Aspects of equivariant KK -theory in its generators and relations picture
aa r X i v : . [ m a t h . K T ] D ec ASPECTS OF EQUIVARIANT KK -THEORY IN ITSGENERATORS AND RELATIONS PICTURE BERNHARD BURGSTALLER
Abstract.
We give a new proof of the universal property of KK G -theory withrespect to stability, homotopy invariance and split-exactness for G a locallycompact group, or a locally compact (not necessarily Hausdorff) groupoid,or a countable inverse semigroup which is relatively short and conceptual.Morphisms in the generators and relations picture of KK G -theory are broughtto a particular simple form. Introduction
In [11] Kasparov has introduced KK -theory for C ∗ -algebras, a bivariant K -theory fusing K -homology with K -theory. Afterwards, Cuntz found another de-scription of KK -theory by interpreting KK -theory elements as quasihomomor-phisms and showing that KK -theory elements act on stable, homotopy invariant,split-exact functors from the C ∗ -category to abelian groups, see the relevant papers[7, 8, 9]. Based on this categorial finding, Higson [10] proved that every stable, ho-motopy invariant, split-exact functor from the C ∗ -category to an additive categoryuniquely ‘extends’ to KK -theory. Actually, Kasparov considered the C ∗ -algebras tobe G -equivariant with respect to a compact group G , and generalized this in [12] tolocally compact, second-countable groups. Cuntz and Higson’s findings were donenon-equivariantly, and in [15] Thomsen generalized Higson’s result to G -equivariant KK -theory.By Cuntz and Higson’s findings it is evident that the category of equivariant KK -theory restricted to the category of separable C ∗ -algebras may be expressedby generators and relations, where the generators are the C ∗ -homomorphisms andother synthetical inverse morphisms. We have described this in more details in [4]and called the category GK for simplicity. In [2] we have shown that S -equivariant KK -theory for a countable discrete inverse semigroup S also satisfies the univer-sal property. The proof works also almost unchanged for a locally compact (notnecessarily Hausdorff) groupoid, see corollary 2.2.In [5] it was noted that in GK -theory morphims may be written as a shortproduct ae − ∆ s b ∆ t f − where a, b are homomorphisms, e, f are corner embeddings and ∆ s , ∆ t are syntheti-cal splits of split exact sequences. To show this one takes a morphism in GK -theory,interprets it as a Kasparov element in KK -theory and goes back to GK -theory bythe functor constructed in the proof of the universal property of KK -theory. Mathematics Subject Classification.
Key words and phrases. KK -theory, universal property, split exact, equivariant, groupoid,inverse semigroup. In this paper we explore the category GK further by simplifying the expressionof morphisms in GK directly in GK . This is done in an equivariant setting withrespect to a locally compact (not necessarily Hausdorff) groupoid or inverse semi-group G . The whole machinery we present is category theoretically very visualand close to ordinary C ∗ -theory. On the other hand, the picture in GK -theory isstill very close to the Kasparov picture. Actually we solve all the harder problemsby using KK -theory, particularly the Kasparov technical theorem encoded in theKasparov product. Together with many ideas taken from KK -theory, for examplethe Kasparov stabilization theorem.Beside these benefits, we get a new proof of the universal property of equivariant KK -theory as a byproduct. Moreover we can improve the word length of the aboveproduct to ae − ∆ s f − Note that we have not optimized the exposition to achieve as fast as possible themain result theorem 14.3 and as a corollary the universal property of KK -theory,corollary 14.6. If one is interested in these proofs one may save several pages.Indeed one would need of sections 5 to 8 only a few lemmas (lemmas 5.4, 7.1, 7.8,8.1, 8.2, 8.5, 8.10, 8.11). As remarked in section 11, its contained proofs could beextremely cut short. The proof of the universal property actually really begins insection 9.We give a short overview of the paper. In section 2 we remark that KK G fora locally compact (not necessarily Hausdorff) groupoid G satisfies the universalproperty by the same proof as for inverse semigroups. In section 3 we brieflyrecall GK -theory, and in section 4 we introduce the functor from GK -theory to KK G -theory. In section 5 we define the important concept of double split exactsequences in the equivariant setting. The idea to use non-equivariant double splitexact sequences in KK -theory is essentially due to Cuntz [8], and the additionalmatrix trick to handle equivariance goes back at least to Connes [6] and was usedby Thomsen [15]. Fundamental is the construction of split exact sequences insection 6, also used by Kasparov [11]. In section 7 we discuss G -actions on a2 × GK -theory, including sideways which are not relevant for the mainresults. Actually, in lemmas 7.8 and 8.12 we see how Kasparov’s definition of the KK -groups come out naturally and suggest itself in our framework. In section 9we introduce the functor from KK G -theory to GK -theory, and in section 10 wedetect the first relations to the functor in the other direction. Section 11 shows animportant concept by Kasparov, and technically simplified by Connes-Skandalis, toprepare a pushout-construction used in the two consecutive sections. In section 12we use the Kasparov product for the fusion of a synthetical split with a double splitexact sequence. We do not need the Kasparov product any more in section 12 to fuseanalogously a double split exact sequence with the inverse of a corner embedding.In section 13 we show by induction on the length of a word of a morphism in GK -theory that it can be simplified to the simple form as stated above. As a corollarywe obtain the universal property of equivariant KK -theory.2. The universal property of KK G for groupoids G Let G be a second countable locally compact group, a second countable locallycompact (not necessarily Hausdorff) groupoid, or a countable inverse semigroup. SPECTS OF EQUIVARIANT KK -THEORY 3 The category of separable G -equivariant C ∗ -algebras and their G -equivariant ho-momorphisms is denoted by C ∗ . We often use the term ‘non-equivariant’ when wewant to ignore any G -action or G -equivariance. All Hilbert modules are assumedto be countably generated, and all C ∗ -algebras are separable.The C ∗ -algebra of adjoint-able operators on a Hilbert B -module E is denoted by L B ( E ) or L ( E ) and its two-sided closed ideal of ‘compact’ operators by K B ( E ).The reference for group equivariant KK -theory is Kasparov [12], for groupoidequivariant KK -theory it is Le Gall [13], and for inverse semigroup equivariant KK -theory it is [3], or see [2] for a summary of the definitions. (We use the slightlyadapted ‘compatible’ version of equivariant KK -theory as in [2].) The category of G -equivariant C ∗ -algebras with the Kasparov groups as morphisms is denoted by KK G .In this paper we write compositions of morphisms in a category and compositionsof functions from left to right. That is, for instance, if f : A → B and