Equivariant higher twisted K-theory of SU(n) for exponential functor twists
aa r X i v : . [ m a t h . K T ] J un EQUIVARIANT HIGHER TWISTED K -THEORY OF SU ( n ) FOR EXPONENTIAL FUNCTOR TWISTS
DAVID E. EVANS AND ULRICH PENNIG
Abstract.
We prove that each exponential functor on the categoryof finite-dimensional complex inner product spaces and isomorphismsgives rise to an equivariant higher (ie. non-classical) twist of K -theoryover G = SU ( n ). This twist is represented by a Fell bundle E → G ,which reduces to the basic gerbe for the top exterior power functor.The groupoid G comes equipped with a G -action and an augmentationmap G → G , that is an equivariant equivalence. The C ∗ -algebra C ∗ ( E )associated to E is stably isomorphic to the section algebra of a locallytrivial bundle with stabilised strongly self-absorbing fibres. Using a ver-sion of the Mayer-Vietoris spectral sequence we compute the equivarianthigher twisted K -groups K G ∗ ( C ∗ ( E )) for arbitrary exponential functortwists over SU (2), and also over SU (3) after rationalisation. Introduction
In three groundbreaking articles [13, 15, 14] Freed, Hopkins and Telemanproved a close connection between the Verlinde algebra of a compact Liegroup G and its twisted equivariant K -theory, where G acts on itself byconjugation. In case G is simply connected their theorem boils down tothe following statement: Let R k ( G ) be the Verlinde ring of positive energyrepresentations of the loop group LG at level k ∈ Z . Then the following R ( G )-modules are naturally isomorphic R k ( G ) ∼ = τ ( k ) K dim( G ) G ( G ) . (1)This identification turns into an isomorphism of rings if the left hand sideis equipped with the fusion product and the right hand side with the prod-uct induced by Poincar´e duality and the group multiplication. The repre-sentation theory of loop groups also dictates the fusion rules of sectors inconformal field theories associated to these groups. In joint work with Gan-non the first named author proved that it is in fact possible to recover thefull system of modular invariant partition functions of these CFTs from thetwisted K -theory picture [9, 10]. This approach has been particularly suc-cessful in the case of the loop groups of tori, where the modular invariantscan be expressed as KK -elements. Even exotic fusion categories, like theones constructed by Tambara-Yamagami have elegant descriptions in termsof K -theory as shown in the upcoming paper [11]. For a simple and simply connected Lie group G the classical equivarianttwists of K -theory over G are classified up to isomorphism by the equivariantcohomology group H G ( G ; Z ) ∼ = Z . The twist τ ( k ) in the FHT theorem cor-responds to ( k + ˇ h ( G )) times the generator, where ˇ h ( G ) is the dual Coxeternumber of G . There are several ways to represent the generator of H G ( G ; Z )geometrically: As a Dixmier-Douady bundle (a locally trivial bundle of com-pact operators) [21], as a bundle of projective spaces [4], in terms of (graded)central extensions of groupoids [13] or as a (bundle) gerbe [20, 23].From a homotopy theoretic viewpoint (and neglecting the group actionfor a moment) twisted K -theory is an example of a twisted cohomologytheory. If R denotes an A ∞ ring spectrum, then it comes with a space ofunits GL ( R ) and has a classifying space of R -lines BGL ( R ), which turnsout to be an infinite loop space for E ∞ ring spectra [3, 2]. In this situationthe twists of R -theory are classified by [ X, BGL ( R )]. If KU denotes a ringspectrum representing K -theory, then the group [ X, BGL ( KU )] splits off H ( X ; Z / Z ) × H ( X ; Z )equipped with the multiplication( ω , δ ) · ( ω , δ ) = ( ω + ω , δ + δ + β ( ω ∪ ω )) , where β denotes the Bockstein homomorphism. The twists classified by H ( X ; Z / Z ) can easily be included in the classical picture for example byusing graded central extensions as in [13] or graded projective bundles [4].However, it was already pointed out by Atiyah and Segal in [4] that thegroup [ X, BGL ( KU )] is in general more subtle than ordinary cohomol-ogy. In joint work with Dadarlat the second author found an operator-algebraic description of the twists of K -theory which covers the full group[ X, BGL ( KU )] and is based on locally trivial bundles of stabilised stronglyself-absorbing C ∗ -algebras [7]. This picture is also easily adapted to includegroups of the form [ X, BGL ( KU [ d ])], i.e. the twists of the localisation of K -theory away from an integer d .Motivated by the isomorphism (1) in the FHT theorem, the operator alge-braic model in the non-equivariant case [7] and the bundle gerbe descriptionof the basic gerbe developed by Murray and Stevenson [23] we introducehigher (i.e. non-classical) equivariant twists over G = SU ( n ) in this paper.Our construction takes an exponential functor F : V iso C → V iso C on the cate-gory of complex finite-dimensional inner product spaces and isomorphismsas input and produces an equivariant Fell bundle E → G . The groupoid G comes with an action of G and an augmentation map G → G , which is anequivariant equivalence, if G is equipped with the adjoint action.Our main examples of exponential functors are the top exterior powerfunctor V top and the full exterior algebra V ∗ . For F = (cid:0)V top (cid:1) ⊗ m with m ∈ N our construction reproduces the basic gerbe from [23]. A familyof non-classical equivariant twists we will sometimes focus on arises from QUIVARIANT HIGHER TWISTED K -THEORY OF SU ( n ) 3 F = ( V ∗ ) ⊗ m . Further examples of exponential functors and a classificationresult in terms of involutive R -matrices are discussed in [24].Each exponential functor F gives rise to a strongly self-absorbing C ∗ -algebra M ∞ F , which carries a canonical G -action and is isomorphic to C (with the trivial action) in the classical case and to an infinite UHF-algebrafor higher twists. The C ∗ -algebra C ∗ ( E ) associated to the Fell bundle E is a C ( G )-algebra that is stably C ( G )-isomorphic to the section algebra ofa locally trivial bundle A → G with fibre M ∞ F ⊗ K . Neglecting the G -action, this bundle is classified by a continuous map G → BGL (cid:0) KU [ d ] (cid:1) ,where d = dim( F ( C n )). At this point our work makes contact with [7]. Weconjecture that the classifying map agrees up to homotopy with the map τ nF : SU ( n ) → SU ≃ BBU ⊕ → BBU ⊗ [ d ] → BGL (cid:0) KU [ d ] (cid:1) considered in [24], but defer the proof to future work. We expect an anal-ogous statement to be true in an equivariant setting, but since the units ofgenuine G -equivariant ring spectra are a matter of current research in equi-variant stable homotopy theory (see for example [26, Ex. 5.1.17]) we willcome back to this question in future work as well.The G -equivariant K -theory of C ∗ ( E ) is a module over the localisation K G ( M ∞ F ) ∼ = R F ( G ) = R ( G )[ F ( ρ ) − ] of the representation ring R ( G ) at F ( ρ ), where ρ denotes the standard representation of SU ( n ). In generalthese R F ( G )-modules are computable via a spectral sequence similar to theone used in [21]. Our computations for SU (2) are summarised in Thm. 5.3.In this case the spectral sequence reduces to the Mayer-Vietoris sequence.We also compute the rationalised higher equivariant twisted K -theory of SU (3) for general exponential functor twists, see Thm. 5.14. Here we adaptthe approach developed in [1] to our situation. In particular, we identify therationalised chain complex on the E -page of the spectral sequence as theone computing Bredon cohomology of the maximal torus T of SU (3) withrespect to a certain local coefficient system.Even though to our knowledge equivariant exponential functor twists ofthe form studied here have not been considered in the literature before,similar exponential morphisms played a crucial role in [33]. Instead of alocalisation of KU , the ring spectrum considered by Teleman is KU [[ t ]], thepower series completion of K -theory and he shows that τ K dim( G ) G ( G ) ⊗ C [[ t ]]is a Frobenius algebra and therefore extends to a 2D topological field theoryfor admissible higher twists τ .We expect the same to be true for the equivariant higher twisted K -groups K G dim( G ) ( C ∗ ( E )) and find evidence for this conjecture in the followingremarkable properties of these groups (which are to be understood afterrationalisation for n = 3): • The spectral sequence collapses on the E -page for all exponentialfunctor twists. DAVID E. EVANS AND ULRICH PENNIG • The R F ( G )-module K G dim( G ) ( C ∗ ( E )) is a quotient of R F ( G ) by anideal. In particular, it carries a ring structure. • The local coefficient system in the SU (3)-case over the Lie algebra t of T is determined by a homomorphism π ( T ) → GL ( R F ( T ))similar to the one constructed in [1, Prop. 3.4], which is reminiscentof the appearance of the flat line bundles in [12, (3.4)].The paper is structured as follows: Section 2 contains preliminary ma-terial about Morita-Rieffel equivalences between C ∗ -algebras. Exponentialfunctors (Def. 2.2) are revisited as well.In Section 3 we describe the construction of the equivariant Fell bundle E → G from an exponential functor F in several steps: The groupoid G de-composes into three disjoint connected components G − , G and G + that arecompatible with composition. A saturated half-bundle is essentially a satu-rated Fell bundle over the subcategory G ∪ G + . In Section 3.1 we prove thetechnical result that any saturated half-bundle (Def. 3.1) extends uniquelyup to isomorphism to a saturated Fell bundle (Thm. 3.3). Moreover, ifthe half-bundle carries a G -action in an appropriate sense, then this actionextends uniquely to one on the Fell bundle (Cor. 3.7). We then focus onthe construction of the half-bundle E , + associated to an exponential func-tor F in Sec. 3.2 (see Lemma 3.10) and discuss the group action on it inSec. 3.3 (see in particular Cor. 3.12). Combining the results from all threesections we obtain a saturated G -equivariant Fell bundle E → G describingthe exponential functor twist over SU ( n ) associated to F .In Sec. 4 we look at the C ∗ -algebra C ∗ ( E ) associated to E and prove that itis a continuous C ( G )-algebra (see Lemma 4.2) with fibre C ∗ ( E g ) over g ∈ G Morita equivalent to M ∞ F (Lem. 4.6). Since C ∗ ( E ) satisfies the generalisedFell condition, it is stably isomorphic to a locally trivial bundle classified bya continuous map to BGL ( KU [ d − ]) where d = dim( F ( C )) by Cor. 4.7.The spectral sequence which allows us to compute the equivariant highertwisted K -groups is introduced in Cor. 4.9 in Sec. 4.1. Using results fromstrongly self-absorbing C ∗ -dynamical systems we then prove in Prop. 4.11in Sec. 4.2 that its terms are in fact modules over R F ( G ) ∼ = R ( G )[ F ( ρ ) − ]and so is K G ∗ ( C ∗ ( E )).The final section contains the computations of the equivariant highertwisted K -groups in the cases SU (2) (Sec. 5.1) and SU (3) (Sec. 5.2). Inthe first case the result is summarised in Thm. 5.3. For SU (3) we firstdetermine the differentials in Lemma 5.7 and compare the resulting chaincomplex with the one computing the Bredon cohomology of t in Sec. 5.2.1.This allows us to compute the rational equivariant higher twisted K -theoryin Thm. 5.14. Acknowledgements.
The second author would like to thank Dan Freedand Steffen Sagave for helpful discussions. Part of this work was completed
QUIVARIANT HIGHER TWISTED K -THEORY OF SU ( n ) 5 while the authors were staying at the Newton Institute during the pro-gramme “Operator algebras: subfactors and their applications”. Both au-thors would like to thank the institute for its hospitality. Their research wassupported in part by the EPSRC grants EP/K032208/1 and EP/N022432/1.2. Preliminaries
Bimodules and Morita-Rieffel equivalences.
In this section wecollect some well-known facts about Hilbert bimodules and Morita-Rieffelequivalences. This is mainly to fix notation. A detailed introduction toHilbert C ∗ -modules can be found in [18]. Let A, B be separable unital C ∗ -algebras. Definition 2.1. An A - B -bimodule is a right Hilbert B -module V togetherwith a ∗ -homomorphism ψ A : A → K ◦ B ( V ) , where K ◦ B ( V ) denotes the compact adjointable right B -linear operatorson V . An A - B -bimodule is called a (Morita-Rieffel) equivalence bimodule if V is full and ψ A is an isomorphism.Given a right Hilbert B -module V with inner product h· , ·i B we can asso-ciate a left Hilbert B -module V op to it in a natural way. The vector spaceunderlying V op is V , i.e. V equipped with the conjugate linear structure.For a given element v ∈ V we denote the corresponding element in V op by v ∗ . The left multiplication by b ∈ B is defined by b v ∗ = ( v b ∗ ) ∗ and the left B -linear inner product is h v ∗ , v ∗ i B = ( h v , v i B ) ∗ .The space hom B ( V, B ) of right B -linear adjointable morphisms is a leftHilbert B -module via the left multiplication ( b · ϕ )( v ) = bϕ ( v ) and the innerproduct h ϕ , ϕ i B = ϕ ◦ ϕ ∗ ∈ hom B ( B, B ) ∼ = B . The map V op → hom B ( V, B ) , v ∗
7→ h v, · i B provides a canonical isomorphism of left Hilbert B -modules and we willsometimes identify the two. Note that there is a conjugate linear bijection V → V op , v v ∗ (2)which satisfies ( v b ) ∗ = b ∗ v ∗ .The definition of A - B equivalence bimodules may seem asymmetric in A and B . It is actually not: Let V be an A - B equivalence bimodule. It carriesa left multiplication by a ∈ A defined by a v = ψ A ( a ) v and a left A -linearinner product given by h v , v i A = ψ − A ( v h v , · i B ) . With respect to this multiplication and inner product V is a full left Hilbert A -module. The rank 1 operator h · , v i A v agrees with the right multipli-cation by h v , v i B . Since V is full, the compact left A -linear operators DAVID E. EVANS AND ULRICH PENNIG therefore agree with B , but the multiplication is reversed, i.e. we obtain anisomorphism ψ B : B op → K A ◦ ( V )that sends b to right multiplication by b . Thus, we could alternatively definean A - B equivalence bimodule as a full left Hilbert A -module together withan isomorphism ψ B as above.If V is an A - B equivalence bimodule, then V op is a full left Hilbert B -module. Let ψ op A : A op → K B ◦ ( V op )be the ∗ -homomorphism given by ψ op A ( a )( v ∗ ) = ( ψ A ( a ∗ ) v ) ∗ . Conjugation by v v ∗ induces a conjugate linear isomorphism K ◦ B ( V ) ∼ = K B ◦ ( V op ). Fromthis we deduce that ψ op A is a (linear) ∗ -isomorphism. Thus, V op is a B - A equivalence bimodule.Note that the left A -linear and right B -linear inner product on an A - B -equivalence bimodule V satisfy the compatibility condition v h v , v i B = h v , v i A v (3)for all v , v , v ∈ V .Let A , B and C be separable unital C ∗ -algebras and let V be an A - B equivalence bimodule and W be a B - C equivalence bimodule. The tensorproduct over B gives an A - C equivalence bimodule that we will denote by V ⊗ B W .
For details about this construction we refer the reader to [18, Chap. 4] or[25]. The left A -linear inner product provides an A - A bimodule isomorphism h· , ·i A : V ⊗ B V op → A .
