:4) The dual group of G is b G : = Hom ( G , U ( )) . With compact-open topologyit becomes again a Hausdorff, locally compact group, and if G is second count-able, then also b G is. For parings of a group and its dual we will use bracketnotation h χ , g i , h α , n i , · · · ∈ U ( ) , for g ∈ G , χ ∈ b G , n ∈ N , α ∈ b N .We recall part of the classical duality theorems [Ru]. Pontrjagin Dualitystates that the canonical map G to bb G is an isomorphism of topological groups.Moreover, if N ⊥ : = { χ ∈ b G | χ | N = } is the annihilator of N , then there is acanonical isomorphism [ G / N ∋ α (cid:0) g
7→ h α , gN i (cid:1) ∈ N ⊥ , and by the samemeans b G / N ⊥ ∼ = b N . Further, the dual group of a discrete group is compact andvice versa, so N ⊥ ⊂ b G is a discrete cocompact subgroup, thus the situationis completely symmetric under exchange of N , G by N ⊥ , b G . Let us denote theintegration of a (compactly supported, continuous) function f : G → C against88he Haar measure of G simply by R G f ( g ) dg . For the Fourier transform ˆ f of f we use the convention ˆ f ( χ ) : = R G h χ , g i f ( g ) dg , χ ∈ b G . It extends to anisomorphism L ( G ) ˆ → L ( b G ) . A.2 Group and ˇC ech Cohomology For a topological G -module M let us denote by C k cont ( G , M ) (resp. C k Bor ( G , M ) )the continuous (resp. Borel) maps G k → M , k =
0, 1, 2, . . . ( C ( G , M ) : = M ).They are topological spaces with the compact-open topology. The differential d : C k ? ( G , M ) → C k + ( G , M ) , given by d f ( g , . . . , g k + ) : = ( − ) k + f ( g , . . . , g k )+ k ∑ i = ( − ) i f ( g , . . . , g i + g i + , . . . , g k + )+ g · f ( g , . . . , g k + ) makes C ∗ ? ( G , M ) a cochain complex with cohomology groups H k ? ( G , M ) : = Z k ? ( G , M ) / B k ? ( G , M ) , for the cocycles Z k ? ( G , M ) : = ker ( d k ) and the boundaries B k ? ( G , M ) : = im ( d k − ) .Let U • = { U i | i ∈ I } be an open covering of a space B . By U i ... i n we denotethe intersection U i ∩ · · · ∩ U i n . Let F be any abelian sheaf and let ˇ C k ( U • , F ) : = ∏ F ( U i ... i k ) . The boundary operator δ : ˇ C k ( U • , F ) → ˇ C k + ( U • , F ) is given by ( δϕ ) i ... i k + : = ϕ i ... i k + | U i ik + − ϕ i i ... i k + | U i ik + + · · · + ( − ) k + ϕ i ... i k | U i ik + ,and we use the standard notation for the cohomology groups ˇ H k ( U • , F ) .We also use the notation A to denote the locally constant sheaf of continu-ous functions to A , for any abelian topological group A . A.3 The Unitary and the Projective Unitary Group