g : B → C are maps, then we write f g for g ◦ f , where composition operator ◦ is used in theusual sense from right to left. This will go as far as that we write f g ( x ) for g ( f ( x )).In spaces of operators like L ( E ) we use the multiplication in the usual sense, thatis, ST means S ◦ T for S, T ∈ L ( E ), but to avoid confusion, we mostly write S ◦ T .For a G -action S on a Hilbert module E we write Ad( S ) for the G -action γ g ( T ) = S g ◦ T ◦ S g − on L ( E ). For a unitary U ∈ A we write Ad( U ) for the ∗ -automorphism f ( a ) = U aU ∗ on A .If we notate grading in a Kasparov element as for example in [ π ⊕ π , E ⊕E , F ] ∈ KK G ( A, B ), then the first notated summand E always means the odd graded part,and the second summand E the even graded part. We also write [ σ + σ , E ⊕E , F ],where then σ ( a ) = π ( a ) ⊕ σ ( a ) = 0 ⊕ π ( a ).A map into multiplication operators like the canonical embedding f : A → L A ( A )is often sloppily denoted by id, or written as f ( a ) = a . The identity map is oftendenoted by 1 (for example in T ⊗ C ∗ -algebra A we write e , e : A → M ( A ) for the twocorner embeddings into the upper left and lower right corner respectively.We denote A ⊗ ( C ([0 , , triv) by A [0 , G -action.In [2] we have proven the universal property of G -equivariant KK -theory when G is a countable discrete inverse semigroup. In this section we remark that theproof works verbatim also when G is a locally compact, not necessarily Hausdorffgroupoid.Indeed, let G be a locally compact groupoid with base space X . At first we mayconsider it as a discrete inverse semigroup S by adjoining a zero element to G , i.e.set S := G ∪ { } .A G -action α on A is then fiber-wise just like an inverse semigroup S -action on A (the zero element 0 ∈ S acts always as zero), with the additional property thatit is continuous in the sense that it forms a map α : s ∗ A → r ∗ A . We cannot, as ininverse semigroup theory, say that α ss − ( A ) is a subalgebra of A ( s ∈ S ), becausethis instead we would interpret as a fiber A ss − of A . But all computations donefor inverse semigroups would be the same if we did it for a groupoid on fibers. Thatis why we need only take care that every introduced G -action is continuous.But the introduced actions, or similar constructions are just: • Cocycles: The definition [2, def. 5.1] has to be replaced by the analogousdefinition 2.1 below.
B. BURGSTALLER • Unitization: One replaces [2, def. 3.3] by [13]. • Direct sum, internal, external tensor product: It is clear that these con-structions are also continuous for groupoids. • For an element [ T, E ] ∈ KK G ( A, B ) one has the condition that the bundle g g ( T s ( g ) ) − T r ( g ) ∈ K ( E r ( g ) ) is in r ∗ K ( E ). Here one has also additionallyto check continuity. Definition 2.1.
Let (
A, α ) be a G -algebra. Set the G -action on L A ( A ) ∼ = M ( A )to be α := Ad( α ). An α -cocycle is a unitary u in r ∗ (cid:0) M ( A ) (cid:1) such that u gh = α g ( u h )in M ( A ) r ( g ) for all g, h ∈ G with s ( g ) = r ( h ).In this way it is (almost) clear that the results of [2] hold also in the locallycompact (not necessarily Hausdorff) groupoid equivariant setting. Corollary 2.2.
Let G be a locally compact (not necessarily Hausdorff ) groupoid.Then KK G is the universal stable, homotopy invariant and split exact categorydeduced from the category of G -equivariant, separable, ungraded C ∗ -algebras. From now on, if nothing else is stated, we assume that G is an inverse semigroup.This is almost invisible, except at least in corollary 7.8.(iv) and it is obvious howto adapt it to groupoids keeping the above remarks in mind.3. GK -theory We are going to recall the definition of GK -theory (“Generators and relations KK -theory”, the group G is not indicated, instead we may also write GK G ) forwhich we refer for more details to [4]. The split exactness axiom is slightly butequivalently altered, see [5, Lemma 3.7]. Definition 3.1.
Let GK be the following category. Object class of GK is the classof all separable G -algebras.Generator morphism class is the collection of all G -equivariant ∗ -homomorphisms f : A → B (with obvious source and range objects) and the collection of thefollowing “synthetical” morphisms: • For every equivariant corner embedding e : ( A, α ) → ( A ⊗ K , δ ) ( δ neednot be diagonal but can be any G -action) add a morphism called e − :( A ⊗ K , δ ) → ( A, α ). • For every equivariant short split exact sequence(1) S : 0 / / ( B, β ) j / / ( M, δ ) f / / ( A, α ) / / s o o C ∗ add a morphism called ∆ S : ( M, δ ) → ( B, β ) or ∆ s if S is understood.Form the free category of the above generators together with free addition andsubstraction of morphims having same range and source (formally this is like the freering generated by these generator morphisms, but one can only add and multiply ifsource and range fit together) and divide out the following relations to turn it intothe category GK : • ( C ∗ -category ) Set g ◦ f = f g for all f ∈ C ∗ ( A, B ) and g ∈ C ∗ ( B, C ). • ( Unit ) For every object A , id A is the unit morphism. SPECTS OF EQUIVARIANT KK -THEORY 5 • ( Additive category ) For all diagrams A i A / / A ⊕ B p B / / p A o o A i B o o (canonical projections and injections) set 1 A ⊕ B = p A i A + p B i A . • ( Homotopy invariance ) For all homotopies f : A → B [0 ,
1] in C ∗ ( A, B ) set f = f . • ( Stability ) All corner embeddings e are invertible with inverse e − . • ( Split exactness ) For all split exact sequences (1) set1 B = j ∆ s M = ∆ s j + f s Remark 3.2. (i) If we have given a split exact sequence (1) then it splits completelyas linear maps, that is, j has a linear split t : M → B with t ( x ) = j − ( x − f s ( x )),and the split exactness relations of definition 3.1 are satisfied for such a split ∆ s := t .Thus ∆ s may be viewed as a substitute for this linear split j , and it is often usefulto think about ∆ s as j in heuristical considerations.(ii) Set-theoretically M = j ( B ) + s ( A ) in (1), and this is a direct linear sum bythe last point.(iii) If (1) has the flaw that it is not exact in the middle but only j ( B ) ⊆ ker( f )then this can be repaired by restricting M to the G -subalgebra N := j ( B ) + s ( A ).(iv) j, s, f in (1) all influence ∆ s . This is clear for the linear split t , and so thismust be even more true for the free generator ∆ s .(v) If we have given an additional homomorphism u : A → M in (1) then this isa second split for f if and only if u ( a ) − s ( a ) ∈ j ( B ) ∀ a ∈ A (vi) Given (1), we have s ∆ s = 0 because s ∆ s = s ∆ s j ∆ s = s (1 − f s )∆ s = 0.(vii) If f : ( A, α ) → M n ( A, δ ) is a corner embedding, then it is invertible in GK .In fact, g : ( M n ( A ) , δ ) → ( M n ( A ) ⊗ K , δ ⊗ triv) is an invertible corner embedding,as well as f g , so f itself must be invertible.4. The functor A Since equivariant KK -theory is stable, homotopy invariant and split exact, thereis a functor from the univerally defined GK -theory to KK -theory. It can be con-cretely constructed as follows, see [15] in the group equivariant case, or [2, section4] for the inverse semigroups equivariant setting. Definition 4.1.