Similarly, h· , ·i B : V op ⊗ A V → B is a bimodule isomorphism as well. Con-cerning the opposite bimodule of a tensor product, there is a canonicalisomorphism ( V ⊗ B W ) op ∼ = W op ⊗ B V op (4)given on elementary tensors by ( v ⊗ w ) ∗ w ∗ ⊗ v ∗ .Let G be a compact group and let α : G → Aut( B ) be a continuous actionof G on B , where Aut( B ) is equipped with the pointwise-norm topology. Wewill call B a G -algebra for short. A G -equivariant right Hilbert B -module [17, Def. 1 and Def. 2] is defined to be a right Hilbert B -module V togetherwith an action of G (denoted by g · v for g ∈ G, v ∈ V ) that satisfiesa) g · ( vb ) = ( g · v ) α g ( b ),b) h g · v, g · w i B = α g ( h v, w i B ),c) ( g, v ) g · v is continuous.If V is a G -equivariant right Hilbert B -module, then V op equipped withthe action g · v ∗ = ( g · v ) ∗ is a G -equivariant left Hilbert B -module. Thegroup G acts continuously on the C ∗ -algebra K ◦ B ( V ) by conjugation. If A denotes another G -algebra, then a G -equivariant A - B -bimodule is an A - B -bimodule V where the structure map ψ : A → K ◦ B ( V ) is G -equivariant. QUIVARIANT HIGHER TWISTED K -THEORY OF SU ( n ) 7 Exponential functors.
Let V fin C be the category of finite-dimensionalcomplex inner product spaces and linear maps and denote by V iso C ⊂ V fin C the subgroupoid with the same objects but unitary isomorphisms as itsmorphisms. The higher twists we are going to construct will depend onthe choice of an exponential functor on V iso C . In the context of higher twiststhese were first considered in [24], which also contains a classification of thoseexponential functors that arise from restrictions of polynomial exponentialfunctors on V fin C in terms of involutive solutions to the Yang-Baxter equation(involutive R -matrices). The following definition is taken from [24, Def. 2.1]and we refer the reader to that paper for a detailed description of the threeconditions a), b) and c) stated below. Definition 2.2. An exponential functor on V fin C (resp. V iso C ) is a tripleconsisting of a functor F : V fin C → V fin C (resp. F : V iso C → V iso C ) together withnatural unitary isomorphisms τ V,W : F ( V ⊕ W ) → F ( V ) ⊗ F ( W )and ι : F (0) → C that satisfy the following conditionsa) F preserves adjoints,b) τ is associative,c) τ is unital with respect to ι .For an exponential functor F (on V fin C or V iso C ) let d ( F ) = dim( F ( C )). Wedefine the dimension spectrum of F to beDim( F ) := { dim( F ( V )) | V ∈ obj( V iso C ) } = { d ( F ) n | n ∈ N } . The exterior algebra functor F ( V ) = V ∗ V provides a natural example ofan exponential functor. The symmetric algebra Sym ∗ ( V ) of a vector space V comes with natural transformations τ and ι as above. It is, however,ruled out by the fact that Sym ∗ ( V ) is infinite-dimensional. The exterioralgebra functor can be modified as follows: Let W be a finite-dimensionalinner product space and consider F W ( V ) = ∞ M k =0 W ⊗ k ⊗ ^ k V As outlined in [24, Sec. 2.2] this provides a polynomial exponential functor F W : V fin C → V fin C . 3. Higher twists via Fell bundles
In this section we will consider a groupoid G that carries an action of G = SU ( n ) and comes with a surjection G → G that is equivariant withrespect to the conjugation action of G on itself. In fact, G will be Moritaequivalent to G . We will then construct a Fell bundle π : E → G such thatits total space E comes with an action of G such that π is equivariant. Thegroupoid G decomposes into a subcategory G , + and we will construct theanalogue of a saturated Fell bundle over this category first, before extending DAVID E. EVANS AND ULRICH PENNIG it to all of G . To achieve this we will need the extension theorem proven inthe next section.3.1. An extension theorem for saturated Fell bundles.
In this sectionwe consider the following situation: Let G be a topological groupoid withobject space G (0) . Suppose that we have a decomposition G = G − ∪ G ∪ G + (5)into disjoint open and closed subspaces. Let G (2) be the space of composablearrows. For U , V ⊂ G define
U · V = n g · g ∈ G | ( g , g ) ∈ G (2) and g ∈ U , g ∈ V o and U − = (cid:8) g − ∈ G | g ∈ U (cid:9) . We will assume that the decomposition (5)satisfies the following conditions( G + ) − = G − (6)( G ) − = G (7) G + · G + ⊆ G + (8) G · G + = G + (9) G + · G = G + (10) G · G = G (11)Since the identities on the objects of G are fixed points of the inversion andthe decomposition is disjoint, we obtain from (6) and (7) that they must becontained in G . Therefore (10) is actually equivalent to G + · G ⊆ G + andlikewise for (9). Taking inverses we also obtain G − · G − ⊆ G − G · G − = G − G − · G = G − Definition 3.1.
Let A be a separable unital C ∗ -algebra. A saturated (Fell)half-bundle is given by the following data: A Banach bundle E , + → G , + with the property that E , + | G = G × A and a continuous multiplication map µ : E (2)0 , + → E , + where E (2)0 , + = n ( e , e ) ∈ E , + | ( π ( e ) , π ( e )) ∈ G (2) o ⊂ ( E , + ) is equipped with the subspace topology. This data has to satisfy the follow-ing conditions: QUIVARIANT HIGHER TWISTED K -THEORY OF SU ( n ) 9 a) The multiplication µ is bilinear and associative. It extends the canonicalone on E , + | G = G × A and fits into a commutative diagram E (2)0 , + E , + G (2)0 , + G , + π × π µ π in which the lower horizontal map is the groupoid multiplication. Wewill use the abbreviated notation e · e := µ ( e , e ).b) There is a continuous inner product h · , · i A : E , + × G , + E , + → A × G (0) that is right A -linear with respect to the multiplication ( e, a ) e · a induced by µ with π ( e ) = g ∈ G , + and π ( a ) = id s ( g ) ∈ G . It fits intothe commutative diagram E , + × G , + E , + A × G (0) G , + G (0) h · , · i A π πs and restricts to h ( a , g ) , ( a , g ) i A = ( a ∗ a , s ( g )) for ( a i , g ) ∈ E , + | G . It iscompatible with the norm in the sense that kh e, e i A k = k e k (12)and turns each fibre ( E , + ) g into a right Hilbert A -module. The leftmultiplication ( a, e ) a · e with π ( e ) = g and π ( a ) = id r ( g ) is compactadjointable with respect to this inner product with adjoint given by a ∗ .Moreover, this left multiplication induces a ∗ -isomorphism ψ A, g : A → K ◦ A (cid:16) ( E , + ) g (cid:17) between A and the compact A -linear operators on each fibre. In otherwords, each fibre ( E , + ) g is an A - A equivalence bimodule.c) The Hilbert A -bimodule structure on the fibres is compatible with themultiplication in the sense that µ induces an A - A bimodule isomorphism( E , + ) g ⊗ A ( E , + ) g ( E , + ) g g µ ∼ = for each composable pair ( g , g ) ∈ G (2)0 , + . Remark 3.2.
Note that Def. 3.1 c) and (12) imply that the norm on E , + is submultiplicative in the sense that k e · e k ≤ k e k · k e k . Theorem 3.3.
Let ( E , + , µ, h · , · i A ) be a saturated half-bundle. There is asaturated Fell bundle π : E → G with the property that E| G , + = E , + , themultiplication of E restricts to the one of E , + and for e , e ∈ E g we have e ∗ · e = h e , e i A . Moreover, π : E → G is unique up to (a canonical)isomorphism of Fell bundles.Proof.
Let inv :
G → G be given by inv( g ) = g − . We first extend E , + over G − by defining π : E − → G − as E − = (inv ∗ E + ) op . This means that we have( E − ) g = ( E + ) op g − fibrewise for all g ∈ G − . Let E = E − ∪ E , + . Together withthe canonical quotient map to G this is a Banach bundle. The conjugatelinear bijection e e ∗ from (2) yields a well-defined ∗ -operation on E . Itfits into the commutative diagram E EG G ∗ π π inv Next we have to extend the multiplication map µ to all of E , i.e. we have toconstruct a bimodule isomorphism µ : E g ⊗ A E g → E g g for ( g , g ) ∈ G (2) .Depending on which subset g , g and g g are contained in, there are sixcases to consider: g g g · g − +3 + − − − + +5 − + − − − − A +, respectively − , refers to the case that the groupoid element is in G , + ,respectively G − . We need the relation( e · e ) ∗ = e ∗ · e ∗ to hold in a Fell bundle. This implies that if we have defined the multiplica-tion in case k , then we have fixed it in case (7 − k ) as well for k ∈ { , . . . , } .This reduces the number of cases to consider to the first three.We will use the following graphical representation of groupoid elements:A morphism in G , + will be drawn as an arrow pointing right, a morphismin G − corresponds to an arrow pointing left. Let g , g ∈ G (2) . If g · g endsup in G , + , then the concatenation of the two corresponding arrows will endin a point to the right of the base of g . Similarly, a composition endingup in G − will end up in a point left of the base of g . Cases 1 , Contrary to the usual notation of morphisms we will draw the arrows from thecodomain to the domain. This way the order of composition agrees with the compositionof the arrows.
QUIVARIANT HIGHER TWISTED K -THEORY OF SU ( n ) 11 • • • g g • • • g g • • • g g The multiplication is already defined in case 1. For case 2, let ( g , g ) ∈ G (2) with g ∈ G , + , g ∈ G − , g · g ∈ G , + . Observe that g = g g · g − is adecomposition of g into elements that are contained in G , + . This can beeasily read off from the above graphical representation. Since µ : ( E , + ) g g ⊗ A ( E , + ) g − → ( E , + ) g is an isomorphism, we can extend µ by defining it to be the upper horizontalarrow in the diagram below, in which the vertical arrow on the right handside is given by right multiplication:( E , + ) g ⊗ A ( E − ) g ( E , + ) g g ( E , + ) g g ⊗ A ( E , + ) g − ⊗ A ( E , + ) op g − ( E , + ) g g ⊗ A A µµ ⊗ id ∼ = id ⊗ h· , ·i A ∼ = ∼ = If we label the arrows by Fell bundle elements instead of groupoid mor-phisms, this definition will graphically be represented as follows: • • • e e ∗ • • • e ′ e ′ e ∗ where e ′ · e ′ = e and the inner product is used to replace the loop on theright hand side with an element in A .Consider case 3, i.e. ( g , g ) ∈ G (2) with g ∈ G , + , g ∈ G − , g · g ∈ G − .Using (4) and the multiplication µ we obtain an isomorphism µ op : ( E , + ) op g ⊗ A ( E , + ) op( g g ) − → ( E , + ) op g − and we can extend the multiplication to this case using the upper horizontalarrow in the diagram below:( E , + ) g ⊗ A ( E − ) g ( E − ) g g ( E , + ) g ⊗ A ( E , + ) op g ⊗ A ( E , + ) op( g g ) − A ⊗ ( E , + ) op( g g ) − µ id ⊗ µ op ∼ = h· , ·i A ⊗ id ∼ = ∼ = Graphically this definition is represented as follows: • • • e e ∗ • • • e ( e ′ ) ∗ ( e ′ ) ∗ i.e. we decompose e ∗ = ( e ′ ) ∗ · ( e ′ ) ∗ for some ( e ′ ) ∗ ∈ ( E , + ) op g and ( e ′ ) ∗ ∈ ( E − ) g g and define e · e ∗ = h e , e ′ i A · ( e ′ ) ∗ . This finishes the extension of the multiplication map µ .Next we have to prove that the extended multiplication is still associative.Let g , g , g ∈ G such that ( g , g ) ∈ G (2) and ( g , g ) ∈ G (2) . Let e i ∈ E g i for i ∈ { , , } . We have to show that( e · e ) · e = e · ( e · e )Each g i could be in G , + or in G − . Thus, if we neglect the compositionsfor a moment, this leaves us with six cases to consider. However, the aboveequality implies e ∗ · ( e ∗ · e ∗ ) = ( e ∗ · e ∗ ) · e ∗ . Therefore we can without loss of generality assume that g ∈ G , + . Thediagrams of all remaining cases are shown in Figure 1. To prove the asso-ciativity condition in each case we make the following observations:The given multiplication µ on G , + is fibrewise a bimodule isomorphism.Thus, whenever we have to decompose an element of e ∈ E , we may withoutloss of generality assume that it is maximally decomposed with respect tothe diagram. This means that in terms of the graphical representation wemay make the following replacements: • • • e • • • e e • • • • e ′ • • • • e e e where e = e · e in the first case and e ′ = e · e · e in the second. Notethat associativity of µ over G , + ensures that we may drop the brackets inthe second case. Likewise, we may assume the analogous decomposition forthe mirror images of these diagrams with arrows pointing to the left.By our definition of the extension of µ to G − associativity is also satisfiedin diagrams that have one of the following forms: • • • • • • • • • • • • QUIVARIANT HIGHER TWISTED K -THEORY OF SU ( n ) 13 • • • • g g g • • • • g g g • • • • g g g • • • • g g g • • • • g g g • • • • g g g • • • • g g g • • • • g g g • • • • g g g • • • • g g g • • • • g g g • • • • g g g Figure 1.
The 12 compositions to consider in the proof of associativity.Using this it follows that associativity holds in the cases 1, 2, 5 and 8 inFig. 1. The multiplication isomorphism µ : ( E , + ) g ⊗ A ( E , + ) g → ( E , + ) g g preserves inner products, hence we obtain h e · e , e ′ · e ′ i A = h e · h e , e ′ i A , e ′ i A , which is diagrammatically represented by the following relation: • • • e · e ( e ′ · e ′ ) ∗ • • • e e ( e ′ ) ∗ ( e ′ ) ∗ Again an analogous relation is true for the mirror images of the above di-agrams. Our observation shows we can drop the brackets in expressionsrepresented by diagrams of the form depicted on the right hand side. Thisimplies that associativity holds in the cases 3, 4, 6, 7 in Fig. 1.The last relation needed is the compatibility of the two inner products ineach fibre. Let e , e , e ∈ ( E , + ) g for some g ∈ G , + . Then (3) implies inour context that: e h e , e i A = h e , e i A e ,e ∗ h e , e i A = h e , e i A e ∗ . Expressed graphically this means that associativity holds for the diagrams: • • e e ∗ e • • e ∗ e e ∗ Using this relation it follows that associativity holds in the cases 9, 10, 11and 12 as well. Since the computations are slightly more involved than inthe previous cases, we give the details for diagram 10. Let e ∗ ∈ ( E − ) g , e ∈ ( E , + ) g and e ∗ ∈ ( E − ) g . Let e = e · e and e ∗ = e ∗ · e ∗ · e ∗ bethe maximal decompositions of e and e . We have( e ∗ · e ) · e ∗ = hh e , e i A e , e i A e ∗ · e ∗ e ∗ · ( e · e ∗ ) = e ∗ h e · e , e · e i A e ∗ and by our considerations above we obtain: e ∗ h e · e , e · e i A e ∗ = e ∗ h e h e , e i A , e i A e ∗ = h e , e h e , e i A i A e ∗ · e ∗ = h e , e i A h e , e i A e ∗ · e ∗ = hh e , e i A e , e i A e ∗ · e ∗ The other cases are similar. This finishes the proof of associativity.Thus, we obtain a Banach bundle
E → G with a continuous, bilinear,associative multiplication µ and a compatible continuous conjugate linearinvolution ∗ : E → E . Our definition implies that e ∗ · e = h e, e i A for all e ∈ E . This implies the C ∗ -norm condition k e ∗ e k = kh e, e i A k = k e k ,which also ensures that the norm is submultiplicative for all e ∈ E . Therefore E → G defines a saturated Fell bundle.To address the question about uniqueness let
F → G be another saturatedFell bundle that satisfies the conditions in the theorem. In particular, we
QUIVARIANT HIGHER TWISTED K -THEORY OF SU ( n ) 15 have F , + = E , + . Therefore the involution yields a (linear!) isomorphismof Banach bundles Θ : F − → ( E + ) op = E − The relation ( e · e ) ∗ = e ∗ · e ∗ shows that Θ has to intertwine the restrictions µ F − : F − ⊗ A F − → F − and µ E − . The relation h f ∗ , f ∗ i A = h f , f i A for all f ∗ , f ∗ ∈ ( F − ) g implies that Θ preserves the inner product as well. We havealready seen above that our extension of the multiplication map was forcedupon us by associativity considerations, the fact that µ induces fibrewisebimodule isomorphisms and the relation f ∗ · f = h f , f i A for f , f ∈ F g .Consequently, if we extend Θ by the identity on F , + = E , + it yields anisomorphism of Fell bundles F → E over G . (cid:3) Remark 3.4.