Let H be some infinite dimensional, separable Hilbert space with unitary groupU ( H ) which we equip with the strong (or equivalently weak) operator topol-ogy. We denote by Ad : U ( H ) → PU ( H ) : = U ( H ) /U ( ) the quotient map from the unitary onto the projective unitary group, and weendow the latter with the quotient topology.Let U ⊂ U ( H ) be any subset, e.g. U = U ( ) · and let M be some measurespace, e.g. G or G / N , then we denote by L ∞ ( M , U ) ⊂ U ( L ( M ) ⊗ H ) (51)89he set of unitary operators which are given by (equivalence classes of) mea-surable functions f : M → U which act as multiplication operators. In partic-ular L ∞ ( M , U ) has the subspace topology of U ( L ( M ) ⊗ H ) which is usuallyreferred as the weak topology on L ∞ ( M , U ) . The image of L ∞ ( M , U ) underAd : U ( L ( M ) ⊗ H ) → PU ( L ( M ) ⊗ H ) is denoted by P L ∞ ( M , U ) .Let us equip the Borel functions Bor ( G , U ) with the compact-open topology. Lemma A.2
The natural map
Bor ( G , U ) → L ∞ ( G , U ) is continuous. Proof :
A standard ε /3-argument. Let f α → f be a converging net in Bor ( G , U ) ,i.e. for each compact K ⊂ G and any w ∈ H we have k f α − f k K , w : = sup g ∈ K k f α ( g ) w − f ( g ) w k H → v ∈ L ( G , H ) k f α v − f v k L ( G , H ) → C c ( G ) are dense in L ( G ) . So for any v ∈ L ( G , H ) and ε > h i ∈ C c ( G ) , vectors w j ∈ H and numbers a ij ∈ C such that k v − ∑ Ni , j = a ij h i ⊗ w j k L ( G , H ) < ε /3.Choose K : = S Ni = supp h i ⊂ G and C : = + ∑ Ni , j = | a ij | R K | h i ( g ) | dg > α such that k f α − f k K , w j < ε N √ C , for all α > α and all j =
1, . . . , N .We can estimate now k f α v − f v k L ( G , H ) ≤ k f α ( v − N ∑ i , j = a ij h i ⊗ w j ) k L ( G , H ) + k ( f α − f )( N ∑ i , j = a ij h i ⊗ w j ) k L ( G , H ) + k f ( N ∑ i , j = a ij h i ⊗ w j − v ) k L ( G , H ) ≤ k f α k Op · ε /3 + (cid:16) Z K k N ∑ i , j = a ij h i ( g )( f α ( g ) w j − f ( g ) w j ) k H dg (cid:17) + k f k Op · ε /3 ≤ ε /3 + (cid:16) N N ∑ i , j = | a ij | Z K | h i ( g ) | k ( f α ( g ) w j − f ( g ) w j ) k H dg (cid:17) + ε /3 ≤ ε /3 + (cid:16) N N ∑ i , j = | a ij | Z K | h i ( g ) | ε N C dg (cid:17) + ε /3 ≤ ε /3 + p ε /9 + ε /3 = ε , 90or all α > α . (cid:4) For PU ( H ) -principal bundles we have the following well-known classifi-cation theorem. (See e.g. [Di, Thm. 10.8.4] for the first and of [PR] for thesecond statement; although therein it is not stated as below, we can carryover the proofs.) To state the theorem we introduce some notation. Let usdenote by Iso ( E ) the set of isomorphism classes of PU ( H ) -principal bundlesover E , and if P → E is a PU ( H ) -principal bundle, we denote by Aut ( P , E ) the group of bundle automorphisms (over the identity of E ). There is the sub-group Null ( P , E ) ⊂ Aut ( P , E ) which consists of all null-homotopic bundleautomorphisms. Theorem A.1
Let E be a paracompact Hausdorff space and P → E be any fixed PU ( H ) -principal bundle.1. PU ( H ) -bundles over E are classified by the second ˇC ech cohomology with valuesin the locally constant sheaf of continuous functions to U ( ) , Iso ( E ) ∼ = ˇ H ( E , U ( )) .