Define A : GK → KK G to be the additive functor which isidentical on objects and as follows on generator morphisms:(i) For an equivariant ∗ -homomorphism f : A → B we put A ( f ) = f ∗ ([id , A, ∈ KK G ( A, B )(ii) For a corner embedding e : ( A, α ) → ( A ⊗ K , δ ) in C ∗ we set A ( e − ) = [id , (cid:0) ( A ⊗ K ) E, δ (cid:1) , ∈ KK G ( A ⊗ K , A )where E := e ( A ) ⊆ A ⊗ K is the G -invariant corner G -algebra.(iii) For a split exact sequence (1) we define the equivariant ∗ -homomorphism(2) χ : M → L B ( B ) : χ ( x )( b ) = j − ( xj ( b )) B. BURGSTALLER and A (∆ s ) = [ f sχ ⊕ χ, ( B ⊕ B, β ⊕ β ) , F ] ∈ KK G ( X, A )where B ⊕ B has the grading − ⊕ + and F is the flip operator.5. Double split exact sequences
Throughout, (
A, α ) and (
B, β ) are G -algebras. Definition 5.1. A double split exact sequence is a diagram of the form0 / / ( B, β ) j / / ( M, γ ) f / / e & & ▲▲▲▲▲▲▲▲▲▲ ( A, α ) s o o t z z ✈✈✈✈✈✈✈✈✈ / / M ( M ) , θ ) ( M, δ ) e o o where all morphisms in the diagram are equivariant ∗ -homomorphisms, the firstline is a split exact sequence in C ∗ , t is another split in the sense that tf = 1 A innon-equivariant C ∗ , and e ii are the corner embeddings. Definition 5.2.
Consider a double split exact sequence as above. We denote by µ θ , or µ if θ is understood, the morphism µ : ( M, δ ) → ( M, γ ) : µ = e e − in GK . The morphism in GK associated to the double split exact sequence is tµ ∆ s We use sloppy language and say for example “the diagram is tµ ∆ s in GK ”,or two double split exact sequences are said to be “equivalent” if their associatedmorphisms are. Throughout, the short notation for the above double split exactsequence will be ( B, β ) j / / ( M, γ ) f / / ( A, α ) s,t o o Notating such a diagram, it is implicitely understood that this is a double splitexact sequence as above if nothing else is said. Often s, t is stated as s ± , which hasto be read as s − , s + . The G -action θ of definition 5.1 will sometimes be called the“ M -action of the double split exact sequence” for simplicity. Example 5.3.
Assume G is the trivial group. Then µ = 1 is the identity in GK because e and e are homotopic by a rotation in C ∗ .Consequently, we have double split exact sequences in the more usual sense and tµ ∆ s = t ∆ s . Moreover, t ∆ s = ( t − s )∆ s . SPECTS OF EQUIVARIANT KK -THEORY 7 Lemma 5.4.
Consider two double split exact sequences which are connected bythree morphisms b, Φ , a in GK as in this diagram: B i / / b (cid:15) (cid:15) M f / / e ●●●●●●●●● φ (cid:15) (cid:15) A s − o o s + (cid:127) (cid:127) ⑦⑦⑦⑦⑦⑦⑦⑦ a (cid:15) (cid:15) M ( M ) Φ (cid:15) (cid:15) M e o o ψ (cid:15) (cid:15) M ( N ) N f o o D j / / N g / / f ; ; ✇✇✇✇✇✇✇✇✇ C t − o o t + _ _ ❅❅❅❅❅❅❅❅ Here we have defined φ := e Φ f − ψ := e Φ f − (i) Then for the commutativity of the left rectangle of the diagram we note∆ s − b = φ ∆ t − ⇐ iφ = bj and f s − φ = φgt − (ii) For commutativity within the right big square of the diagram we observe s + µφ = at + µ ⇔ s + ψ = at + (iii) Consequently, commutativity of double split exact sequences in this diagramcan be decided as s + µ ∆ s − b = at + µ ∆ t − ⇐ Conditions of (i) and (ii) hold true
Proof. (i) We compute φ ∆ t − = (∆ s − i + f s − ) φ ∆ t − = ∆ s − bj ∆ t − + φgt − ∆ t − = ∆ s − b (ii) This is clear by commutativity of involved rectangles in the diagram andinvertibility of all corner embeddings. (iii) Also clear. (cid:3) We will exclusively encounter this situation:
Remark 5.5.
Assume that Φ = φ ⊗ M is equivariant for a non-equivariant ∗ -homomorphism φ : M → M .Then this φ is the φ and ψ in the above diagram as non-equivariant maps, andboth are automatically equivariant as maps as entered in the diagram.6. The M (cid:3) A -construction We shall use the following standard procedure to produce split exact sequences,and this is in fact key:
Definition 6.1.