Let A be a unital separable C ∗ -algebra and let MR ( A ) bethe 2-groupoid that has A as its objects, the A - A equivalence bimodules as1-morphisms and bimodule isomorphisms as 2-morphisms . The groupoid G is a 2-groupoid with just identity 2-morphisms. If we forget about thetopology, then a saturated Fell bundle is a functor E : G → MR ( A ) , whereas saturated half-bundles correspond to functors E : G , + → MR ( A ) . Theorem 3.3 can be rephrased by saying that the restriction functorres : F un ( G , MR ( A )) → F un ( G , + , MR ( A ))induced by the inclusion G , + → G is an isomorphism of functor categories.This seems to suggest that there should be a proof of Theorem 3.3 basedon category theory. However, a first step would require identifying the righttopology on MR ( A ) to obtain a bijection between saturated Fell bundlesand continuous functors.Since we want to construct an equivariant Fell bundle from an equivari-ant half-bundle, we need to understand group actions on both structures.Restricting to the following kind of actions is natural in this context: Definition 3.5.
Let G be a compact group and let G be a groupoid witha decomposition as in (5) that has the properties (6) – (11). We call anaction of G on G by groupoid automorphisms admissible if it preserves thedecomposition from (5), i.e. each group element yields a homeomorphism G − → G − and similarly for G and G + , respectively. Definition 3.6.
Let G be a compact group that acts admissibly on G andlet A be a separable unital G -algebra. A G -equivariant saturated half-bundle is a saturated half-bundle E , + carrying a continuous G -action such that theprojection map E , + → G , + is equivariant and the following properties hold: The notation MR is for Morita-Rieffel. a) On E = E , + | G = G × A the action restricts to the diagonal action of G on G and A .b) The multiplication map µ : E (2)0 , + → E , + is G -equivariant (where the do-main is equipped with the diagonal G -action) and the inner productsatisfies h g · e , g · e i A = α g ( h e , e i A )for all g ∈ G . Corollary 3.7.
Suppose that G carries an admissible action by a compactgroup G . Let ( E , + , µ, h · , · i A ) be a G -equivariant saturated half-bundle. Let E be the extension of E , + to a saturated Fell bundle as in Thm. 3.3.Then the G -action on E , + extends to a continuous G -action on E insuch a way that the multiplication map and the projection π : E → G areequivariant and g · e ∗ = ( g · e ) ∗ for all e ∈ E and g ∈ G . This extension isunique.Proof. The condition g · e ∗ = ( g · e ) ∗ uniquely fixes the group action on E − and has all properties stated in the corollary. (cid:3) Construction of the Fell bundle over SU ( n ) . The groupoid G al-luded to in the introduction to this section is now constructed as follows:Let G = SU ( n ), T ⊂ C the unit circle and let Z = T \ { } ∼ = (0 , g ∈ G denote the set eigenvalues of g (in its standard representation on C n )by EV( g ). Let Y be the space Y = { ( g, z ) ∈ G × Z | z / ∈ EV( g ) } . There is a canonical quotient map π : Y → G . The groupoid G is now givenby the fibre product Y [2] of Y with itself over G , i.e. G = Y [2] = { ( y , y ) ∈ Y × Y | π ( y ) = π ( y ) } equipped with the subspace topology . Note that we can identify this spacewith Y [2] = { ( g, z , z ) ∈ G × Z × Z | z i / ∈ EV( g ) for i ∈ { , }} . Since Z is homeomorphic to an open interval via e : (0 , → Z with e ( ϕ ) =exp(2 πi ϕ ) we can equip it with a total ordering by defining z = e ( ϕ ) ≥ z = e ( ϕ ) if and only if ϕ ≥ ϕ . Now we can decompose Y [2] into disjointsubspaces Y [2] = Y [2]+ ∪ Y [2]0 ∪ Y [2] − with Y [2]+ = { ( g, z , z ) ∈ Y [2] | z > z and ∃ λ ∈ EV( g ) , z > λ > z } ,Y [2]0 = { ( g, z , z ) ∈ Y [2] | ∄ λ ∈ EV( g ) , max( z , z ) > λ > min( z , z ) } ,Y [2] − = { ( g, z , z ) ∈ Y [2] | z < z and ∃ λ ∈ EV( g ) , z < λ < z } . We will view an element ( g, z , z ) ∈ Y [2] as a morphism from ( g, z ) to ( g, z ). Thus,the composition is ( g, z , z ) · ( g, z , z ) = ( g, z , z ). QUIVARIANT HIGHER TWISTED K -THEORY OF SU ( n ) 17 Fix a (continuous) exponential functor (
F, τ, ι ) on V iso C as in Def. 2.2.Consider the standard representation of G on C n and let M F = End( F ( C n )).Let M ∞ F be the UHF-algebra given by the infinite tensor product M ∞ F = ∞ O i =1 M F . We will construct a saturated half-bundle E , + over G , + = Y [2]+ ∪ Y [2]0 suchthat over G = Y [2]0 it coincides with the trivial C ∗ -algebra bundle E = G × M ∞ F . To understand the fibre of E , + over G + fix g ∈ G and let ( g, z , z ) ∈ G + .Consider the following subspaces of C n : E ( g, z , z ) = M z <λ 1) = End( F (Eig( g, M ≺ F ( g, z ) = End( F ( E ≺ ( g, z ))) , M ≻ F ( g, z ) = End( F ( E ≻ ( g, z ))) . Just as in [23, Sec. 3] it follows that the bundle E → G + with fibre E ( g, z , z )over ( g, z , z ) ∈ G + is a locally trivial vector bundle. Therefore F ( E ) is aswell. Observe that the endomorphism bundle of F ( E ) has fibre M F ( g, z , z )over ( g, z , z ) ∈ G + . The fibre of our half-bundle E + will be given by thefollowing locally trivial bundle of right Hilbert M ∞ F -modules: E ( g,z ,z ) = F ( E ( g, z , z )) ⊗ M ∞ F where the right multiplication is given by right multiplication on M ∞ F . Thetransformation τ induces a ∗ -isomorphism for ( g, z , z ) , ( g, z , z ) ∈ G + ofthe form M F ( g, z , z ) ⊗ M F ( g, z , z ) → M F ( g, z , z ) (14)To define the multiplication on the fibres of E + we need the next lemma. Lemma 3.8. There is an isomorphism ϕ g,z ,z : M F ( g, z , z ) ⊗ M ∞ F → M ∞ F (constructed in the proof ) which is associative in the sense that for ( g, z , z ) , ( g, z , z ) ∈ G + the following diagram commutes: M F ( g, z , z ) ⊗ M F ( g, z , z ) ⊗ M ∞ F M F ( g, z , z ) ⊗ M ∞ F M F ( g, z , z ) ⊗ M ∞ F M ∞ F id ⊗ ϕ g,z ,z ϕ g,z ,z ϕ g,z ,z where the vertical arrow on the left is the isomorphism from (14) .Proof. To construct ϕ g,z ,z first note that the decomposition (13) yields acorresponding decomposition of the algebra M F , which we will also denoteby τ by a slight abuse of notation: M F M F ( g, ⊗ M ≺ F ( g, z ) ⊗ M F ( g, z , z ) ⊗ M ≻ F ( g, z ) . τ ∼ = (15)Let M , ≺ F ( g, z ) = M F ( g, ⊗ M ≺ F ( g, z ) and define ϕ ( k ) g,z ,z : M F ( g, z , z ) ⊗ M ⊗ kF → M ⊗ ( k +1) F to be the following composition M F ( g, z , z ) ⊗ M ⊗ kF M F ( g, z , z ) ⊗ ( M , ≺ F ( g, z ) ⊗ M F ( g, z , z ) ⊗ M ≻ F ( g, z )) ⊗ k ( M ≺ F ( g, z ) ⊗ M F ( g, z , z ) ⊗ M ≻ F ( g, z )) ⊗ ( k +1) M ⊗ ( k +1) F id ⊗ τ ⊗ k α ( k ) g,z ,z ( τ − ) ⊗ ( k +1) with the endomorphism α ( k ) g,z ,z given by α ( k ) g,z ,z ( T ⊗ ( A ⊗ B ⊗ C ) ⊗ · · · ⊗ ( A k ⊗ B k ⊗ C k ))= ( A ⊗ T ⊗ C ) ⊗ ( A ⊗ B ⊗ C ) ⊗ · · · ⊗ ( A k ⊗ B k − ⊗ C k ) ⊗ (1 ⊗ B k ⊗ ϕ ( k ) g,z ,z fits into the following commutative diagram M F ( g, z , z ) ⊗ M ⊗ kF M F ( g, z , z ) ⊗ M ⊗ ( k +1) F M ⊗ ( k +1) F M ⊗ ( k +2) FA ⊗ B A ⊗ B ⊗ ϕ ( k ) g,z ,z ϕ ( k +1) g,z ,z B B ⊗ ψ ( k +1) g,z ,z QUIVARIANT HIGHER TWISTED K -THEORY OF SU ( n ) 19 where ψ ( k ) g,z ,z is an endomorphism constructed analogously to ϕ ( k ) g,z ,z byconjugating the endomorphism( M , ≺ F ( g, z ) ⊗ M F ( g, z , z ) ⊗ M ≻ F ( g, z )) ⊗ k M F ( g, z , z ) ⊗ ( M , ≺ F ( g, z ) ⊗ M F ( g, z , z ) ⊗ M ≻ F ( g, z )) ⊗ kβ ( k ) g,z ,z given by β ( k ) g,z ,z (( A ⊗ B ⊗ C ) ⊗ · · · ⊗ ( A k ⊗ B k ⊗ C k ))= B ⊗ ( A ⊗ B ⊗ C ) ⊗ · · · ⊗ ( A k − ⊗ B k ⊗ C k − ) ⊗ ( A k ⊗ ⊗ C k )with the corresponding tensor products of τ . We define the homomorphisms ϕ g,z ,z : M F ( g, z , z ) ⊗ M ∞ F → M ∞ F ψ g,z ,z : M ∞ F → M F ( g, z , z ) ⊗ M ∞ F as the ones induced by ϕ ( k ) g,z ,z and ψ ( k ) g,z ,z on the colimits. The diagramabove shows that ϕ g,z ,z and ψ g,z ,z are inverse to each other. The associa-tivity condition stated above can be seen from the colimit of the followingcommutative diagram M F ( g, z , z ) ⊗ M F ( g, z , z ) ⊗ M ⊗ kF M F ( g, z , z ) ⊗ M ⊗ ( k +1) F M F ( g, z , z ) ⊗ M ⊗ ( k +1) F M ⊗ ( k +2) F id ⊗ ϕ ( k ) g,z ,z ϕ ( k +1) g,z ,z ϕ ( k +1) g,z ,z in which the vertical arrow on the left is the map from (14) tensored withthe homomorphism A A ⊗ (cid:3) Corollary 3.9. Let E → G + be the vector bundle with fibre E ( g, z , z ) over ( g, z , z ) ∈ G + . The isomorphisms ϕ g,z ,z constructed in Lemma 3.8 yielda continuous isomorphism of C ∗ -algebra bundles of the form ϕ + : End( F ( E )) ⊗ M ∞ F → G + × M ∞ F . Proof. Since M F ( g, z , z ) = End( F ( E ( g, z , z ))), the isomorphisms fromLemma 3.8 indeed piece together to give a map ϕ + as described in thestatement. Therefore the only issue left to prove is continuity of ϕ + . Firstobserve that E is by definition a subbundle of the trivial bundle G + × C n . Itsorthogonal complement E ⊥ is a locally trivial vector bundle as well. By con-tinuity of F we obtain locally trivial bundles F ( E ) and F ( E ⊥ ). Let M ∞ F ( E ),respectively M ∞ F ( E ⊥ ), be the UHF-algebras obtained as the fibrewise infi-nite tensor product of End( F ( E )), respectively End( F ( E ⊥ )), and note thatthe ∗ -homomorphism τ from (15) translates into a continuous isomorphismof C ∗ -algebra bundles G + × M ∞ F M ∞ F ( E ) ⊗ M ∞ F ( E ⊥ ) τ The maps α ( k ) g,z ,z from Lemma 3.8 induce another continuous isomorphismof C ∗ -algebra bundles: α ∞ : End( F ( E )) ⊗ M ∞ F ( E ) ⊗ M ∞ F ( E ⊥ ) → M ∞ F ( E ) ⊗ M ∞ F ( E ⊥ )which shifts End( F ( E )) into the tensor factor M ∞ F ( E ). By definition ϕ + isobtained by conjugating α ∞ by τ and therefore is continuous. (cid:3) Let E → G + be the vector bundle from Cor. 3.9. Let E = G × M ∞ F , E + = F ( E ) ⊗ M ∞ F and let E , + = E ∪ E + . This is a locally trivial bundle of full right Hilbert M ∞ F -modules, where M ∞ F acts by right multiplication on itself. The bundleof compact adjointable right M ∞ F -linear operators on E + agrees withEnd( F ( E )) ⊗ M ∞ F , which we can identify with M ∞ F using ϕ + to define a left M ∞ F -module struc-ture on the fibres of E + given by a · ( ξ ⊗ b ) := ϕ − ( a )( ξ ⊗ b ). To turn E , + intoa saturated half-bundle we need to equip it with a bilinear and associativemultiplication µ . On E + we define µ by the following diagram:( F ( E g,z ,z ) ⊗ M ∞ F ) ⊗ M ∞ F ( F ( E g,z ,z ) ⊗ M ∞ F ) F ( E g,z ,z ) ⊗ M ∞ F F ( E g,z ,z ) ⊗ F ( E g,z ,z ) ⊗ M ∞ F F ( E g,z ,z ⊕ E g,z ,z ) ⊗ M ∞ Fµκ ∼ = τ ⊗ id ∼ = where the map κ is given by κ (( ξ ⊗ a ) ⊗ ( η ⊗ b )) = ξ ⊗ a · ( η ⊗ b ) . This isan isomorphism with inverse ξ ⊗ η ⊗ a ( ξ ⊗ ⊗ ( η ⊗ a ). Let ℓ z i ,z j : M ∞ F ⊗ F ( E z i ,z j ) ⊗ M ∞ F → F ( E z i ,z j ) ⊗ M ∞ F be defined by left multiplication. The associativity condition in Lemma 3.8implies that the following diagram commutes: M ∞ F ⊗ F ( E z ,z ) ⊗ M ∞ F ⊗ F ( E z ,z ) ⊗ M ∞ F M ∞ F ⊗ F ( E z ,z ) ⊗ M ∞ F F ( E z ,z ) ⊗ M ∞ F ⊗ F ( E z ,z ) ⊗ M ∞ F F ( E z ,z ) ⊗ M ∞ F ℓ z ,z ⊗ id ℓ z ,z ( τ ⊗ id) ◦ (id ⊗ ℓ z ,z ) where we eliminated the group element g from the notation for clarity, andwhere the map 1 is given by id M ∞ F ⊗ (( τ ⊗ id M ∞ F ) ◦ (id F ( E z ,z ) ⊗ ℓ z ,z )). Asa consequence we obtain that the multiplication µ is associative on E + .On E we define the multiplication by the composition of the groupoidelements and the multiplication in M ∞ F , which is clearly bilinear and asso-ciative. If ( g, z , z ) ∈ G + and ( g, z , z ) ∈ G , then by definition E ( g, z , z )and E ( g, z , z ) agree. This identification and the right multiplication by M ∞ F defines the multiplication on elements from the set { ( e , e ) ∈ E , + | π ( e ) ∈ G + , π ( e ) ∈ G , ( π ( e ) , π ( e )) ∈ G (2) } . QUIVARIANT HIGHER TWISTED K -THEORY OF SU ( n ) 21 Using the left multiplication by M ∞ F we can also extend µ over { ( e , e ) ∈ E , + | π ( e ) ∈ G , π ( e ) ∈ G + , ( π ( e ) , π ( e )) ∈ G (2) } . in an analogous way. The resulting multiplication map is still associative.The fibrewise inner products on the right Hilbert A -modules yield a globalcontinuous inner product, i.e. for ξ ⊗ a, η ⊗ b ∈ F ( E ( g, z , z )) ⊗ M ∞ F wedefine h ξ ⊗ a, η ⊗ b i A = ( h ξ, η i C a ∗ b, ( g, z ))This ensures that all of the properties in Def. 3.1 b) hold. The multiplicationalso satisfies Def. 3.1 c) by construction. Thus, we have proven: Lemma 3.10. The triple ( E , + , µ, h · , · i A ) constructed above is a saturatedhalf-bundle. The