2. There is a short exact sequence → Null ( P , E ) → Aut ( P , E ) → ˇ H ( E , U ( )) → P → E is the trivial bundle P = E × PU ( H ) we iden-tify Aut ( P , E ) with the continuous functions C ( E , PU ( H )) . If we keep thisin mind, a corollary of the classification theorem of bundle automorphisms isthe following statement which is sometimes called homotopy commutativityof the projective unitary group. Corollary A.1
Let E be a paracompact Hausdorff space and f , g : E → PU ( H ) twocontinuous functions. Then there extists a continuous function V : E → U ( H ) suchthat f ( x ) g ( x ) = g ( x ) f ( x ) Ad ( V ( x )) , x ∈ E . Proof : x f ( x ) g ( x ) and x g ( x ) f ( x ) define the same ˇCech class. (cid:4) We do not give a proof of Theorem A.1, but we remark that it dependsheavily on the the fact that the unitary group U ( H ) is contractible. For thestrong topology on U ( H ) this is not difficult to prove and may be found in [Di,10.8]. We line out the proof for convenience. Assume H = L ([
0, 1 ]) , then let ϕ t : L ( t ) ∼ = L (
0, 1 ) be the isometric isomorphism defined by ϕ t ( f )( x ) : = √ t f ( tx ) , for t >
0. We define H : [
0, 1 ] × U ( L (
0, 1 )) → U ( L (
0, 1 )) by H ( U ) : = and ( H ( t , U )( f ))( x ) : = ((cid:0) ϕ − t ◦ U ◦ ϕ t ( f | ( t ) ) (cid:1) ( x ) , if 0 < x < t , f ( x ) , if t < x <
1, for t > H is a homotopy connecting the identity on U ( H ) and the constant func-tion with value ∈ U ( H ) , thus H is a contraction.91 emark A.1 For each fixed time slice t ∈ [
0, 1 ] H ( t , . ) is a group homomorphism.So U ( H ) is contractible as group Remark A.2
Theorem A.1 also holds when we replace the unitary group U ( H ) byany contractible, abelian (sub)group U ab ( H ) such that U ( ) ⊂ U ab ( H ) . I.e. the sec-ond ˇC ech cohomology ˇ H ( E , U ( )) also classifies PU ab ( H ) -bundles over paracom-pact Hausdorff spaces E and the first ˇC ech cohomology ˇ H ( E , U ( )) also classifies PU ab ( H ) -bundle automorphisms, for PU ab ( H ) : = U ab ( H ) /U ( ) . The next lemma shows that there exists a appropriate commutative versionU ( ) → U ab ( H ) → PU ab ( H ) of U ( ) → U ( H ) → PU ( H ) . Lemma A.3
There exists a contractible, commutative subgroup U ab ( H ) ⊂ U ( H ) such that U ( ) · ⊂ U ab ( H ) . Proof :
We let H = L (
0, 1 ) and U ab ( H ) : = L ∞ ([
0, 1 ] , U ( )) . It is a commuta-tive subgroup and U ( ) · ⊂ U ab ( H ) . The contraction H from above restrictsto h : = H | [ ] × U ab ( H ) : [
0, 1 ] × U ab ( H ) → U ab ( H ) and is given by h ( t , g )( s ) : = ( g ( t s ) , if s < t ,1, if s > t . (cid:4) A.4 Crossed Products
Let ( A , G , α ) be a C ∗ -dynamical system, i.e. A is some C ∗ -algebra equippedwith a (strongly) continuous action α : G → Aut ( A ) . Let ( π , H ) be some faith-ful representation of A . We embed C c ( G , A ) , the compactly supported contin-uous functions G → A , into L ( L ( G , H )) : define f × : L ( G , H ) → L ( G , H ) by ( f × F )( g ) : = Z G π ( α − g ( f ( h ))) F ( g − h ) dh ,for f ∈ C c ( G , A ) , F ∈ L ( G , H ) . The adjoint operator of f × is easily calculatedand is ( f × ) ∗ = f × × , wherein f × ( g ) : = α g ( f ( − g )) ∗ . Clearly f × has com-pact support, and f × is continuous (all α g have norm 1). So C c ( G , A ) is closedunder the × -operation. Furthermore one has f × ( f × F ) = ( f × f ) × F ,wherein ( f × f )( g ) : = Z G f ( h ) α h ( f ( g − h )) dh We use the same symbol × for different maps. C c ( G , A ) . Thus C c ( G , A ) ֒ → L ( L ( G , H )) is a ∗ -subalgebra. The crossed product of G and A is then defined as the norm com-pletion G × α A : = (cid:18) C c ( G , A ) k . k , × , × (cid:19) .