Let i : ( B, β ) → ( M, γ ) be an equivariant injective ∗ -homomorphismsuch that the image of i is an ideal in M . Let s : ( A, α ) → ( M, γ ) be an equivariant ∗ -homomorphism. Then we define the equivariant G -subalgebra M (cid:3) A := { ( s ( a ) + i ( b ) , a ) ∈ M ⊕ A | a ∈ A, b ∈ B } B. BURGSTALLER of ( M ⊕ A, γ ⊕ α ). The G -action on M (cid:3) A is denoted by γ (cid:3) α . In particular wehave a split exact sequence0 / / B j / / M (cid:3) A f / / A / / s (cid:3) o o , where j ( b ) = ( i ( b ) , f ( m, a ) = a and ( s (cid:3) a ) := ( s ( a ) , a ) for all a ∈ A, b ∈ B, m ∈ M .If we have given a double split exact sequence as in definition 5.1 with M of theform M (cid:3) A then it is understood that j, f and s (cid:3) M (cid:3) A refers always to the firstnotated split s , or the split indexed by minus (e.g. s − (cid:3)
1) if it appears in a doublesplit exact sequence. We also write s (cid:3) instead of s (cid:3) M (cid:3) A by m (cid:3) a := ( m, a ). The operator (cid:3) binds weakly, that is forexample, m + n (cid:3) a = ( m + n ) (cid:3) a .Non-equivariantly we have(3) M ( M (cid:3) A ) ∼ = M ( M ) (cid:3) M ( A ) ⊆ M ( M ) ⊕ M ( A )with respect to i ⊗ s ⊗ G -algebras ( M ( M ) , γ ) and ( M ( A ) , δ ) and ( M ( M (cid:3) A ) , θ ) is canon-ically a G -invariant G -subalgebra of ( M ( M ) ⊕ M ( A ) , γ ⊕ δ ) then we call θ also γ (cid:3) δ . Lemma 6.2.
Consider definition 6.1. If we have G -algebras ( M ( M ) , γ ) and ( M ( A ) , δ ) and M ( i ( B )) is G -invariant under γ then M ( M (cid:3) A ) is G -invariantunder the G -action γ ⊕ δ if and only if γ g (( s ⊗ a )) − ( s ⊗ δ g ( a )) ∈ M ( i ( B )) ∀ a ∈ M ( A ) Proof.
We apply the isomorphism (3) and may work with i ⊗ s ⊗ (cid:3) One may observe that the non-equivariant splits of the exact sequence of defi-nition 6.1 are exactly the maps of the from t (cid:3)
1. We may bring any double splitexact sequence to this form:
Lemma 6.3. (i)
Any double split exact sequence as in the first line of this diagramcan be completed to this diagram B i / / (cid:15) (cid:15) M f / / φ (cid:15) (cid:15) A s ± o o (cid:15) (cid:15) B j / / M (cid:3) A g / / A t ± o o such that the first line ist the second line in GK, that is, s + µ ∆ s − = t + µ ∆ t − . (ii) If the G -action on M ( M ) is θ , then the G -action on M ( M (cid:3) A ) is θ (cid:3) δ ifand only if the G -action θ (cid:3) δ exists if and only if f ⊗ M ( M ) , θ ) → ( M ( A ) , δ ) is equivariant. (iii) The G -action on M ( M (cid:3) A ) is of the form θ (cid:3) δ . SPECTS OF EQUIVARIANT KK -THEORY 9 Proof. (i) Define the second line of the diagram as in definition 6.1, that is, put g ( m (cid:3) a ) = a , j ( b ) = i ( b ) (cid:3) t ± ( a ) = s ± ( a ) (cid:3) a . Define φ ( m ) = m (cid:3) f ( m )We are going to apply lemma 5.4 for B = D , A = C , a = 1, b = 1, andΦ = φ ⊗ M . Note that φ is bijective. We define the G -action on M ( M (cid:3) A ) insuch a way that Φ becomes equivariant. By remark 5.5, φ = ψ in the diagram oflemma 5.4 and both maps are G -equivariant. We have iφ ( b ) = i ( b ) (cid:3) j ( b ) f s − φ ( m ) = f s − ( m ) (cid:3) f ( m ) = φg ( s − (cid:3) m ) , which is the condition of lemma 5.4.(i). Further s + φ ( a ) = s + ( a ) (cid:3) a = t + ( a ) yieldsthe condition of lemma 5.4.(ii). Hence the claim follows from lemma 5.4.(iii).(ii) We assume that ( M ( A ) , δ ) exists and want to see when θ (cid:3) δ is valid:Note that m ⊕ a ∈ M (cid:3) A if and only if f ( m ) = a . Hence θ g ( m ) ⊕ δ g ( a ) ∈ M ( M (cid:3) A ) if and only if ( f ⊗ θ g ( m )) = δ g ( a ). Set a = ( f ⊗ m ).(iii) This follows from (ii) and corollary 7.5, which is independent from thislemma. (cid:3) Actions on M ( A )In this section we want to inspect closer how a M -action of a double split exactsequences looks like. This is a key lemma: Lemma 7.1.
Let
S, T be two G -actions on a Hilbert ( B, β ) -module E . Then α g (cid:18) x yz w (cid:19) = (cid:18) S g xS g − S g yT g − T g zS g − T g wT g − (cid:19) defines a G -action α on M (cid:0) L B ( E ) (cid:1) This is actually the inner action
Ad( S ⊕ T ) on L B (cid:0) ( E , S ) ⊕ ( E , T ) (cid:1) . Definition 7.2.
The α of the last lemma is also denoted by Ad( S ⊕ T ) or Ad( S, T ). Lemma 7.3.