group action on E , + . The group G = SU ( n ) acts on G = Y [2] by conjugation, i.e. for h ∈ G and ( g, z , z ) ∈ G we define h · ( g, z , z ) = ( hgh − , z , z ) . Observe that conjugation is a group automorphism and does not change theset of eigenvalues. Therefore this action is admissible in the sense of Def. 3.5.Let ( g, z , z ) ∈ G , + . Any element h ∈ G defines an isomorphism h : E ( g, z , z ) → E ( hgh − , z , z ) , ξ hξ , where h acts on Eig( g, λ ) ⊂ C n using the standard representation of SU ( n ).The exponential functor F turns this into a unitary isomorphism F ( h ) : F ( E ( g, z , z )) → F ( E ( hgh − , z , z )) . The naturality of the structure isomorphism τ of F ensures that the followingdiagram commutes: F ( E ( g, z , z )) ⊗ F ( E ( g, z , z )) F ( E ( g, z , z )) F ( E ( hgh − , z , z )) ⊗ F ( E ( hgh − , z , z )) F ( E ( hgh − , z , z )) F ( h ) ⊗ F ( h ) τ F ( h ) τ Similarly, G acts by conjugation on M F and therefore also on the infinitetensor product M ∞ F . Denote this action by α : G → Aut( M ∞ F ). This turns M ∞ F into a G -algebra. Combining F ( h ) and α we obtain isomorphisms of A - A bimodules F ( E ( g, z , z )) ⊗ M ∞ F → F ( E ( hgh − , z , z )) ⊗ M ∞ F (16)inducing a continuous action of G on E , + covering the action of G on G , + . Lemma 3.11. Let E → G + be the vector bundle with fibre E ( g, z , z ) over ( g, z , z ) ∈ G + . The isomorphism ϕ + : End( F ( E )) ⊗ M ∞ F → G + × M ∞ F constructed in Cor. 3.9 is G -equivariant (where h ∈ G acts on End( F ( E )) via Ad F ( h ) and on M ∞ F via α h ). In particular, (16) is an isomorphism ofbimodules and µ from Lem. 3.10 is G -equivariant. Proof. We use the notation introduced in Cor. 3.9. The action of h ∈ SU ( n )maps the eigenspace Eig( g, λ ) unitarily onto Eig( hgh − , λ ). This induces thegiven action of G on E and another unitary action of G on E ⊥ in such a waythat G + × C n = E ⊕ E ⊥ is an equivariant direct sum decomposition. Withrespect to the induced actions on M ∞ F ( E ) and M ∞ F ( E ⊥ ) the isomorphism τ : G + × M ∞ F → M ∞ F ( E ) ⊗ M ∞ F ( E ⊥ ) from Cor. 3.9 is G -equivariant. Since G acts in the same way on each tensor factor of the infinite tensor product M ∞ F ( E ) the shift isomorphism α ∞ from Cor. 3.9 is equivariant as well. Butthese are the building blocks of ϕ + . Thus, this implies the statement. (cid:3) Combining Thm. 3.3 and Cor. 3.7 we obtain the main result of this section: Corollary 3.12. The triple ( E , + , µ, h · , · i A ) together with the G -action de-fined above is a G -equivariant saturated half-bundle in the sense of Def. 3.6.In particular, there is a saturated Fell bundle π : E → G with the propertiesa) E| G , + = E , + ,b) e ∗ · e = h e , e i A for all e , e ∈ E lying in the same fibre,c) the group G = SU ( n ) acts continuously on E such that π : E → G isequivariant and g · e ∗ = ( g · e ) ∗ .The Fell bundle E is unique up to isomorphism. Remark 3.13. Note that for F = V top the algebra M ∞ F agrees with C and F ( E ) is the determinant line bundle of E . Moreover, the fibre ( E − ) ( g,z ,z ) can be identified with ( E ∗ + ) ( g,z ,z ) . Thus, our definition generalises the equi-variant basic gerbe as constructed in [23].4. The C ∗ -algebra associated to E In this section we will review the construction of the C ∗ -algebra C ∗ ( E )associated to the Fell bundle E . A priori there are several C ∗ -completions ofthe section algebra of E , but an amenability argument shows that all of themhave to agree. We will also see that C ∗ ( E ) is a continuous C ( G )-algebra,which is stably isomorphic to a section algebra of a locally trivial bundleof C ∗ -algebras. As a consequence we obtain a Mayer-Vietoris sequence inequivariant K -theory.We start by reviewing the construction of the reduced C ∗ -algebra associ-ated to the Fell bundle E . Let A = C ( Y, M ∞ F ). We can equip the space ofcompactly supported sections C c ( Y [2] , E ) with an A -valued inner product asfollows: h σ, τ i A ( g, z ) = Z T \{ } σ ( g, w, z ) ∗ · τ ( g, w, z ) dw , where σ, τ ∈ C c ( Y [2] , E ), the dot denotes the Fell bundle multiplication andwe used the Lebesgue measure on T \ { } with respect to which the subsetEV( g ) ∩ ( T \ { } ) is of measure zero. The space C c ( Y [2] , E ) also carries anatural right A -action given for a ∈ A and σ ∈ C c ( Y [2] , E ) by( σ · a )( g, z , z ) = σ ( g, z , z ) · a ( g, z ) . QUIVARIANT HIGHER TWISTED K -THEORY OF SU ( n ) 23 Denote by L ( E ) the completion of C c ( Y [2] , E ) to a right Hilbert A -modulewith respect to the norm k σ k = sup ( g,z ) ∈ Y kh σ, σ i A ( g, z ) k . The space C c ( Y [2] , E ) can also be equipped with a convolution product,which assigns to σ, τ ∈ C c ( Y [2] , E ) the section( σ ∗ τ )( g, z , z ) = Z T \{ } σ ( g, z , w ) · τ ( g, w, z ) dw Likewise, the ∗ -operation on the Fell bundle induces an involution that maps σ ∈ C c ( Y [2] , E ) to σ ∗ ( g, z , z ) = σ ( g, z , z ) ∗ and we have h σ ∗ τ , τ i A = h τ , σ ∗ ∗ τ i A , i.e. convolution by σ is an adjointable and hence bounded operator on L ( E ).Let L ◦ A (cid:0) L ( E ) (cid:1) be the adjointable right A -linear operators on the Hilbert A -module L ( E ). By the above considerations we obtain a well-defined ∗ -homomorphism C c ( Y [2] , E ) → L ◦ A (cid:0) L ( E ) (cid:1) . Definition 4.1. We define C ∗ r ( E ) to be the C ∗ -algebra obtained as thenorm-closure of C c ( Y [2] , E ) in L ◦ A (cid:0) L ( E ) (cid:1) . It is called the reduced C ∗ -algebra associated to the Fell bundle E .Denote by C ∗ max ( E ) the maximal cross-sectional C ∗ -algebra of E . Thegroupoid G = Y [2] is equivalent in the sense of Renault to the trivial groupoid G G idid which is (topologically) amenable. Since amenability is preserved by equiv-alence, the same is true for Y [2] . Therefore [28, Thm. 1] implies that thereduced and the universal norm agree on C c ( Y [2] , E ) and thus C ∗ max ( E ) ∼ = C ∗ r ( E ). Hence, we will drop the subscript from now on and write C ∗ ( E ) forthis C ∗ -algebra.Observe that C ∗ ( E ) carries a continuous G -action defined on sections σ ∈ C c ( Y [2] , E ) by ( g · σ )( h, z , z ) = g · σ ( g − hg, z , z ) . It is also a C ( G )-algebra in a natural way via the action that is defined onsections σ ∈ C c ( Y [2] , E ) with f ∈ C ( G ) as follows( f · σ )( g, z , z ) = f ( g ) σ ( g, z , z ) . Note that this is indeed central and therefore provides a ∗ -homomorphism C ( G ) → Z ( M ( C ∗ ( E ))) Lemma 4.2. The multiplication by elements in C ( G ) defined above turns C ∗ ( E ) into a continuous C ( G ) -algebra. For g ∈ G let Y [2] g be the subgroupoiddefined by Y [2] g = (cid:0) π − ( g ) (cid:1) [2] . and let E g = E| Y [2] g . Then the fibre of C ∗ ( E ) over g is given by C ∗ ( E g ) .Proof. We can identify G with the orbit space of the action of Y [2] on Y .Thus, [28, Cor. 10] implies that C ∗ ( E ) is a C ( G )-algebra with fibres C ∗ ( E g ).The only statement left to show is that g 7→ k a g k is lower semi-continuousfor every a ∈ C ∗ ( E ), where a g denotes the image of a in C ∗ ( E g ). Withoutloss of generality we may assume that a is a section σ ∈ C c ( Y [2] , E ). Let g ∈ G and ǫ > 0. Denote by τ g ∈ L ( E g ) the restriction of τ ∈ L ( E ). Notethat k τ g k L ( E g ) = sup z ∈ T kh τ, τ i A ( g, z ) k . Take τ ∈ L ( E ) with k τ k L ( E ) = 1, k τ g k L ( E g ) = 1 and k ( σ ∗ τ ) g k L ( E g ) ≥ k σ g k C ∗ ( E g ) − ǫ . Since the inner product on L ( E ) takes values in A = C ( Y, M ∞ F ), the func-tion f : Y → R given by f ( h, z ) = kh σ ∗ τ, σ ∗ τ i A ( h, z ) k is continuous andextends to G × T . Since T is compact, there is z ∈ T with ( g, z ) ∈ Y and f ( g, z ) = sup z ∈ T f ( g, z ). By continuity of h f ( h, z ) there is an openneighbourhood U of g such that for all h ∈ U kh σ ∗ τ, σ ∗ τ i A ( h, z ) k ≥ kh σ ∗ τ, σ ∗ τ i A ( g, z ) k − ǫ ≥ k σ g k C ∗ ( E g ) − ǫ . But kh σ ∗ τ, σ ∗ τ i A ( h, z ) k ≤ k ( σ ∗ τ ) h k L ( E h ) and since k τ h k L ( E h ) ≤ k σ h k C ∗ ( E h ) ≥ k σ g k C ∗ ( E g ) − ǫ for all h ∈ U , which shows that the map is lower semi-continuous. (cid:3) We are going to prove that C ∗ ( E ) is stably isomorphic to the sectionalgebra of a locally trivial bundle with fibre M ∞ F ⊗ K . The following lemmawill provide a first step and shows that local sections of π : Y → G give riseto trivialisations via Morita equivalences. Lemma 4.3. Let σ : V → Y be a continuous section of π : Y → G over aclosed subset V ⊂ G . Let Y V = π − ( V ) . Denote the corresponding restric-tion of G (respectively E ) by G V (respectively E V ). Let p T : Y → T be therestriction of the projection map to Y , let t = p T ◦ σ and let ι : Y V → G V , ( g, z ) ( g, z, t ( g )) The Banach bundle C V = ι ∗ E V gives rise to a Morita equivalence X V between C ∗ ( E V ) and C ( V, M ∞ F ) . If V is G -invariant, then X V is a G -equivariantMorita equivalence. QUIVARIANT HIGHER TWISTED K -THEORY OF SU ( n ) 25 Proof. We will prove the first part of the statement by showing that C V provides an equivalence of Fell bundles in the sense of [22, Sec. 6] between E V and the C ∗ -algebra bundle V × M ∞ F over V .First note that the space Y V is an equivalence between G V and the trivialgroupoid over V , which we will also denote V by a slight abuse of notation.Let β : V → G be given by β ( g ) = ( g, t ( g ) , t ( g )). We can and will identify V × M ∞ F with β ∗ E V . The bundle κ : C V → Y V carries a left action of E V anda right action of β ∗ E V = V × M ∞ F → V such that [22, Def. 6.1 (a)] holds.The two sesquilinear forms C V × C V → E V , ( c, d ) 7→ h c, d i E V := c · d ∗ , C V × κ C V → V × M ∞ F , ( c, d ) 7→ h c, d i β ∗ E V := c ∗ · d satisfy the conditions listed in [22, Def. 6.1 (b)]. Since E V → G V is a sat-urated Fell bundle, [22, Def. 6.1 (c)] is also true for the bundle C V → Y V .Therefore, by [22, Thm. 6.4] the completion X V of C c ( Y V , C V ) with respectto the norms induced by the above inner products is an imprimitivity bi-module between the C ∗ -algebras C ∗ ( E V ) and C ( V, M ∞ F ).For the proof of the second part suppose that V is G -invariant. Note thatthe adjoint action lifts to Y V . Moreover, the action described in Sec. 3.3restricts to C ∗ ( E V ). Let α : G → Aut( M ∞ F ) be the action of G on M ∞ F induced by Ad F ( ρ ) where ρ : G → U ( n ) is the standard representation. Then G acts on C ( V, M ∞ F ) via the adjoint action on V and by α on M ∞ F . It isstraightforward to check on sections that these definitions turn X V into anequivariant Morita equivalence. (cid:3) Remark 4.4. By [28, Lem. 9] the C ( G )-algebra structure is compatiblewith the Fell bundle restriction in the sense that restricting the sectionsto G V induces a natural ∗ -isomorphism C ∗ ( E )( V ) ∼ = C ∗ ( E V ). Definition 4.5. Let X be a locally compact metrisable space. A contin-uous C ( X )-algebra A whose fibres are stably isomorphic to strongly self-absorbing C ∗ -algebras is said to satisfy the (global) generalised Fell condition if for each x ∈ X there exists a closed neighbourhood V of x and a projection p ∈ A ( V ) such that [ p ( v )] ∈ GL ( K ( A ( v ))) for all v ∈ V . Lemma 4.6. The fibre algebra C ∗ ( E g ) is Morita equivalent to M ∞ F and thecontinuous C ( G ) -algebra C ∗ ( E ) satisfies the generalised Fell condition.Proof. Let V = { g } ⊂ G and choose z ∈ T such that ( g, z ) ∈ Y . Thefirst statement is now a consequence of Lemma 4.3 for V = { g } ⊂ G and σ : { g } → Y given by σ ( g ) = ( g, z ). For any g ∈ G let X g be the resultingMorita equivalence between C ∗ ( E g ) and M ∞ F .It remains to be proven that C ∗ ( E ) satisfies the generalised Fell condition.Let g ∈ G and choose an open neighbourhood U of g with the property that S = { z ∈ T \ { } | z / ∈ EV( h ) for any h ∈ U } contains an open interval J ⊂ S . Note that U × J ⊂ Y [2] . Since thereare no eigenvalues in between any two points of J , the restriction of E to this subspace is just the trivial bundle with fibre M ∞ F . Thus, extension of asection by 0 produces an inclusion of convolution algebras C c ( U × J , M ∞ F ) → C c ( Y [2] , E )and the completion of the left hand side in the representation on L ( E ) isisomorphic to C ( U, K ( L ( J )) ⊗ M ∞ F ). The resulting ∗ -homomorphism C ( U, K ⊗ M ∞ F ) → C ∗ ( E )is an inclusion of C ( G )-algebras. Pick a closed neighbourhood V ⊂ U of g .If we restrict both sides to V we obtain C ( V, K ⊗ M ∞ F ) → C ∗ ( E )( V ). Let e ∈ K be a rank 1-projection and define p ∈ C ∗ ( E )( V ) to be the image of1 C ( V, M ∞ F ) ⊗ e with respect to this inclusion. Fix v ∈ V . The isomorphism K ( C ∗ ( E )( v )) ∼ = K ( C ∗ ( E v )) ∼ = K ( M ∞ F )induced by the Morita equivalence X v maps the K -theory element [ p ( v )] ∈ K ( C ∗ ( E )( v )) to the class of the right Hilbert M ∞ F -module p ( v ) X v in K ( M ∞ F ).We can choose the value of z ∈ T used to define X v such that z ∈ J . More-over, we can without loss of generality assume that e ∈ K ( L ( J )) is theprojection onto the subspace spanned by a compactly supported function f ∈ C c ( J ) ⊂ L ( J ). Then we have p ( v ) X v ∼ = p ( v ) C c ( J, M ∞ F ) k·k L ∼ = eL ( J ) ⊗ M ∞ F ∼ = M ∞ F , which represents the unit in K ( M ∞ F ) and is therefore invertible. (cid:3) Corollary 4.7. The continuous C ( G ) -algebra C ∗ ( E ) is stably isomorphicto the section algebra of a locally trivial bundle A → G of C ∗ -algebras withfibre M ∞ F ⊗ K . In particular, it is classified by a continuous map G → B Aut( M ∞ F ⊗ K ) ≃ BGL (cid:0) KU (cid:2) d − F (cid:3)(cid:1) where d F = dim( F ( C )) .Proof. By Lemma 4.6 the algebra C ∗ ( E ) satisfies the generalised Fell con-dition and its fibres are Morita equivalent to the infinite UHF-algebra M ∞ F .Therefore the statement follows from [7, Cor. 4.9]. (cid:3) The spectral sequence. For G = SU ( n ) let ℓ = n − G . Choose a maximal torus T ℓ of G with Lie algebra t . Let Λ ⊂ t bethe integral lattice with dual lattice Λ ∗ . Denote by h · , · i g the basic innerproduct on g . Choose a collection α , . . . , α ℓ ∈ Λ ∗ of simple roots and let t + = (cid:8) ξ ∈ t | h α j , ξ i g ≥ ∀ j ∈ { , . . . , ℓ } (cid:9) be the corresponding positive Weyl chamber. Let ∆ ℓ be the standard ℓ -simplex defined as∆ ℓ = ( ( t , . . . , t ℓ ) ∈ R ℓ +1 | ℓ X i =0 t i = 1 and t j ≥ ∀ j ∈ { , . . . , ℓ } ) QUIVARIANT HIGHER TWISTED K -THEORY OF SU ( n ) 27 This simplex can be identified with the fundamental alcove of G , which isthe subset cut out from t + by the additional inequality h α , ξ i g ≥ − 1, where α is the lowest root. The alcove parametrises conjugacy classes of G in thesense that each such class contains a unique element exp( ξ ) with ξ ∈ ∆ ℓ .Denote the corresponding continuous quotient map by q : G → ∆ ℓ . A sketch of the situation in the case n = 3 is shown in Fig. 2. α α α ∆ Figure 2. The root system of G = SU (3) with positiveroots α and α and lowest root α . The blue dots mark theweights inside the Weyl chamber and the fundamental alcove∆ is shown in blue.For a non-empty subset I ⊂ { , . . . , ℓ } we let ∆ I ⊂ ∆ ℓ be the closedsubsimplex spanned by the vertices in I . Let ξ I ∈ g be the barycentre of ∆ I and let G I be the centraliser of exp( ξ I ). In fact, the isomorphism class of G I does not depend on our choice of ξ I as long as it is an element in the interiorof ∆ I . For J ⊂ I we have G I ⊂ G J , which induces a group homomorphism G/G I → G/G J . Let G n = a | I | = n +1 G/G I . Denote the set { , . . . , n } by [ n ]. Let f : [ m ] → [ n ] be an order-preservinginjective map. For each I ⊂ { , . . . , ℓ } with | I | = n + 1 there is a uniqueorder-preserving identification [ n ] ∼ = I . Let J ⊂ I be the subset correspond-ing to f ([ m ]) ⊂ [ n ] in this way. The above construction induces a continuousmap f ∗ I : G/G I → G/G J and those maps combine to f ∗ : G n → G m . This turns [ n ] G n with f f ∗ into a contravariant functor. Therefore G • is a semi-simplicial space. The group G can be identified with its geometric realisation, i.e. G ∼ = k G • k = a I G/G I × ∆ I ! / ∼ where the equivalence relation identifies the faces of ∆ I using the maps G/G I → G/G J in the other component. The map q : G → ∆ ℓ is induced bythe projection maps G/G I × ∆ I → ∆ I in this picture. Let A i = ( t , . . . , t ℓ ) ∈ ∆ ℓ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X k = i t k ≤ δ n ⊂ ∆ ℓ , where 0 < δ n < A i ) i ∈{ ,...,ℓ } cover ∆ ℓ .Then ( V i ) i ∈{ ,...,ℓ } with V k = q − ( A k ) is a cover of G by closed sets. Notethat each V k is G -homotopy equivalent to the open star of the k th vertex.For each non-empty subset I ⊂ { , . . . , ℓ } let A I = \ i ∈ I A i and V I = q − ( A I ) = \ i ∈ I V i . Note that the barycentre ξ I of ∆ I is contained in A I . Therefore there is acanonical embedding ι I : G/G I → V I . Lemma 4.8. The embedding ι I : G/G I → V I defined above is a G -equiva-riant deformation retract.Proof. Observe that ∆ { ,...,ℓ }\{ i } ∩ A i = ∅ , which implies that ∆ K ∩ A j = ∅ if j / ∈ K . Hence, the intersection ∆ K ∩ A I can only be non-empty if I ⊂ K .Therefore V I = a I ⊂ K G/G K × (∆ K ∩ A I ) ! / ∼ and the quotient maps G/G K → G/G I induce a well-defined G -equivariantcontinuous map r I : V I → G/G I with the property that r I ◦ ι I = id G/G I .Note that the set A I is convex and consider the contraction H A given by H A : A I × [0 , → A I , ( η, s ) ξ I + (1 − s )( η − ξ I ) . Since ξ I ∈ ∆ I , each H As maps A I ∩ ∆ K to itself for all sets K with I ⊂ K .Thus, we can lift H A to a G -equivariant continuous map H : V I × [0 , → V I which provides a homotopy between ι I ◦ r I and id V I that leaves ι I ( G/G I )invariant. (cid:3) Corollary 4.9. Let ρ : G → U ( n ) be the standard representation of G . Foreach non-empty subset I ⊂ { , . . . , ℓ } let ρ I : G I → U ( n ) be the restrictionof ρ to G I . There is a cohomological spectral sequence with E -page E p,q = M | I | = p +1 K G I q ( M ∞ F ) ∼ = (L | I | = p +1 R ( G I ) (cid:2) F ( ρ I ) − (cid:3) for q even , for q odd , QUIVARIANT HIGHER TWISTED K -THEORY OF SU ( n ) 29 where R ( H ) denotes the representation ring of H . It converges to the asso-ciated graded of a filtration of K G ∗ ( C ∗ ( E )) .Proof. The cover of G by closed sets gives rise to a semi-simplicial space V • with V n = a | I | = n +1 V I . Let f : [ m ] → [ n ] be an injective order-preserving map. Given I with | I | = n + 1 there is a unique order-preserving identification [ n ] ∼ = I andan inclusion V I ⊂ V J , where J ⊂ I is the subset corresponding to f ([ m ]).This construction gives a map f ∗ : V n → V m which turns [ n ] V n intoa contravariant functor. Let A → G be the C ∗ -algebra bundle found inCor. 4.7 and denote by A I → V I its restriction to V I . The spaces A I can beassembled to form a simplicial bundle of C ∗ -algebras A • → V • with A n = a | I | = n +1 A I . Replacing each V I by G and each A I by A in the definitions of V • and A • we also obtain two ‘constant’ semi-simplicial spaces A c • and G c • , respectively.Their geometric realisations are k G c • k ∼ = G × ∆ ℓ and k A c • k ∼ = A × ∆ ℓ . Thecanonical morphisms A • → A c • and V • → G c • give rise to the followingdiagram: k A • k A × ∆ ℓ Ak V • k G × ∆ ℓ G pr A pr G The second square is a pullback. It is not hard to check that the firstsquare is a pullback as well (compare this with [16, Rem. 2.23]). Moreover,the composition q : k V • k → G × ∆ ℓ → G in the diagram is a G -homotopyequivalence.For each pair ( X, f ) where X is a compact Hausdorff G -space and f : X → G is a G -equivariant continuous map, consider the contravariant functor( X, f ) K G ∗ ( C ( X, f ∗ A ))from the category of compact Hausdorff G -spaces over G to abelian groups.This functor satisfies the analogues of conditions (i) – (iv) in [27, § 5] in thiscategory. Using the same argument as in the proof of [27, Prop. 5.1] wetherefore end up with a spectral sequence with E -page E p,q = M | I | = p +1 K Gq ( C ( V I , A I )) ∼ = M | I | = p +1 K Gq ( C ∗ ( E )( V I )) , whose termination is K G ∗ ( C ∗ ( E )) ∼ = K G ∗ ( C ( G, A )), since the ∗ -homomor-phism C ( G, A ) → C ( k V • k , q ∗ A ) induces an isomorphism in equivariant K -theory by G -homotopy invariance. What remains to be done is to identifythese K -theory groups. By Lemma 4.8 the map G/G I → V I induces an isomorphism K Gq ( C ∗ ( E )( V I )) → K Gq ( C ∗ ( E )( G/G I )). By the same lemmaeach V k is G -equivariantly contractible. Therefore A| G/G I is equivariantlytrivialisable and K Gq ( C ∗ ( E )( G/G I )) ∼ = K Gq ( C ( G/G I , M ∞ F )) ∼ = K G I q ( M ∞ F ) . The matrix algebra M ⊗ kF is G -equivariantly Morita equivalent via the im-primitivity bimodule F ( C n ) ⊗ k to C with the trivial G -action. Therefore K G I ( M ⊗ kF ) ∼ = K G I ( C ) ∼ = R ( G I ) and K G I ( M ⊗ kF ) = 0. The ∗ -homomorphism M ⊗ kF → M ⊗ ( k +1) F given by a a ⊗ G I -representation F ( C n ) on K G I . This implies K G I q ( M ∞ F ) ∼ = R ( G I ) (cid:2) F ( ρ I ) − (cid:3) and K G I q +1 ( M ∞ F ) = 0for the K -theory of the direct limit. (cid:3) The module structure of K G ∗ ( E ) . An important consequence ofCor. 4.7 and Cor. 4.9 is that K G ∗ ( C ∗ ( E )) has a canonical module structureover the ring K G ( M ∞ F ) ∼ = R ( G )[ F ( ρ ) − ]. To see this, we need the followingobservation about strongly self-absorbing C ∗ -dynamical systems. Lemma 4.10. Let G be a compact Lie group and let σ : G → U ( n ) be aunitary representation of G . Let D = M n ( C ) ⊗∞ be the infinite UHF-algebraobtained from M n ( C ) and let α : G → Aut( D ) be the action of G , which actson each tensor factor of D via Ad σ . Let X be a compact Hausdorff G -space.Then the first tensor factor embedding ι X : C ( X, D ) → C ( X, D ) ⊗ D , f f ⊗ D is strongly asymptotically G-unitarily equivalent to a ∗ -isomorphism.Proof. By [32, Prop. 6.3], ( D, α ) is strongly self-absorbing in the sense of[31, Def. 3.1]. Using the notation introduced in [32, Def. 2.4] we see that( D ∞ ,α ) α ∞ = ( D α ) ∞ by integrating over G . The fixed-point algebra D α is an AF-algebra and thushas a path-connected unitary group. By [32, Prop. 2.19] the action α is uni-tarily regular. The result will therefore follow from [30, Thm. 3.2] if we canconstruct a unital equivariant ∗ -homomorphism θ : D → F ∞ ,α ( C ( X, D )).Let s k : D → D be an approximately central sequence of unital equivariant ∗ -homomorphisms and define θ ( d ) k = 1 C ( X ) ⊗ s k ( d ) . This ∗ -homomorphism satisfies all conditions. (cid:3) There is a canonical isomorphism K G ( C ) ∼ = R ( G ). As we have seenabove, we also have K G ( M ∞ F ) ∼ = R ( G )[ F ( ρ ) − ], where the isomorphism canbe chosen in such a way that the unit map C → M ∞ F induces the localisa-tion homomorphism R ( G ) → R ( G )[ F ( ρ ) − ]. Note that the multiplication QUIVARIANT HIGHER TWISTED K -THEORY OF SU ( n ) 31 in R ( G ) corresponds to the tensor product in K G ( C ). Likewise, the iden-tification K G ( M ∞ F ) ∼ = R ( G )[ F ( ρ ) − ] is also an isomorphism of rings. Themultiplication in R ( G )[ F ( ρ ) − ] corresponds to K G ( M ∞ F ) ⊗ K G ( M ∞ F ) K G ( M ∞ F ⊗ M ∞ F ) K G ( M ∞ F ) ∼ = where the second map is induced by the first factor embedding. To see whythis is true it suffices to note that the following diagram commutes K G ( M ∞ F ) ⊗ K G ( M ∞ F ) K G ( M ∞ F ⊗ M ∞ F ) K G ( M ∞ F ) K G ( C ) ⊗ K G ( C ) K G ( C ) ∼ = where the vertical homomorphism are induced by unit maps and turn intoisomorphisms after localisation. Proposition 4.11. The first factor embedding C ∗ ( E ) → C ∗ ( E ) ⊗ M ∞ F givenby a a ⊗ M ∞ F induces an isomorphism in equivariant K -theory and turns K G ∗ ( C ∗ ( E )) into a module over the ring K G ( M ∞ F ) ∼ = R ( G )[ F ( ρ ) − ] via K G ∗ ( C ∗ ( E )) ⊗ K G ( M ∞ F ) K G ∗ ( C ∗ ( E ) ⊗ M ∞ F ) K G ∗ ( C ∗ ( E )) ∼ = where the first homomorphism is induced by the tensor product in K -theory.The sequence constructed in Cor. 4.9 is a spectral sequence of modules.Proof. Fix a non-empty subset I ⊂ { , . . . , ℓ } . Let X I = ι I ( G/G I ) ⊂ G ,denote the barycentre of ∆ I by ξ I and note that X I = q − ( ξ I ). We willfirst show that the embedding C ∗ ( E )( X I ) → C ∗ ( E )( X I ) ⊗ M ∞ F induces anisomorphism in equivariant K -theory. The group elements g ∈ X I share thesame eigenvalues. Thus, there exists z ∈ T with the property that ( g, z ) ∈ Y for one (and hence all) g ∈ X I . Define σ I : X I → Y by σ I ( g ) = ( g, z ) andlet X I be the G -equivariant Morita equivalence resulting from Lemma 4.3using the section σ I . The claimed isomorphism is then a consequence of thefollowing commutative diagram K G ∗ ( C ∗ ( E )( X I )) K G ∗ ( C ∗ ( E )( X I ) ⊗ M ∞ F ) K G ∗ ( C ( X I , M ∞ F )) K G ∗ ( C ( X I , M ∞ F ⊗ M ∞ F )) ∼ = ∼ = ∼ = in which the vertical maps are induced by X I and the horizontal isomorphismfollows from Lemma 4.10.The first factor embedding C ∗ ( E ) → C ∗ ( E ) ⊗ M ∞ F induces a natural trans-formation between the spectral sequences from Cor. 4.9 associated to thefunctors ( X, f ) K G ∗ ( C ∗ ( f ∗ E )) and ( X, f ) K G ∗ ( C ∗ ( f ∗ E ) ⊗ M ∞ F ), whichis an isomorphism on all pages by our previous observation. This impliesthat K G ∗ ( C ∗ ( E )) → K G ∗ ( C ∗ ( E ) ⊗ M ∞ F ) is also an isomorphism, which gives rise to the module structure as described. A diagram chase shows that thisstructure is compatible with the K -theoretic description of the multiplica-tion in K G ( M ∞ F ) described above. (cid:3) Remark 4.12. It would be interesting to know whether the first factor em-bedding C ∗ ( E ) → C ∗ ( E ) ⊗ M ∞ F itself is strongly asymptotically G -unitarilyequivalent to a ∗ -isomorphism. The analogous non-equivariant statement istrue by [7, Lemma 3.4] and Cor. 4.7 .5. The equivariant higher twisted K -theory of SU ( n )In this section we will compute the equivariant higher twisted K -theoryof G = SU ( n ) for n ∈ { , } with respect to the adjoint action of G onitself and the equivariant twist described by the Fell bundle E constructedin Cor. 3.12. This is defined to be the G -equivariant operator algebraic K -theory of the G - C ∗ -algebra C ∗ ( E ), i.e. K G ∗ ( C ∗ ( E )).5.1. The case SU (2) . For G = SU (2) we have ℓ = 1. The map q : G → ∆ can be described as follows: Since the eigenvalues of any g ∈ SU (2) areconjugate to one another, each g = ± λ g withnon-negative imaginary part. Note that g arg( λ g ) extends to all of G .Let q : SU (2) → [0 , 1] be given by q ( g ) = arg( λ g ) π ∈ [0 , . If we pick δ n = the spectral sequence from Cor. 4.9 boils down to theMayer-Vietoris sequence for the G -equivariant closed cover of SU (2) by V = q − ([0 , ]) and V = q − ([ , K G -terms: K G ( C ∗ ( E )) R F ( SU (2)) ⊕ R F ( SU (2)) R F ( T )0 0 K G ( C ∗ ( E )) d where R F ( T ) = R ( T )[ F ( ρ { , } ) − ] and R F ( SU (2)) = R ( SU (2))[ F ( ρ ) − ].Let ρ be the standard representation of SU (2), then R F ( SU (2)) ∼ = Z [ ρ ][ F ( ρ ) − ] ,R F ( T ) ∼ = Z [ t, t − ][ F ( t + t − ) − ] . Lemma 5.1. As a module over R F ( SU (2)) the ring R F ( T ) is free of rank and β = { , t } is a basis.Proof. Let q = t + t − . It suffices to show that any f ∈ Z [ t, t − ] can beuniquely written as f = g + tg with g i ∈ Z [ q ] ⊂ Z [ t, t − ]. Let α ∈ Aut( Z [ t, t − ]) be given by α ( t ) = t − . Let g = t − f − tα ( f ) t − − t , g = α ( f ) − ft − − t . QUIVARIANT HIGHER TWISTED K -THEORY OF SU ( n ) 33 Note that α ( g i ) = g i for i ∈ { , } and f = g + tg . Let m ∈ Z and consider f = t m . In this case the numerator is divisible by the denominator and g i ∈ Z [ q ]. Using the linearity of the expressions in f we see that g i ∈ Z [ q ]holds in general. Suppose that g + tg = g ′ + tg ′ for another pair g ′ i ∈ Z [ q ].Applying α to both sides we obtain (cid:18) t t − (cid:19) (cid:18) g g (cid:19) = (cid:18) t t − (cid:19) (cid:18) g ′ g ′ (cid:19) . Multiplication by the matrix (cid:16) t − − t − (cid:17) yields ( t − − t ) g i = ( t − − t ) g ′ i and acomparison of coefficients gives g i = g ′ i . (cid:3) Observe that F ( C ) is a representation of T , which we will identify withits corresponding polynomial in Z [ t, t − ] and denote by F ( t ). Let α ∈ Aut( Z [ t, t − ]) be the ring automorphism given by α ( t ) = t − and note that α ( F ( t )) = F ( t − ) . Lemma 5.2. If we identify R F ( T ) with R F ( SU (2)) ⊕ R F ( SU (2)) using thebasis β = { , t } from Lemma 5.1, then the homomorphism d : R F ( SU (2)) ⊕ R F ( SU (2)) → R F ( T ) is represented by the matrix (cid:18) − g ( F )0 − g ( F ) (cid:19) for polynomials g i ( F ) ∈ R F ( SU (2)) given by g ( F ) = t − F ( t ) − tF ( t − ) t − − t and g ( F ) = F ( t − ) − F ( t ) t − − t . (17) (Note that g i ( F ) satisfies α ( g i ( F )) = g i ( F ) and therefore describes an ele-ment in R F ( SU (2)) .Proof. We can identify K G ( M ∞ F ) ∼ = R F ( SU (2)) and K T ( M ∞ F ) ∼ = R F ( T ).Using these isomorphisms, the homomorphism d fits into the commutativediagram K G ( C ∗ ( E )( V )) ⊕ K G ( C ∗ ( E )( V )) K G ( C ∗ ( E )( V ∩ V )) K G ( M ∞ F ) ⊕ K G ( M ∞ F ) K T ( M ∞ F ) ∼ = ∼ = d (18)where the upper horizontal map is induced by the inclusions V ∩ V → V i for i ∈ { , } . The two vertical isomorphisms are constructed as follows:Both X i ∼ = G/G i = ∗ are one-point spaces. By Lemma 4.8 the inclusions X i → V i are G -equivariant homotopy equivalences. We will choose specificMorita equivalences between C ∗ ( E )( V i ) and C ( V i , M ∞ F ), which give rise tothe following isomorphisms K G ( C ∗ ( E )( V i )) K G ( C ( V i , M ∞ F )) K G ( M ∞ F ) ∼ = ∼ =4 DAVID E. EVANS AND ULRICH PENNIG where the first map is induced by the Morita equivalence and the secondby the inclusion X i → V i . Let ω = − ω = exp( πi ). Consider thecontinuous sections σ i : V i → Y given by σ i ( g ) = ( g, ω i ) for i ∈ { , } , whichare well-defined by the definition of V i . By Lem. 4.3 they induce equivariantMorita equivalences X V i between C ∗ ( E )( V i ) and C ( V i , M ∞ F ).For the vertical isomorphism on the right hand side we restrict the Moritaequivalence induced by X V to V ∩ V and use K G ( C ∗ ( E )( V ∩ V )) K G ( C ( V ∩ V , M ∞ F )) K G ( C ( G/ T , M ∞ F )) ∼ = ∼ = with the first map induced by the equivalence and the second by G/ T ∼ = X { , } → V ∩ V , [ g ] g (cid:18) i − i (cid:19) g − . We obtain the following description of d : If d ( H , H ) = d (0) ( H ) + d (1) ( H ),then d ( i ) fits into the following commutative diagram: K G ( C ∗ ( E )( V i )) K G ( C ∗ ( E )( V ∩ V )) K G ( C ( V i , M ∞ F )) K G ( C ( V ∩ V , M ∞ F )) K G ( M ∞ F ) K G ( C ( G/ T , M ∞ F )) res ∼ = X Vi ∼ = X V | V ∩ V ∼ = X op Vi ⊗ X V ∼ = d ( i ) where the tensor product on the middle horizontal arrow is taken over C ∗ ( E )( V ∩ V ) and the bimodules have to be restricted to V ∩ V . In thecase i = 0 the bimodule X op V ⊗ X V is trivial. Let H ∈ R ( G ) ⊂ K G ( M ∞ F ).Using the isomorphism K G ( C ( G/ T , M ∞ F )) ∼ = K T ( M ∞ F ) the element d (0) ( H )agrees with the restriction of H to T . This gives the first column of thematrix in the statement.Let I = { , } and define σ I : V I → G by σ I ( g ) = ( g, ω , ω ). Note thatthis is well-defined and we have the following isomorphism of bimodules: X op V ⊗ X V ∼ = C ( V I , σ ∗ I E ) . All elements of X I have eigenvalues ± i . Therefore over X I this bimodulerestricts to continuous sections of the bundle with fibre E g ⊗ M ∞ F , where E g = F (Eig( g, i )) , i.e. it corresponds to taking the tensor product with the vector bundle E → G/ T , which is isomorphic to F ( L ), where L → G/ T is the canoni-cal line bundle associated to the principal T -bundle G → G/ T . The fibreof F ( L ) over [ e ] ∈ G/ T is the representation F ( C ), where T acts on C byleft multiplication. By Lemma 5.1 the decomposition of F ( t ) ∈ R ( T ) withrespect to the basis β is given by g ( F ) and g ( F ). Together with the signconvention in the exact sequence this explains the second column. (cid:3) QUIVARIANT HIGHER TWISTED K -THEORY OF SU ( n ) 35 Theorem 5.3. Let F be an exponential functor with F ( C ) ≇ F ( C ∗ ) as T -representations and let g ( F ) ∈ R F ( SU (2)) be the polynomial from (17) .The equivariant higher twisted K -theory of G = SU (2) with twist describedby the Fell bundle E constructed from F is given by K G ( C ∗ ( E )) = 0 ,K G ( C ∗ ( E )) = R F ( SU (2)) / ( g ( F )) In particular, note that K G ( C ∗ ( E )) is always a quotient ring of R F ( SU (2)) .Proof. The localisation of an integral domain at any multiplicative subsetcontinues to be an integral domain. By hypothesis we have g ( F ) = 0. Thus,the matrix representation of d from Lemma 5.2 implies that d is injective,which proves K G ( C ∗ ( E )) = 0. The group K G ( C ∗ ( E )) is isomorphic to thecokernel of d and the matrix representation of d implies that it has theclaimed form. (cid:3) We will conclude this section with explicit computations for some expo-nential functors. Example 5.4. Let b , . . . , b k ∈ N and let W j = C b j . Let F j ( V ) = M m ∈ N W ⊗ mj ⊗ ^ m ( V ) . and define F = F ⊗ · · · ⊗ F k . By [24, Sec. 2.2] each F j is an exponentialfunctor and so is F . The character of the irreducible representation ρ m of SU (2) with highest weight m in R ( T ) is t m + t m − + ... + t − m +1 + t − m . Fromthis we compute g ( F ) = g ( F ) · · · · · g ( F k ) with g ( F i ) = 1 + b i t as follows: g ( F ) = Q ki =1 (1 + b i t ) − Q ki =1 (1 + b i t − ) t − t − = k X ℓ =1 X I ⊂{ ,...,k }| I | = ℓ b I ρ ℓ − where b I is the product over all b i with i ∈ I . In case b = · · · = b k = 1, i.e.for F ( V ) = V ∗ ( V ) ⊗ k we obtain g ( F ) = k X ℓ =1 (cid:18) kℓ (cid:19) ρ ℓ − . Using the fusion rules for SU (2) the corresponding rings can be computedexplicitly by expressing the ideals in terms of ρ = ρ . For example, k = 3 : g ( F ) = 3 + 3 ρ + ρ = ρ + 3 ρ + 2 = ( ρ + 2)( ρ + 1) ,k = 4 : g ( F ) = 4 + 6 ρ + 4 ρ + ρ = ρ ( ρ + 2) ,k = 5 : g ( F ) = 5 + 10 ρ + 10 ρ + 5 ρ + ρ = ( ρ + 2) ( ρ + ρ − ,k = 6 : g ( F ) = 6 + 15 ρ + 20 ρ + 15 ρ + 6 ρ + ρ = ( ρ + 2) ( ρ − . Note that the element ρ + 2 = V ∗ ( ρ ) is a unit in R F ( SU (2)). The case k = 5seems to be particularly interesting as the following corollary shows. Corollary 5.5. The equivariant higher twisted K -theory of SU (2) with twistgiven by the exponential functor F = ( V ∗ ) ⊗ satisfies K G ( C ∗ ( E )) ∼ = Z ⊕ Z with basis { , x } , where x is the class of − ρ . It carries a ring structure givenby the Yang-Lee fusion rules x = x + 1 .Proof. Note that K G ( C ∗ ( E )) ∼ = R F ( SU (2)) / ( g ( F )) carries a ring structureand in this quotient ring the relation ( ρ + 2)(1 − ρ ) = 1 holds, which impliesthat the localisation is not necessary, since ρ + 2 is already invertible with( ρ + 2) − = (1 − ρ ) . The second statement follows directly from the relation ρ + ρ − (cid:3) Example 5.6. The classical case at level k ∈ N corresponds to the choice F = (cid:0)V top (cid:1) ⊗ k . In this situation we have M ∞ F ∼ = C . This implies R F ( SU (2)) ∼ = R ( SU (2)) . Together with g ( F ) = t k − t − k t − t − = ρ k − we obtain K G ( C ∗ ( E )) = R ( SU (2)) / ( ρ k − ).5.2. The case SU (3) . The group G = SU (3) has rank ℓ = 2. Let F bean exponential functor and let ρ be the standard representation of G on C .Consider the following localisations of representation rings: R F ( G I ) = R ( G I ) h F ( ρ | G I ) − i . For | I | = 1 we have G I = SU (3) and we will denote G { i } by G i . In case | I | = 2 the group G I is isomorphic to U (2) and the choice of I determinesan embedding U (2) ⊂ SU (3). If I = { , , } , then G I is the subgroup of alldiagonal matrices, which is our choice of maximal torus T ⊂ SU (3). The E -page of the spectral sequence from Cor. 4.9 vanishes in odd rows and hasthe following chain complex in the even rows: R F ( SU (3)) R F ( U (2)) R F ( T ) d d The generators for the representation rings are chosen as follows: R ( SU (3)) ∼ = Z [ s , s ] ,R ( U (2)) ∼ = Z [ s, d, d − ] ,R ( T ) ∼ = Z [ t ± , t ± , t ± ] / ( t t t − , where s = ρ is the standard representation of SU (3), s = V s , s de-notes the standard representation of U (2) on C and d is its determinantrepresentation. The characters t i are obtained by restricting the standardrepresentation of SU (3) to the maximal torus T and projecting to the i th QUIVARIANT HIGHER TWISTED K -THEORY OF SU ( n ) 37 diagonal entry. Let r : R F ( SU (3)) → R F ( U (2)) be the restriction homomor-phism . Then we have r ( s ) = s + d − ,r ( s ) = d − s + d . Let λ F = F ( d − ) and µ F = F ( s ). Note that F ( s + d − ) = F ( s ) · F ( d − )is a unit in R F ( U (2)). Hence, the same is true for λ F , µ F ∈ R F ( U (2)). Toexpress the differential d in terms of the r , λ F