(This is in fact well-defined, since the operator norm of f × is independent ofthe faithful representation ( π , H ) .)For χ ∈ b G , f ∈ C c ( G , A ) we set ˆ α χ ( f )( g ) : = h χ , g i f ( g ) which extends to astrongly continuous action ˆ α : b G → Aut ( G × α A ) , so ( G × α A , b G , ˆ α ) again de-fines a C ∗ -dynamical system. Going once more through the process of buildingthe crossed product gives a C ∗ -dynamical system ( b G × ˆ α ( G × α A ) , G , ˆˆ α ) , and akey statement in the analysis of crossed products is the following Takai DualityTheorem (see e.g. [Pe1]). Theorem A.2
There is an isomorphism of C ∗ -dynamical systems (cid:0) b G × ˆ α ( G × α A (cid:1) , G , ˆˆ α ) ∼ = (cid:0) A ⊗ C ∗ K ( L ( G )) , G , α ⊗ Ad ◦ ̺ (cid:1) , for the right regular representation ̺ : G → U ( L ( G )) , i.e. ( ̺ g f )( h ) : = f ( h + g ) and Ad ( ̺ g )( K ) = ̺ g K ̺ − g , for K ∈ K ( L ( G )) . A.5 Some Topology
We will sometimes use the word space as abbreviation for topological space.Let X , Y be topological spaces. By Bor ( X , Y ) we denote the set of Borel func-tions from X to Y . We endow this space with the compact-open topology, i.e.we define a basis of the topology by all sets of the form U K , V : = { f : X → Y | f ( K ) ⊂ V } for compact K ⊂ X and open V ⊂ Y . The subspace of contin-uous maps will be denoted by Map ( X , Y ) ⊂ Bor ( X , Y ) . We use the notation C ( X , Y ) for the continuous functions if we do not want to specify the topologyon it, or in case Y is a normed space, then we put the supremums norm on C ( X , Y ) . Recall the exponential law [Sch] for Map ( ., .. ) . Lemma A.4
Let X , Y , Z be topological spaces. Assume X and Y to be Hausdorffand Y locally compact. Then we have a homoeomorphism
Map ( X × Y , Z ) ∼ = Map ( X , Map ( Y , Z )) f (cid:0) x f ( x , ) (cid:1) .Let E → B be a surjective fibration and assume Y is locally compact andHausdorff. By use of the exponential law it is then immediate that Map ( Y , E ) → Map ( Y , B ) still has the homotopy lifting property with respect to all Haus-dorff spaces; we denote this property by T HLP . For contractible E the mapMap ( Y , E ) → Map ( Y , B ) : = { f ≃ const. } becomes a surjection having the T HLP . 93e now restrict ourselves to the particular case of E = U ( H ) , B = PU ( H ) .Since U ( H ) is Polish [ Ke ] the quotient map U ( H ) → PU ( H ) admits a Borelsection s : PU ( H ) → U ( H ) [Ke, Thm. 12.7].Let f ∈ Map ( Y , PU ( H )) be a map not homotopic to a constant map. TheBorel function s ∗ f = s ◦ f : Y → U ( H ) gives rise to a bijection of setsMap ( Y , U ( H )) → Map ( Y , U ( H )) · s ∗ f . g g · s ∗ f We turn this map into a homoeomorphism by defining the topology of the righthand side to be the image of the topology of the left hand side. ThenMap ( Y , U ( H )) · s ∗ f → Map ( Y , PU ( H )) · f .as well as the assembled map ∐ f ∈ π (cid:0) Map ( Y , U ( H )) · s ∗ f (cid:1) → ∐ f ∈ π (cid:0) Map ( Y , PU ( H )) · f (cid:1) becomes a surjection satisfying the T HLP . Here π : = { f : Y → PU ( H ) } is aset of representatives of the first homotopy group π ( Map ( Y , PU ( H ))) . If Y iscompact Map ( Y , PU ( H )) ⊂ Map ( Y , PU ( H )) is open, i.e.Map ( Y , PU ( H )) = ∐ f ∈ π (cid:0) Map ( Y , PU ( H )) · f (cid:1) .To consider non compact Y the following easy statement is helpful. Lemma A.5
Let Z be any (pointed) space, and let Z z denote the path-connectedcomponent of z ∈ Z. Then ∐ [ z ] ∈ π ( Z ) Z z → [ [ z ] ∈ π ( Z ) Z z = Zis a fibration.