Let ( A, α ) and ( A, δ ) be G - C ∗ -algebras.Let ( M ( A ) , θ ) be a G -algebra and the corner embeddings e : ( A, α ) → ( M ( A ) , θ ) and e : ( A, δ ) → ( M ( A ) , θ ) be equivariant.Then θ is of the form θ g (cid:18) x yz w (cid:19) = (cid:18) α g ( x ) β g ( y ) γ g ( z ) δ g ( w ) (cid:19) Also: (i)
One has the relations γ g ( ax ) = δ g ( a ) γ g ( x ) γ g ( xb ) = γ g ( x ) α g ( b )(4) β g ( ax ) = α g ( a ) β g ( x ) β g ( xb ) = β g ( x ) δ g ( b )(5) α g ( xy ) = β g ( x ) γ g ( y ) δ g ( xy ) = γ g ( x ) β g ( y )(6) β g ( y ) = γ g ( y ∗ ) ∗ γ gh = γ g γ h β gh = β g β h (7)(ii) ( A, γ ) is an imprimitivity Hilbert (( A, δ ) , ( A, α )) -bimodule, where the bimod-ule structure is multiplication in A , and the right inner product is h a, b i = a ∗ b andthe left one is h a, b i = ab ∗ . (iii) Analogously, ( A, β ) is an imprimitivity Hilbert (( A, α ) , ( A, δ )) -bimodule. (iv) Let χ : A → L A ( A ) be the natural embedding.Then α and γ are G -actions on the Hilbert ( A, α ) -module A .Consequently we have the G -action Ad( α ⊕ γ ) on the matrix algebra M ( L ( A,α ) ( A )) .The map χ ⊗ M : ( M ( A ) , θ ) → ( M ( L ( A,α ) ( A )) , Ad( α ⊕ γ )) is a G -equivariant injective ∗ -homomorphism. (v) γ determines θ uniquely and completely. (vi) θ determines α, β, γ and δ uniquely. (vii) In general, α and δ do not determine γ and thus not θ . (viii) If we drop all assumptions then we may add:A G -algebra ( A, α ) and a Hilbert ( A, α ) -module action γ on A alone ensure theexistence of the above θ with all assumptions and assertions of this lemma.Proof. If a ∈ A , ( a i ) ⊆ A an approximate unit of A , and we apply θ g to (cid:18) a i
00 0 (cid:19) (cid:18) a (cid:19) (cid:18) a i (cid:19) then we see that the θ g applied to the middle matrix has again the form of themiddle matrix.(i) One computes expressions like (cid:18) γ g ( z ) 0 (cid:19) (cid:18) α g ( x ) 00 0 (cid:19) (cid:18) δ g ( x ) (cid:19) (cid:18) γ g ( z ) 0 (cid:19) θ (cid:18) z (cid:19) ∗ and uses the fact that θ is a G -action on a C ∗ -algebra.(ii) Put the first relation of (7) into line (6) and then use lines (4) and (6).(iv) By (ii), α and γ are G -actions as claimed, so that the existence of Ad( α ⊕ γ )is by lemma 7.1. By relations (i) one can deduce(8) (cid:18) α g ( x ) x ′ β g ( y ) y ′ γ g ( z ) z ′ δ g ( w ) w ′ (cid:19) = (cid:18) α g ( xα g − ( x ′ )) α g ( yγ g − ( y ′ )) γ g ( zα g − ( z ′ )) γ g ( wγ g − ( w ′ )) (cid:19) for all x, ..., w ′ ∈ A , which shows G -equivarinace of χ ⊗
1. In fact, the second matrixline follows directly from (4), and the upper right corner from the first relation of(6).(v) α, δ and β are determined by γ by (6) and the first relation of (7).(viii) By (iv), we can construct Ad( α ⊕ γ ) and aim to define θ by its restriction.To show that the image of χ ⊗ G -invariant, we consider the right hand side of(8) and want to construct identity with the left hand side. For the first column thisis clear. Setting β as in the first identity of (7) we get the upper right corner. Thelower right corner follows from γ g ( aa ∗ γ g − ( x )) = γ g ( a ) α g ( a ∗ γ g − ( x )) = γ g ( a ) γ g ( a ) ∗ x (vii) Take for example G = Z / A = C (or any A ), α = δ the trivial action.Then γ g ( x ) = x and γ g ( x ) = ( − g x are two valid choices. (cid:3) Corollary 7.4.
Let ( M ( A i ) , θ i ) be two G -algebras as in lemma 7.3 ( i = 1 , ). Let φ : A → A be a non-equivariant ∗ -homomorphism. Then φ ⊗ M is G -equivariantif and only if it is G -equivariant on the lower left corner space ( A , γ ) .Proof. By lemma 7.3.(i), confer also (v). (cid:3)
SPECTS OF EQUIVARIANT KK -THEORY 11 Corollary 7.5.
Consider the double split exact sequence of definition 5.1. (i)
Then the ideal M ( j ( B )) is invariant under the action θ . (ii) The map f ⊗ M ( M ) , θ ) → ( M ( A ) , δ ) is equivariant for the quotient G -action δ on M ( A ) ∼ = M ( M ) /M ( j ( B )) .Proof. Write θ as in lemma 7.3. The corner action α leaves j ( B ) invariant. Hence,by the formulas of lemma 7.3.(i) we see that γ, β, δ leave an element b ∈ j ( B )invariant. (cid:3) Definition 7.6.
Let U ∈ A be a unitary in a C ∗ -algebra A . Then we define the ∗ -homomorphism κ U : M ( A ) → M ( A ) : κ U (cid:18) x yz w (cid:19) = (cid:18) x yU ∗ U z U wU ∗ (cid:19) In other words, κ U = Ad(1 ⊕ U ) on L D ( D ⊕ D ) ∼ = M ( L D ( D )) for A = L D ( D ).Notice that κ − U = κ U ∗ . Definition 7.7.
Let U ∈ A be a unitary in a G -algebra ( A, α ). Then we write θ U for the G -action on M ( A ) defined by θ Ug = κ U ◦ ( α g ⊗ M ) ◦ κ − U Lemma 7.8.
Let
S, T be two G -actions on a Hilbert ( B, β ) -module E .Consider a diagram K B ( E ) / / L B ( E ) (cid:3) A / / A s ± (cid:3) o o which is double split exact except that we have not found a M -action yet. But weknow that s − is equivariant with respect to Ad( S ) on L ( E ) , and s + is equivariantwith respect to Ad( T ) on L ( E ) .Equip M ( L B ( E ) ⊕ A ) ∼ = L B ( E ⊕ E ) ⊕ M ( A ) with the G -action Ad( S ⊕ T ) ⊕ ( α ⊗ M ) . Then the following assertions are equivalent: (i) s − ( a ) (cid:0) S g T g − − S g S g − (cid:1) ∈ K B ( E ) for all g ∈ G , a ∈ A (ii) S g s − ( a ) T g − − s − ( α g ( a )) ∈ K B ( E ) for all g ∈ G , a ∈ A (iii) M ( L B ( E ) (cid:3) A ) is a G -invariant subalgebra.In case that there is a unitary U ∈ L ( E ) such that T g ◦ U = U ◦ S g for all g ∈ G ,that is if Ad( S ⊕ T ) = θ U for the G -action Ad( S ) on L ( E ) , these conditions are also equivalent to (iv) s − ( a ) (cid:0) g ( U ) − U g (1) (cid:1) ∈ K B ( E ) for all g ∈ G , a ∈ A ( G -action is Ad( S ) ).Proof. (ii) ⇒ (iii): Let x := s − ( a ) + k (cid:3) a ∈ X := L ( B ) (cid:3) A for a ∈ A, k ∈ K ( B ).Put x into the lower left corner of M ( X ) and apply the G -action and see whatcomes out:(9) S g (cid:0) s − ( a ) + k (cid:1) T g − (cid:3) α g ( a ) = s − ( α g ( a )) + S g kT g − (cid:3) α g ( a ) ∈ L ( E ) (cid:3) A Similarly we get it for the upper right corner by taking the adjoint in (ii). Forthe lower right corner we observe T g ( s − ( a ) + k ) T g − (cid:3) α g ( a ) = T g (cid:0) s + ( a ) + k ′ + k (cid:1) T g − (cid:3) α g ( a ) ∈ L ( E ) (cid:3) A for a certain k ′ ∈ K ( B ) by remark 3.2.(iii) ⇒ (ii): By (9) for k = 0. (i) ⇒ (ii): S g s − ( a ) T g − = S g s − ( a ) S g S g − T g − ≡ s − ( α g ( a )) mod K ( E )Since T = U ◦ S ◦ U ∗ , (i) ⇔ (iv) is obvious. (cid:3) By using corollary 7.5, the equivalence between (ii) and (iii) of the last lemmamay be analogously generalized to diagrams of the form B / / M (cid:3) A / / A s ± (cid:3) o o .8. Computations with double split exact sequences
From now on, if nothing else is said, the M -action on M (cid:3) A is always understoodto be of the form γ (cid:3) ( α ⊗
1) for G -algebras ( M ( M ) , γ ) and ( A, α ).Actions on L B ( E ) will always be of the form Ad( S ) for a G -action S on E . Lemma 8.1.