and µ F we first need to givean explicit description of the map q : SU (3) → ∆ : The eigenvalues of each g ∈ SU (3) can be uniquely written in the formexp(2 πiκ ) , exp(2 πiκ ) , exp(2 πiκ ) , where κ , κ , κ ∈ R satisfy P j =0 κ j = 0 and κ ≥ κ ≥ κ ≥ κ − 1. Let µ = diag (cid:0) , − , − (cid:1) ∈ t and µ = diag (cid:0) , , − (cid:1) ∈ t be the (duals of the)fundamental weights. For any triple ( κ , κ , κ ) as above, there are uniquevalues s, t ≥ s + t ≤ κ , κ , κ ) = s µ + t µ . (19)The map q : SU (3) → ∆ sends g ∈ SU (3) to the point (1 − ( s + t ) , s, t )in the simplex, i.e. if { e , e , e } denotes the standard basis of R , then q (exp(2 πiµ j )) agrees with e j . We choose δ n = as the constant for theclosed cover of ∆ given by A , A , A . The result is shown in Fig. 3.0 2 1 A A A Figure 3. The three closed sets A i covering ∆ .To express the differential d in terms of the representation rings, we firstobserve that we have three inclusions ι I : G { , , } → G I for I ⊂ { , , } with | I | = 2. These induce three restriction maps r I : R F ( G I ) → R F ( G { , , } ) ∼ = R F ( T ) . Let ν F = F ( t ) for t ∈ R ( T ) ⊂ R F ( T ) as defined above. In the nextlemma we write r ij for r I with I = { i, j } . The three inclusions G I ⊂ G for | I | = 2 induce the same map on representation rings. Lemma 5.7. The trivialisations R F ( G I ) ∼ = K G ( C ∗ ( E )( X I )) in the spectralsequence can be chosen in such a way that the differential d is given by thefollowing expression d ( x , x , x )=( − r ( x ) + λ F · r ( x ) , − r ( x ) + µ − F · r ( x ) , − r ( x ) + λ − F · r ( x )) where x i ∈ R F ( G i ) = R F ( SU (3)) and the three components on the righthand side correspond to the subsets I = { , } , { , } and { , } respectively.Moreover, d takes the following form d ( y , y , y ) = r ( y ) + ν F · r ( y ) − r ( y ) , where y ij ∈ R F ( G { i,j } ) .Proof. As above we write X i for X { i } and similarly for G i and V i . Observethat G i = G implies that X i is a one-point space for i ∈ { , , } . We willfirst discuss the construction of the differential d . Restriction along the G -equivariant homotopy equivalence X i → V i induces an isomorphism K G ( C ∗ ( E )( V i )) K G ( C ∗ ( E )( X i )) . ∼ = The differential d is an alternating sum of restriction homomorphisms alongthe inclusions of the form V { i,j } → V k with k ∈ { i, j } composed with iso-morphisms as shown in the following diagram: R F ( G k ) K G ( C ∗ ( E )( X k )) K G ( C ∗ ( E )( V k )) R F ( G { i,j } ) K G ( C ∗ ( E )( X { i,j } )) K G ( C ∗ ( E )( V { i,j } )) ∼ = ∼ = ∼ = ∼ = We will fix the isomorphisms on the left hand side by choosing an equivarianttrivialisation of C ∗ ( E )( V k ) via Morita equivalences given by Lemma 4.3. Let ω = − , ω = exp (cid:0) πi (cid:1) , ω = exp (cid:0) πi (cid:1) and define σ k : V k → Y by σ k ( g ) = ( g, ω k ). We claim that this is well-definedand will show this for k = 0. The other cases follow similarly. To see that ω / ∈ EV( g ) for all g ∈ V it suffices to prove that all coordinates κ i of q ( g )are different from ± for all g ∈ V . By (19) we have κ = 23 s + 13 t = 12 ⇔ s = 3 − t s + t ≤ ≤ t ≤ . Likewise, κ = − s ′ − t ′ = − ⇔ s ′ = 3 − t ′ s ′ + t ′ ≤ s ′ ≥ ≤ t ′ ≤ . Note that the coor-dinate κ is never equal to ± , since this would contradict the constraintsimposed by κ ≥ κ ≥ κ ≥ κ − κ + κ + κ = 0. Therefore thematrices with at least one eigenvalue equal to − QUIVARIANT HIGHER TWISTED K -THEORY OF SU ( n ) 39 two line segments described above and shown in Fig. 4. Our choice for δ n was made in such a way that the resulting A avoids this set proving ourclaim for V . The situation will look similar for V and V insofar as Fig. 4just has to be rotated accordingly.0 2 1 A Figure 4. The red lines correspond to SU (3) elements withat least one eigenvalue equal to − 1. As can be seen from thispicture the set A avoids those two lines.By Lem. 4.3 the section σ k constructed above gives an equivariant Moritaequivalence X V k between C ∗ ( E )( V k ) and C ( V k , M ∞ F ). Let I ⊂ { , , } anddenote the minimal element of I by i . The restriction of X V i to V I is aMorita equivalence between C ∗ ( E )( V I ) and C ( V I , M ∞ F ) and the trivialisation R F ( G I ) ∼ = K G ( C ∗ ( E )( X I )) is induced by the restricting further to X I ⊂ V I .The differential d is a signed sum of components of the form d Ik : K G ( M ∞ F ) → K G ( C ( G/G I , M ∞ F ))with k ∈ I . Just as in Lem. 5.2 the d Ik fits into the following commutativediagram: K G ( C ∗ ( E )( V k )) K G ( C ∗ ( E )( V I )) K G ( C ( V k , M ∞ F )) K G ( C ( V I , M ∞ F )) K G ( M ∞ F ) K G ( C ( G/G I , M ∞ F )) res ∼ = X Vk ∼ = X Vi ∼ = X op Vk ⊗ X Vi ∼ = d Ik From this we see that if I = { i, j } with i < j and k = i , then after identifyingthe domain of d Ik with R F ( G ) and the codomain with R F ( G I ) the map agreeswith the restriction homomorphism. Let I = { i, j } with i < j and define σ I : V I → Y by σ I ( g ) = ( g, ω j , ω i ). In this situation we have X op V j ⊗ X V i ∼ = C ( V I , σ ∗ I E ) . Let E → V I be the vector bundle with fibre over g ∈ V I given by E g = F (Eig( g, λ )) where λ ∈ EV( g ) is the eigenvalue of g between ω j and ω i . Then σ ∗ I E → V I is either of the form E ⊗ M ∞ F if ω j < ω i or ( E ⊗ M ∞ F ) op if ω i < ω j .Note that E | X I ∼ = F ( Q ), where Q → X I is the vector bundle associ-ated to the principal G I -bundle G → G/G I either using the inverse de-terminant representation d − or the standard representation s dependingon whether dim(Eig( g, λ )) = 1 or 2 respectively. Using the identifications K G ( M ∞ F ) ∼ = R F ( G ) and K G ( C ( X I , M ∞ F )) ∼ = R F ( G I ), the map d Ij thereforecorresponds to a factor of the form F ( d − ) ± or F ( s ) ± times the restrictionhomomorphism. The resulting factors are listed in Fig. 5. Together withI order eigenvalues representation factor { , } ω < ω e ( ), e ( − ), e ( − ) d − F ( d − ) = λ F { , } ω < ω e ( ), 1, e ( − ) s F ( s ) − = µ − F { , } ω < ω e ( ), e ( ), e ( − ) d − F ( d − ) − = λ − F Figure 5. The table shows the eigenvalue λ of g ∈ X I be-tween ω i and ω j in red with e ( ϕ ) = e πiϕ and the resultingfactor in the right hand column.the sign convention for the exact sequence this explains the form of d .The same reasoning can be used for d . Let I ⊂ { , , } be a subset with | I | = 2 and let J = { , , } . The differential d decomposes into a sum d ( x , x , x ) = d { , } ( x ) + d { , } ( x ) − d { , } ( x )with three maps d I : K G ( C ( X I , M ∞ F )) → K G ( C ( X J , M ∞ F )) that fit into thefollowing commutative diagram: K G ( C ( V I , M ∞ F )) K G ( C ( V J , M ∞ F )) K G ( C ( X I , M ∞ F )) K G ( C ( X J , M ∞ F )) ∼ = X op VI ⊗ X VJ ∼ = d I Just as above we see that d I agrees with the restriction homomorphism r I in the cases I = { , } and I = { , } , since X op V I ⊗ X V J is trivial then. Theonly remaining case is I = { , } , where we have X op V I ⊗ X V J ∼ = X op V ⊗ X V ∼ = C ( V J , σ ∗{ , } E )The eigenvalues for all g ∈ X J are e ( ), 1, e ( − ) and only the first one liesbetween ω and ω . Let λ = e ( ) and let E → X J be the vector bundlewith fibres given by E g = F (Eig( g, λ )) . It is isomorphic to F ( Q ), where Q → X J is the vector bundle associatedto the principal G J -bundle G → G/G J via the representation t . Thus, by QUIVARIANT HIGHER TWISTED K -THEORY OF SU ( n ) 41 the same argument as before the map d { , } agrees with F ( t ) times therestriction homomorphism r . (cid:3) Restriction to maximal torus. As above, denote by t the Lie algebra ofthe maximal torus T ⊂ SU (3). In this section we will prove that the chaincomplex in Lemma 5.7 computes the equivariant (Bredon) cohomology of t with respect to an extended Weyl group action and a twisted coefficient sys-tem. This approach is reminiscent of the method used in [1] to compute the(rational) twisted equivariant K -theory of actions with isotropy of maximalrank and classical twist. We will focus here on the action of G = SU (3)on itself by conjugation with a non-classical twists. An extension of thisapproach to G = SU ( n ) will be part of upcoming work.Let W = S be the Weyl group of SU (3). Our identification of T withthe diagonal matrices induces a corresponding isomorphism t ∼ = { ( h , h , h ) ∈ R | h + h + h = 0 } . The fundamental group π ( T , e ) agrees with the lattice Λ in t obtained asthe kernel of the exponential map. We will identify the two, which gives π ( T , e ) = Λ = { ( k , k , k ) ∈ Z | k + k + k = 0 } (20)The Weyl group acts on t and Λ by permuting the coordinates and we define f W = π ( T ) ⋊ W . Note that W also acts on Z in the same way. Let c W = Z ⋊ W and observethat f W ⊂ c W as a normal subgroup. Given an exponential functor F weobtain a group homomorphism ϕ : π ( T , e ) → GL ( R F ( T )) , ϕ ( k , k , k ) = F ( t ) k · F ( t ) k · F ( t ) k . If we define F ( − t i ) = F ( t i ) − we can rewrite the right hand side as ϕ ( k , k , k ) = F ( k t + k t + k t ) . Combining ϕ with the permutation action of W on R F ( T ) results in anaction of f W on R F ( T ). Just as in [1] this gives rise to local coefficientsystems R and R Q as follows R ( f W /H ) = R F ( T ) H , R Q ( f W /H ) = R F ( T ) H ⊗ Q . The simplex ∆ ⊂ t is a fundamental domain for the action of f W on t andturns it into a f W -CW-complex, in which the k -cells are labelled by subsets I ⊂ { , , } with | I | = k + 1. We have three 0-cells, three 1-cells andone 2-cell. Let f W I be the stabiliser of ξ I . Likewise let W I ⊂ W be thestabiliser of exp(2 πiξ I ). The restriction maps R F ( G I ) → R F ( T ) inducering isomorphisms r I : R F ( G I ) → R F ( T ) W I . As above let q : G → ∆ be the quotient map that parametrises conjugacyclasses. Let ˆ q = q | T ◦ q t , where q t : t → T is the universal covering. Let B I = ˆ q − ( A I ) ⊂ t . Note that { B , B , B } is a f W -equivariant cover of t as shown in Fig. 6. It has the property that the inclusion map f W · ξ I → B I isan equivariant homotopy equivalence. Figure 6. The cover of t induced by the cover of ∆ .For any subset I ⊂ { , , } the Bredon cohomology H k f W ( B I , R ) is onlynon-zero in degree k = 0 where we have a natural isomorphism H f W ( B I ; R ) ∼ = R F ( T ) f W I . For J ⊂ I the restriction homomorphism H f W ( B J ; R ) → H f W ( B I , R ) trans-lates into the natural inclusion R F ( T ) W J ⊂ R F ( T ) W I . The sum over all H q f W ( B I ; R ) with | I | = p + 1 forms the E -page of a spectral sequence thatconverges to H p + q f W ( t ; R ). By our above considerations this E -page boilsdown to the chain complex C k f W ( t ; R ) = M | I | = k +1 R F ( T ) f W I with the differentials d cell k : C k f W ( t ; R ) → C k +1 f W ( t ; R ) given by alternating sumsof inclusion homomorphisms. We can identify W with the subgroup of f W consisting of elements of the form (0 , w ) ∈ π ( T , e ) ⋊ W . Observe that W i = W = f W for i ∈ { , , } , W { , } = f W { , } , W { , } = f W { , } and wehave group isomorphisms f W → W , x (( − , , , e W ) · x · ((1 , , , e W ) , f W → W , x ((0 , , , e W ) · x · ((0 , , − , e W ) . Here, e W denotes the neutral element of W and we used the conjugationaction of c W on W . The first isomorphism restricts to f W { , } → W { , } .These identifications induce corresponding isomorphisms of the fixed point QUIVARIANT HIGHER TWISTED K -THEORY OF SU ( n ) 43 rings e r : R F ( T ) W → R F ( T ) f W , p F ( t ) · p e r : R F ( T ) W → R F ( T ) f W , p F ( t ) − · p e r { , } : R F ( T ) W { , } → R F ( T ) f W { , } , p F ( t ) · p Define e r I : R F ( T ) W I → R F ( T ) f W I for all other I ⊂ { , , } to be theidentity. Let ˆ r I = e r I ◦ r I : R F ( G I ) → R F ( T ) f W I . Lemma 5.8. The isomorphisms ˆ r I fit into the following commutative dia-gram: L | I | =1 R F ( G I ) L | I | =2 R F ( G I ) R F ( T ) C f W ( t ; R ) C f W ( t ; R ) C f W ( t ; R ) d L | I | =1 ˆ r I ∼ = d L | I | =2 ˆ r I ∼ = ˆ r { , , } =id ∼ = d cell0 d cell1 In particular, the chain complex from Lemma 5.7 computes the f W -equivariantBredon cohomology H ∗ f W ( t ; R ) of t with coefficients in R .Proof. Let r ( k ) (respectively e r ( k ) ) for k ∈ { , , } be the sum over all r I (respectively