Proof :
Give any test space X and a diagram X h / / i (cid:15) (cid:15) ∐ Z z (cid:15) (cid:15) X × I h / / Z we define the (unique) homotopy ˜ h : X × I → ∐ Z z that fits into the abovediagram simply by ˜ h ( x , t ) : = h ( x , t ) . One has to check that ˜ h is continuous. Forthis it is sufficient to show that ˜ h − ( Z z ) is open. Since h maps path-connectedcomponents into path-connected components we have h − ( Z z ) = h − ( Z z ) × I .Since h − ( Z z ) ⊂ X is open, it follows that ˜ h − ( Z z ) = h − ( Z z ) = h − ( Z z ) × I ⊂ X × I is open. (cid:4) ∐ f ∈ π (cid:0) Map ( Y , U ( H )) · s ∗ f (cid:1) → Map ( Y , PU ( H )) (52)is surjective and satisfies the T HLP for all locally compact Hausdorff spaces Y , e.g. Y = G × G / N . This implies the next Corollary A.2
Let U be a contractible Hausdorff space, and let µ : U → Map ( G × G / N , PU ( H )) be continuous. Then there exists a continuous lift µ : U → Bor ( G × G / N , U ( H )) such that Ad ∗ ◦ µ = µ , i.e. Ad ◦ µ ( u ) = µ ( u ) for all u ∈ U. Proof : (52) has the T HLP and U is contractible and Hausdorff. (cid:4) For the remainder of this paragraph we stick to compact Y , e.g. Y = G / N .We shall consider the map Ad ∗ : Bor ( Y , U ( H )) → Bor ( Y , PU ( H )) . Lemma A.6 Ad ∗ is continuous and open. Proof :
Note first that Ad : U ( H ) → PU ( H ) is continuous and open.Continuity: Let U K , V : = { g | g ( K ) ⊂ V , K ⊂ Y compact, V ⊂ PU ( H ) open } .Then Ad − ∗ ( U K , V ) = { f | f ( K ) ⊂ Ad − ( V ) } is open in Bor ( Y , U ( H )) .Openness: Let U K , W : = { f | f ( K ) ⊂ W , K ⊂ Y compact, W ⊂ U ( H ) open } .Then the inclusion Ad ∗ ( U K , W ) ⊂ { g | g ( K ) ⊂ Ad ( W ) } is obvious; we showequality. To do so it is sufficient to construct a Borel section s of U ( H ) → PU ( H ) such that s ( Ad ( W )) ⊂ W . PU ( H ) is separable. Take a countable denseset { x i } i ∈ N and a local trivialisation U ⊂ PU ( H ) of U ( H ) → PU ( H ) . Let U i : = U x i ⊂ PU ( H ) and V i : = Ad ( W ) ∩ U i , so S i V i = Ad ( W ) . It suffices toconstruct Borel sections s i : V i → W and puzzling them together by s ( z ) : = s ( z ) , if z ∈ V , s ( z ) , if z ∈ V \ V ,... ... s n ( z ) , if z ∈ V n \ S n − m = V m ... ... ,then s is Borel, since we put together a countable family. The sections s i : V i → W may be obtained by similar manners: Let h i : Ad − ( V i ) → V i × U ( ) be atrivialisation, and let W i : = h i ( Ad − ( V i ) ∩ W ) . Then W i ⊂ V i × U ( ) is open.Thus for each l ∈ L : = { l ∈ N | x l ∈ V i } there is ϕ l ∈ U ( ) and an openneighbourhood V li ∋ x l such that V li × { ϕ l } ⊂ W i . We define s i ( z ) : = h − i ( z , ϕ l ) , if z ∈ V l i , h − i ( z , ϕ l ) , if z ∈ V l i \ V l i ,... ... h − i ( z , ϕ l n ) , if z ∈ V l n i \ S n − m = V l m i ... ... ,95or a counting l , l , . . . of L . This completes the proof. (cid:4) Lemma A.7