Given an equivariant ∗ -homomorphism f : A → B we get a doublesplit exact sequence B i / / B ⊕ A g / / A s ± o o with s − ( a ) = (0 , a ) , s + ( a ) = ( f ( a ) , a ) and one has f = s + µ ∆ s − in GK .Proof. The M -space is ( M ( A ⊕ B ) , ( β ⊕ α ) ⊗ M ), µ = 1 by a rotation homotopy, i ( b ) = ( b, , g ( b, a ) = a and ∆ s − = (1 − gs − ) i − is just the linear split, see remark3.2. (cid:3) Lemma 8.2.
Given the first line and an equivariant ∗ -homomorphism ϕ as in thisdiagram it can be completed to this diagram B i (cid:3) / / M (cid:3) A / / A s ± (cid:3) o o B i (cid:3) / / O O M (cid:3) X / / φ O O X t ± o o ϕ O O such that ϕ ( s + (cid:3) µ ∆ s − (cid:3) = t + µ ∆ t − in GK .We assume here that the G -action on M ( M (cid:3) A ) is of the form θ (cid:3) ( α ⊗ .Proof. Let X = ( X, γ ). If the M -action of the first line is θ (cid:3) ( α ⊗ θ (cid:3) ( γ ⊗ φ = id (cid:3) ϕ and t ± = ϕs ± (cid:3) φ ⊗ (cid:3) Lemma 8.3.
Every split exact sequence as in the first line is isomorphic to theone of the second line as indicated in this diagram: B i / / (cid:15) (cid:15) M φ (cid:15) (cid:15) f / / A s ± o o (cid:15) (cid:15) B j / / L B ( B ) (cid:3) A / / A t ± o o That is, s + µ ∆ s − = t + µ ∆ t − in GK.The G -action on M ( L B ( B ) (cid:3) A ) is of the form Ad( S ⊕ T ) (cid:3) δ , where ( M ( A ) , δ ) and ( M ( L B ( B )) , Ad( S ⊕ T )) are G -algebras. SPECTS OF EQUIVARIANT KK -THEORY 13 Proof.
Set j = iφ . Define χ analogously as in (2). Set φ ( m ) = χ ( m ) (cid:3) f ( m ). Put t ± ( a ) = χ ( s ± ( a )) (cid:3) a . Note that φ is bijective. Set Φ = φ ⊗ M and define the G -action on its range in such a way that Φ becomes G -equivariant. Verify withlemma 5.4.For the sake of simpler notation we assume now that i is the identity embedding.By lemma 7.3.(iv), M ( M ) embedds equivariantly into L M ( M ⊕ M, Ad( S ⊕ T )),confer the formula in lemma 7.1.But S restricts to a G -action on B , and T restricts to a Hilbert ( B, S )-moduleon B (because T ( b ) = T ( b ) S ( b ) ∈ B ).Hence by 7.3.(vii) we can equip L B ( B ⊕ B ) with the G -action Ad( S ⊕ T ) as well,or in other words, use the same formula as in lemma 7.1. The G -action δ comesfrom corollary 7.5. (cid:3) Definition 8.4.
Let (
B, β ) be a G -algebra. Write H B := L ∞ i =1 B for the infinitedirect sum Hilbert ( B, β )-module. We shall equip H B with various G -actions S ,but often require that S is of the form S = β ⊕ T for a G -action T on H B ( S g ( b ⊕ b ⊕ b ⊕ . . . ) = β g ( b ) ⊕ T ( b ⊕ b ⊕ . . . )). The letter R will always stand for sucha G -action and we may pick out R := β ⊕ triv deliberately.If a copy of H B is derived from another construction, say the Kasparov stabi-lization theorem then we always keep the original G -action: Lemma 8.5.
Let ( E , S ) and ( H B , β ⊕ T ) be G -Hilbert ( B, β ) -modules. Then thereis a G -Hilbert ( B, β ) -module isomorphism Y : ( E , S ) ⊕ ( H B , β ⊕ T ) → ( H B , β ⊕ V ) . Proof.
Excluding the first coordinates (
B, β ) of the H B s on which Y is set to bethe identity, we apply Kasparov’s non-equivariant stabilization theorem to obtain Y , and define the G -action V in such a way that Y becomes G -equivariant. (cid:3) Definition 8.6.
Define the C ∗ -algebra isomorphism κ : B ⊗ K → K B ( H B ) ⊆ L B ( H B ) : κ (cid:0) ( b ij ⊗ e ij ) i,j (cid:1)(cid:0) ( ξ k ) k (cid:1) = (cid:16) X j b ij ξ j (cid:17) i where b ij , ξ ∈ B and 1 ≤ i, j, k . We equip B ⊗ K with the G -action such that κ becomes equivariant. That is, if γ is the G -action on L B ( H B ) then δ = κ − ◦ γ ◦ κ is the G -action on B ⊗ K . Definition 8.7.
If the G -action on H B is β ⊕ S then we have an G -invariant cornerembedding ( B, β ) → ( B ⊗ K , δ ) which we denote by e δ or e if δ is understood.In lemma 8.2 we saw how we can merge a homomorphism from the right handside with a split exact sequence. The next lemma is the analogy from the left handside. Lemma 8.8.