e r I ) for all I ⊂ { , , } with | I | = k + 1. Consider b d : M | I | =1 R F ( T ) W I → M | I | =2 R F ( T ) W I , b d : M | I | =2 R F ( T ) W I → R F ( T )with b d ( x , x , x ) = ( − x + F ( t ) x , − x + F ( t + t ) − x , − x + F ( t ) − x )and b d ( y , y , y ) = y + F ( t ) y − y . Then we have r (2) ◦ d = b d ◦ r (1) and b d ◦ r (2) = d . The statement follows from the following two observations:( e r (2) ◦ b d )( x , x , x )= ( − x + F ( t ) x , − F ( t ) x + F ( t ) − x , − x + F ( t ) − x )= ( − e r ( x ) + e r ( x ) , − e r ( x ) + e r ( x ) , − e r ( x ) + e r ( x ))= ( d cell0 ◦ e r (1) )( x , x , x )and ( d cell1 ◦ e r (2) )( y , y , y ) = y + F ( t ) y − y = b d ( y , y , y ) . (cid:3) The above lemma reduces the problem of computing the equivarianthigher twisted K -theory of SU (3) to the computation of Bredon cohomologygroups with local coefficients. We will determine these groups after rational-ising the coefficients, i.e. we compute H ∗ f W ( t ; R Q ). The inclusion R → R Q induces a homomorphism H ∗ f W ( t ; R ) → H ∗ f W ( t ; R Q ) . (21)Even though [1, Thm. 3.11] is only stated for coefficient systems R Q wherethe module structure is induced by a homomorphism π ( T ) → hom( T , S ),the proof works verbatim in our situation, where R F ( T ) ⊗ Q carries a π ( T )-action induced by π ( T ) → GL ( R F ( T ) ⊗ Q ). Hence, we obtain H ∗ f W ( t ; R Q ) ∼ = H ∗ π ( T ) ( t ; R Q ) W . (22) Lemma 5.9. Let F be an exponential functor with deg( F ( t )) > , then ( F ( t ) − F ( t ) , F ( t ) − F ( t )) is a regular sequence in R F ( T ) ⊗ Q .Proof. Let R F = R F ( T ) ⊗ Q , R = R ( T ) ⊗ Q ⊂ R F and let q F = F ( ρ ) = F ( t + t + t ). Let I jk ⊂ R F be the ideal generated by F ( t k ) − F ( t j ). Wehave to show that multiplication by F ( t ) − F ( t ) is injective on R F /I .On this quotient F ( t ) − F ( t ) agrees with F ( t ) − F ( t ). Suppose we haveelements p, q ∈ R F with the property that p is not divisible by F ( t ) − F ( t )and p · ( F ( t ) − F ( t )) = q · ( F ( t ) − F ( t )) . (23)Multiplying both sides by an appropriate power of q F we may assume that p, q ∈ R . Now we can use the relation t = ( t t ) − to express both sidesof (23) in terms of t , t , t − , t − . Since we may multiply both sides by t k t l for appropriate k, l ∈ N , we can without loss of generality assume that p, q ∈ Q [ t , t ]. Let F ( t ) = m X k =0 a k t k with a m = 0. We have deg( F ( t )) = deg( F ( t )) = m and by our assumption m > 0. However, note that deg( F ( t )) ≤ 0. The highest order term of F ( t )can be expressed as follows a m t m = a m t m − m − X k =1 a k ( t k − t k ) + F ( t ) − F ( t ) . Since we are working over Q , we can therefore assume that p is a linearcombination of terms t k t l with l < m by adapting q accordingly. Supposethat p has total degree r and let p r be the corresponding homogeneous part.Comparing the terms of highest degree in (23) we obtain − p r t m = q r ( t m − t m ) , where q r is the homogeneous part of q of degree r . Since the left handside contains no summands t k t l with l ≥ m , this equation implies q r = 0,therefore p r = 0 and hence p = 0. This is a contradiction to our initialdivisibility assumption. Hence p must be divisible by F ( t ) − F ( t ) provingthat multiplication by F ( t ) − F ( t ) is injective on R F /I . (cid:3) QUIVARIANT HIGHER TWISTED K -THEORY OF SU ( n ) 45 Theorem 5.10. Let F be an exponential functor with deg( F ( t )) > . Then H kπ ( T ) ( t ; R Q ) = 0 for k = 2 and H π ( T ) ( t ; R Q ) ∼ = R F ( T ) ⊗ Q / ( F ( t ) − F ( t ) , F ( t ) − F ( t )) . Moreover, W acts on H π ( T ) ( t ; R Q ) by signed permutations.Proof. Let Λ = π ( T ) be as in (20). The two vectors κ = (1 , − , 0) and κ = (0 , , − 1) form a basis of Λ. Note that Q [Λ] ∼ = Q [ r , r , r ] / ( r r r − κ corresponds to r r − under this isomorphism. Likewise κ agreeswith r r − . The action of Λ on R F ( T ) ⊗ Q extends to a ring homomorphism ϕ : Q [Λ] → R F ( T ) ⊗ Q given by ϕ X k,l,m a klm s k s l s m = X k,l,m a klm F ( t ) k F ( t ) l F ( t ) m . In particular, ϕ ( r r − ) = F ( t ) F ( t ) − and similarly for r r − . The equi-variant cohomology groups H ∗ Λ ( t ; R Q ) are computed by the cochain complexhom Q [Λ] ( C ∗ ( t ) ⊗ Q , R F ( T ) ⊗ Q ) (24)(see [5, I.9, (9.3)]), where C ∗ ( t ) is the cellular chain complex of t viewed asa Λ-CW-complex. Note that t has a Λ-CW-structure with one 0-cell givenby the orbit of the origin, two 1-cells corresponding to the orbits of theedge from (0 , , 0) to (1 , − , 0) and to (0 , , − C ∗ ( t ) can be identified with the Koszul complex K n = ^ n Z ⊗ Z [Λ]for the sequence ( r r − − , r r − − C ∗ ( t ) and K ∗ in thisway, the cochain complex in (24) turns into C nF = ^ n R F with d n ( y ) = x ∧ y where R F = R F ( T ) ⊗ Q and x = ( F ( t ) F ( t ) − − , F ( t ) F ( t ) − − R F by Lemma 5.9. The first part of thestatement now follows from [8, Cor. 17.5]. We can identify Z with Λ using κ and κ . This induces an action of W on Z . The group W acts diagonallyon K n using its natural action on Z [Λ]. If we equip the cochain complexhom Q [Λ] ( K n ⊗ Q , R F ( T ) ⊗ Q )with the W -action given by ( g · ϕ )( y ) = gϕ ( g − y ), then the isomorphism ofthis cochain complex with (24) is equivariant. Likewise, W acts diagonallyon C nF ∼ = V n Q ⊗ R F making the last isomorphism equivariant as well. Inparticular, W acts on V Q via the sign representation. This proves thesecond statement. (cid:3) To distinguish the signed permutation action of W on R F ( T ) ⊗ Q fromthe usual one, we denote the two modules by R sgn F and R F respectively asin [1]. We also define I sgn F = ( F ( t ) − F ( t ) , F ( t ) − F ( t )). Over the rationalnumbers taking invariants with respect to a finite group action is an exactfunctor. Hence, Thm. 5.10 gives isomorphisms of R WF -modules H f W ( t ; R Q ) ∼ = H ( t ; R Q ) W ∼ = ( R sgn F ) W / ( I sgn F ) W . The ring R WF is a localisation of the quotient of the ring of symmetric poly-nomials in the variables t , t , t by the ideal generated by ( t t t − R sgn F ) W is a similar quotient of the antisymmetric polynomials bythe submodule ( t t t − R sgn F ) W . Any antisymmetric polynomial is divis-ible by the Vandermonde determinant∆ = ∆( t , t , t ) = ( t − t )( t − t )( t − t ) . This induces an R WF -module isomorphismΨ : ( R sgn F ) W → R WF , p p ∆(compare with [1, Sec. 5.1]). Let θ jk = F ( t j ) − F ( t k ). Lemma 5.11. The submodule ( I sgn F ) W is generated by the two antisymmet-ric polynomials q + = θ t + θ t + θ t and q − = θ t − + θ t − + θ t − .The corresponding submodule Ψ(( I sgn F ) W ) is generated by Ψ( q ± ) = − 1∆ det F ( t ) F ( t ) F ( t ) t ± t ± t ± . (25) Proof. First note that θ + θ + θ = 0. Therefore q ± ∈ ( I sgn F ) W . Themodule R ( T ) is free of rank 6 over R ( T ) W by [29, Thm. 2.2]. Thus, thesame is true for R F as a module over R WF . An explicit basis is given by β = { e, t , t , t − , t − , t − t } . Consider the averaging map α : R sgn F → ( R sgn F ) W , p X g ∈ W g · p . This is a surjective module homomorphism, which maps the submodule I sgn F onto ( I sgn F ) W . Let q ∈ ( I sgn F ) W and choose p ∈ I sgn F such that α ( p ) = q .Then we have p = θ p + θ p for suitable p i ∈ R F . We have to see that q = α ( p ) is in the submodulegenerated by q ± . After decomposing p and p with respect to β we seethat it suffices to check that α ( θ y ) and α ( θ y ) lie in this submodule for QUIVARIANT HIGHER TWISTED K -THEORY OF SU ( n ) 47 all y ∈ β . Since α ( θ ) = α ( θ ) = 0, this is true for y = e . We have α ( θ t ) = 16 (( θ + θ ) t + ( θ + θ ) t + ( θ + θ ) t )= − 16 ( θ t + θ t + θ t ) = − q + α ( θ t ) = 13 ( θ t + θ t + θ t ) = 13 q + The cases α ( θ t ) and α ( θ t ) work in a similar way. The expressions α ( θ t − ), α ( θ t − ), α ( θ t − ) and α ( θ t − ) produce corresponding mul-tiples of q − . In the remaining two cases a short computation shows that α ( θ t − t ) = 16 q + · ( t − + t − + t − ) ,α ( θ t − t ) = 16 q − · ( t + t + t ) . This shows that the submodule ( I sgn F ) W is generated by q + and q − . Thedeterminant formula follows from a straightforward computation. (cid:3) Example 5.12. In case of the classical twist, i.e. for F = ( V top ) ⊗ m we have F ( t i ) = t mi . For Ψ( q + ) equation (25) boils down to the definition of theSchur polynomial for the partition with just one element [19, I.3, p. 40].In this case the Schur polynomial agrees with the complete homogeneoussymmetric polynomial h m − . Using the properties of the determinant wealso haveΨ( q − ) = − 1∆ det t m +11 t m +13 t m +13 t t t = 1∆ det t m +11 t m +13 t m +13 t t t which produces h m − . Altogether we obtainΨ( q + ) = − h m − , Ψ( q − ) = h m − . For m = 0 we have q − = q + = 0, for m = 1 we get q + = 0 and q − = 1and in the case m = 2 the submodule ( I sgn F ) W is generated by q + = 1 and q − = h . Thus, for m ∈ { , } the quotient R WF /I WF vanishes. Example 5.13. For the m th power of the exterior algebra twist F = ( V ∗ ) ⊗ m we have F ( t i ) = (1 + t i ) m and since F ( t j ) = m X l =0 (cid:18) ml (cid:19) t lj the determinant formula for Ψ( q ± ) givesΨ( q + ) = − m X l =2 (cid:18) ml (cid:19) h l − , Ψ( q − ) = m X l =1 (cid:18) ml (cid:19) h l − . We are now in the position to provide a complete computation of theequivariant higher twisted K -theory for SU (3) after rationalisation and sum-marise our results in the following theorem. Theorem 5.14. For the rational equivariant higher twisted K -theory of SU (3) with twist given by an exponential functor F with deg( F ( t )) > wehave the following isomorphism of R F ( SU (3)) -modules: K G ( C ∗ ( E )) ⊗ Q ∼ = ( R F ( SU (3)) ⊗ Q ) /J F , K G ( C ∗ ( E )) ⊗ Q ∼ = 0 where J F is the submodule generated by the two representations σ F and σ F whose characters χ , χ are the symmetric polynomials χ = 1∆ det (cid:18) F ( t ) F ( t ) F ( t ) t t t (cid:19) , χ = 1∆ det (cid:18) F ( t ) t F ( t ) t F ( t ) t t t t (cid:19) . In the case F = ( V ∗ ) ⊗ m the submodule J F is generated by the representations σ F = m X l =2 (cid:18) ml (cid:19) Sym l − ( ρ ) , σ F = m X l =1 (cid:18) ml (cid:19) Sym l − ( ρ ) . Proof. Let Q be the universal UHF-algebra equipped with the trivial action.By continuity of the K -functor the K -theory of C ∗ ( E ) ⊗ Q is the rationalisa-tion of the K -theory of C ∗ ( E ). To compute K G ∗ ( C ∗ ( E ) ⊗ Q ) we can use thecorresponding spectral sequence from Cor. 4.9. The resulting cochain com-plex will have R F ( G I ) ⊗ Q in place of R F ( G I ) in each degree. Lemma 5.8identifies it as the complex computing the f W -equivaraint cohomology of t with respect to the coefficient system R Q . Thus, the homomorphism (21) isnow an isomorphism. Combining Thm. 5.10 with Lemma 5.11 we obtain thefirst part of the statement. The second part follows from Example 5.13 byidentifying the characters given by the homogeneous symmetric polynomialswith symmetric powers of the standard representation. (cid:3) Remark 5.15. The choice of the orientation of T that went into the con-struction of E through the choice of order on T \ { } features in the com-putations of the twisted K -groups as follows: Changing the orientation toits opposite corresponds to replacing all factors of the form F ( t ) by F ( t ) − .Since the ideal I sgn F is invariant under this transformation, we obtain iso-morphic higher twisted K -groups in both cases.We expect Thm. 5.14 also to be true without the rationalisation. Since themodules R F ( G I ) are free over R F ( SU (3)) the differentials from Lemma 5.7in the cochain complex can be expressed in terms of matrices, which allowedus to perform an extensive computer analysis in the case of the full twist F = ( V ∗ ) ⊗ m for m ∈ { , . . . , } . For these levels we thereby confirmed theabove conjecture.The approach presented above should also be seen as a blueprint forthe computation of the rationalised equivariant higher twisted K -theory of SU ( n ). 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Cambridge Univ.Press, Cambridge, 2004. Cardiff University, School of Mathematics, Senghennydd Road, Cardiff,CF24 4AG, Wales, UK E-mail address : [email protected] E-mail address ::