If Y is compact then Ad ∗ : Bor ( Y , U ( H )) → Bor ( Y , PU ( H )) is alocally trivial principal fibre bundle with structure group Bor ( Y , U ( )) . Proof :
Let t : V → U ( H ) be a local section of U ( H ) → PU ( H ) for some open V ⊂ PU ( H ) . Then t ∗ : U V ∋ f t ◦ f ∈ Bor ( Y , U ( H )) is a local sectionon U V : = { f : Y → PU ( H ) | f ( Y ) ⊂ V } which is open, since Y is compact.We can cover the whole of Bor ( Y , PU ( H )) by translates of U V under the actionof Bor ( Y , PU ( H )) on itself. The lemma will be proven if we can show thatBor ( Y , PU ( H )) and Bor ( Y , U ( H )) /Bor ( Y , U ( )) are homoeomorphic. Take aBorel section σ : PU ( H ) → U ( H ) . Then σ ∗ : Bor ( Y , PU ( H )) → Bor ( Y , U ( H )) /Bor ( Y , U ( )) f [ σ ◦ f ] is easily seen to be a bijection such thatBor ( Y , U ( H )) Ad ∗ (cid:15) (cid:15) * * UUUUUUUUUUUUUUUU
Bor ( Y , PU ( H )) σ ∗ / / Bor ( Y , U ( H )) /Bor ( Y , U ( )) commutes. It follows that σ ∗ is a homoeomorphism, for the quotient map andAd ∗ are both continuous and open. (cid:4) We will use the above lemma in combination with the next.
Lemma A.8
Let P → M be a locally trivial fibre bundle, and B a paracompact space.Then for each covering { U i | i ∈ I } of B and maps ζ ij : U i ∩ U j → M, there exists arefinement { V ix | ix ∈ I × B } ( V ix ⊂ U i ) and continuous ζ ix , jy : V ix ∩ V jy → P suchthat the diagram P (cid:15) (cid:15) V ix ∩ V jy ζ ix , jy nnnnnnnnnnnnnn ζ ij | Vix ∩ Vjy / / Mcommutes.
Proof :
Without restriction we can assume that the covering { U i } i ∈ I is locallyfinite, so for each x ∈ X I x : = { k ∈ I |{ x } ∩ U k = ∅ } is finite. If x / ∈ U i , wedefine V ix : = ∅ . If x ∈ U i , then i ∈ I x . For each ζ ik ( x ) , k ∈ I x , choose a localtrivialisation M i , k , x ∋ ζ ik ( x ) of P → M . Then V ix : = T k ∈ I x ζ − ik ( M i , k , x ) ⊂ U i isopen. By construction the image of ζ ij | V ix ∩ V jy is contained in T k ∈ I x M i , k , x , andwe can compose ζ ij | V ix ∩ V jy with any local section, say M i , i , x → P , to define ζ ix , jy . (cid:4) ndex Symbol Page h . , . i [ . , . ]
19, 76 A A A ji α ji α ρ α µ , Aut B BA β ji δ g c i c τ χ , χ ji d Dyn 31Dyn † δ δ s , δ as , δ im E b E
31, 43 E dyn EA F F
46, 80ˆ F f µ G , G / N b G , b G / N ⊥ Γ , Γ
15, 79 γ ji g ji g ji g dyn ji H I κ κ i κ ai , κ bi κ σ , κ σ κ ˆ σ κ top κ top i K ( H ) ?? L ∞ l i , l i , λ i λ G λ G / N m i µ i µ i µ i µ dyn i N N ⊥ ω , ω i P , ( P , E ) P b P
31, 43 P top P dyn P H d Par 31PU ( H ) π ϕ , ϕ kji ϕ ∗ P , φ ji φ , ˆ φ ji φ dyn ji ψ , ψ kji q , Q G , Q G / N ρ ρ τ ρ ρ dyn σ σ T n , ˆ T n ( B ) T µ s , Top as im τ , τ ( B ) u (cid:13) ( H ) ab ( H ) U • , U i , U ji , . . . 18, 89 w ji Z k cont , Z k Bor , ˇ Z k ζ ji ζ ji ζ ji ζ ′ ji , ˆ ζ ′ ji ζ dyn ji d ζ dyn ji
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