Let the first line of the following diagram be given, where f denotesan equivariant ∗ -homomorphism. Then it can be completed to this diagram C e (cid:15) (cid:15) B f o o i / / L B ( B ) (cid:3) A / / φ (cid:15) (cid:15) A s ± (cid:3) o o (cid:15) (cid:15) C ⊗ K κ / / L C (cid:0) B ⊗ f C ⊕ H C (cid:1) (cid:3) A / / A t ± o o such that ( s + (cid:3) µ ∆ s − (cid:3) f e = t + µ ∆ t − in GK .Proof. Let C = ( C, γ ). We set φ ( T (cid:3) a ) = T ⊗ ⊕ (cid:3) a and t ± = ( s ± (cid:3) φ . If the M -action of the first line is Ad( S ⊕ T ) (cid:3) δ (confer lemma8.3), then we set it to Ad( S ⊗ γ ⊕ R , T ⊗ γ ⊕ R ) (cid:3) δ in the second line.We have a G -Hilbert C -module isomorphism B ⊗ f C → C := f ( B ) C ⊆ C : b ⊗ c f ( b ) c (norm closure of sums) into a G -Hilbert C -submodule C of C .We have an equivariant ∗ -homomorphism h : M ( f ( B )) → L C ( C ⊕ C ) ⊆ L C ( C ⊕ H C )by matrix-vector multiplication, where the summand C means here the distin-guished first coordinate ( C, γ ) of H C .That is why we can rotate iφ to g for g ( b ) = (0 ⊗ ⊕ ( f ( b ) ⊕ (cid:3) h , which is in κ ( C ⊗ K ).Thus iφ = g = f eκ in GK . It is now easy to verify with lemma 5.4. (cid:3) Lemma 8.9.
Given the upper right double split exact sequence of this diagram onecan draw this dagram B ⊗ K (cid:15) (cid:15) K B ( E ) h o o j / / L B ( E ) (cid:3) A / / φ (cid:15) (cid:15) A s ± (cid:3) o o (cid:15) (cid:15) B ⊗ K κ / / L B ( E ⊕ H B ) (cid:3) A / / A t ± o o such that the first line is the second line in GK , i.e. ( s + (cid:3) µ ∆ s − (cid:3) h = t + µ ∆ t − .Proof. We set φ ( T (cid:3) a ) = T ⊕ (cid:3) a , t ± = s ± ⊕ (cid:3) id and h = jφκ − . If Ad( S ⊕ T ) (cid:3) δ is the M -action of the first line, we put it to Ad( S ⊕ R , T ⊕ R ) (cid:3) δ on the secondline. Recall that B ⊗ K has the G -action κ − ◦ Ad( S ⊕ R ) ◦ κ . Verify with lemma5.4. (cid:3) Lemma 8.10.
Let the first line of the following diagram be given and φ t be evalu-ation at time t ∈ [0 , . Then it can be completed to this diagram B [0 , φ t (cid:15) (cid:15) e / / B [0 , ⊗ K / / a t (cid:15) (cid:15) L ( E ⊕ H B [0 , ) (cid:3) A / / b t (cid:15) (cid:15) A s ± o o (cid:15) (cid:15) B e t / / B ⊗ K / / L ( E ⊗ φ t B ⊕ H B ) (cid:3) A / / A s ± b t o o such that s + µ ∆ s − e − φ t = s + b t µ ∆ s − b t e t − for all t ∈ [0 , and these elementsdo not depend on t .Proof. Here a t and b t with b t ( T (cid:3) a ) = T ⊗ φ t (cid:3) a are the evaluation maps at time t ∈ [0 , φ t is the evaluation of the identity homotopy on B [0 ,
1] in C ∗ (which, note, a t is not) φ = φ t for all t . SPECTS OF EQUIVARIANT KK -THEORY 15 If the M -action of the first line of the diagram is Ad( S ⊕ R, T ⊕ R ) (cid:3) δ then ofthe second line it is Ad(( S ⊕ R ) ⊗ φ t β, ( T ⊕ R ) ⊗ φ t β ) (cid:3) δ .Now verify the claim with lemma 5.4 and Φ = b t ⊗ (cid:3) Normally, a homotopy runs in a fixed algebra with a fixed G -action. If wecombine homotopy with matrix technique, we can however allow homotopies wherethe G -action of the range algebra, and so the range object changes: Lemma 8.11.
Let s : A → (cid:0) M ( X [0 , , ( θ ( t ) ) t ∈ [0 , (cid:1) be an equivariant homomor-phism into the lower right corner for θ as in lemma 7.3. Assume that the upper leftcorner action θ ( t )11 does not depend on t ∈ [0 , . Then s e (0)11 − = s e (1)11 − : A → X in GK .Proof. Consider the diagram (cid:0) X [0 , , ( θ ( t )11 ) t ∈ [0 , (cid:1) e / / ψ t (cid:15) (cid:15) (cid:0) M ( X [0 , , ( θ ( t ) ) t ∈ [0 , (cid:1) φ t (cid:15) (cid:15) A s o o (cid:15) (cid:15) (cid:0) X, θ ( t )11 (cid:1) e ( t )11 / / (cid:0) M ( X ) , θ ( t ) (cid:1) A s t o o where e and e ( t )11 are the corner embeddings and φ t and ψ t are the evaluationmaps. Since both rectangles of the diagram commute we get se − ψ t = s t ( e ( t )11 ) − .For θ ( t )11 is independent of t , ψ t is evaluation of the identity homotopy, so ψ = ψ t in GK . (cid:3) In lemma 7.8.(iv) we have observed a G -equivariance condition reminiscent of KK -theory. In the next lemma we are going to observe how grading and thecommutator condition [ a, F ] ∈ K ( E ) come into play: Namely, if we start with asingle split exact sequence, how can we construct a second split?: Lemma 8.12.
Let U be a unitary in M and t : A → M a ∗ -homomorphism.Consider a diagram B i / / ( M, γ ) (cid:3) A / / A s (cid:3) o o U ◦ t ◦ U ∗ ⊕ v v ♥♥♥♥♥♥♥♥♥♥♥♥♥♥ ( M, δ ) ⊕ A Then the image of U ◦ t ◦ U ∗ ⊕ is in M (cid:3) A if and only if s ( a ) ◦ U − U ◦ t ( a ) ∈ i ( B ) for all a ∈ A if and only if h s ( a ) ⊕ t ( a ) , (cid:18) UU ∗ (cid:19) i ∈ M ( i ( B )) ∀ a ∈ A Proof.
The proof is straightforward, or see the similar proof of lemma 9.4. (cid:3)
If the condition of the last lemma is satisified, then U ◦ t ◦ U ∗ (cid:3) M -action.Typically t is equivariant with respect to γ and one defines the G -action on X := M ( M (cid:3) A ) by θ U (cid:3) . To this end X must be invariant under this action, and for M = L ( E ) this equivalent to the other condition of Kasparov theory, see lemma7.8. 9. The functor B For the following definition see for example [1, 17.6].
Definition 9.1.
Let z = [ s, ( E , S ) , F ] ∈ KK G ( A, B ) be a Kasparov element. Byfunctional calculus we choose an operator homotopy ( s, ( E , S ) , F ) ∼ ( s, ( E , S ) , F ′ )such that F ′ is self-adjoint and k F ′ k ≤
1. We denote the new F ′ by F again. Set U = (cid:18) F (1 − F ) / (1 − F ) / − F (cid:19) ∈ L B (cid:0) ( E , S ) ⊕ ( E , S ) (cid:1) Since U is a compact perturbation of F ⊕ ( − F ), by adding on zero cycles to z we get z = [ s ⊕ ⊕ , ( E ⊕ E ⊕ H B , S ⊕ S ⊕ R ) , U ⊕
1] =(10) z = [ s ⊕ , ( E ⊕ H B , S ⊕ V ) , U ( F )]where we have written ( E , S ) ⊕ ( H B , R ) ∼ = ( H B , V ) for simplicity, and have set U ( F ) := U ⊕ ∈ L B (cid:0) ( E , S ) ⊕ ( H B , V ) (cid:1) , which is a self-adjoint unitary, but notice that U ( F ) means still the first U ⊕ Lemma 9.2. (i) If ( s, E , F t ) t ∈ [0 , ∈ KK G ( A, B ) is an operator homotopy,then U ( F t ) t ∈ [0 , ∈ L B ( E ⊕ H B ) is homotopy of unitaries. (ii) If F is self-adjoint unitary and s : A → L B ( E ) a non-equivariant homo-morphism, then U ( F ) ◦ (cid:0) s ( a ) ⊕ (cid:1) ◦ U ( F ) ∗ = F ◦ s ( a ) ◦ F ∗ ⊕ If B = D [0 , and φ t : D [0 , → D is the evaluation map at t ∈ [0 , then U ( F ) ⊗ φ t U ( F ⊗ φ t .Proof. These claims follow easily from definition 9.1. Recall that the transitionfrom F to F ′ respects homotopy. (cid:3) Definition 9.3.
Let z = [ s − + s + , ( H B , S ) , F ] ∈ KK G ( A, B ) be given where F isunitary and S is of the form β ⊕ T , see definition 8.4.Then we define B ( z ) = t + µ ∆ t − e − that is in details, the element of GK associated to this diagram read from right toleft:(11) B e / / B ⊗ K κ / / L B (( H B , S )) (cid:3) A / / A t ± o o where t − ( a ) = s − ( a ) (cid:3) at + ( a ) = F ◦ s + ( a ) ◦ F ∗ (cid:3) a where the G -action on M ( L B (( H B , S )) (cid:3) A ) is θ F (cid:3) . The letter e denotes theequivariant corner embedding. SPECTS OF EQUIVARIANT KK -THEORY 17 Here, θ F (cid:3) is an incorrect but suggestive notation for θ F (cid:3) ( α ⊗ M ).We note that the action θ F (cid:3) is just Ad( S ⊕ T ) (cid:3) ( α ⊗ G -action T isdefined in such a way that the unitary F becomes equivariant, that is, T ◦ F = F ◦ S . Lemma 9.4.
The assignment B is well defined.Proof. The image of t + . Since F is odd graded, F = (cid:18) VV ∗ (cid:19) for a unitary V acting on the Hilbert module H B ∼ = E − ⊕ E + (graded parts). Let us also write s − ( a ) = v − ( a ) ⊕ s + ( a ) = 0 ⊕ v + ( a ) acting on E − ⊕ E + .Because [ s − ( a ) + s + ( a ) , F ] is a compact operator in Kasparov theory, we getthat [ v − ( a ) ⊕ v + ( a ) , F ] F ∗ = ( v − ( a ) ⊕ v + ( a )) F F ∗ − F ( v − ( a ) ⊕ v + ( a )) F ∗ = v − ( a ) ⊕ v + ( a ) − V v + ( a ) V ∗ ⊕ V ∗ v − ( a ) V is in K B ( H B ), so the first coordinate v − ( a ) − V v + ( a ) V ∗ is in K B ( E − ). But thismeans s − ( a ) − F s + ( a ) F ∗ ∈ K B ( H B ). Hence t + ( a ) ∈ t − ( A ) + κ ( B ⊗ K ) = L B ( H B ) (cid:3) AM -action. Define a G -action T on H B by T g ◦ F = F ◦ S g . By lemma 7.8.(iii)-(iv), the G -action θ F (cid:3) is valid. Corner embedding.
If we had two choices of corner embeddings e then they wouldbe the same in GK by a rotation homotopy. Homotopy invariance.
Let z = [ s − + s + , ( H B [0 , , S ) , F ] ∈ KK G ( A, B [0 , F , F are unitary. We rewrite z in the form (10) andgo with it into definition 9.3. Then applying the diagram (11) associated to B ( z )to lemma 8.10 we see by lemma 9.2.(ii)-(iii) that B ( z ) = B ( z ) for the evaluations z t of z at time t ∈ [0 , (cid:3) Lemma 9.5.
Let z = [ s − + s + , ( E , S ) , F ] ∈ KK G ( A, B ) be given.Then B ( z ) = t + µ ∆ t − e − for the diagram B e / / B ⊗ K κ / / L B ( E ⊕ H B ) (cid:3) A / / A t ± o o where E ⊕ H B is equipped with a G -action S ⊕ R and the G -action on M ( L B ( E ⊕ H B ) (cid:3) A ) is defined to be θ U ( F ) (cid:3) and t − ( a ) = s − ( a ) ⊕ (cid:3) at + ( a ) = U ( F ) ◦ ( s + ( a ) ⊕ ◦ U ( F ) ∗ (cid:3) a Proof.
We bring z to the form (10) and apply definition 9.3. (cid:3) Lemma 9.6.
Consider lemma 9.5 where z = [ s − + s + , ( E , S ) , ( F m ) m ∈ [0 , ] is anoperator homotopy. Then t (0)+ µ = t (1)+ µ in GK for t ( m )+ ( a ) = U ( F m ) ◦ ( s + ( a ) ⊕ ◦ U ( F m ) ∗ (cid:3) a Proof.
Set X = L B ( E ⊕ H B ) (cid:3) A . We consider M ( X [0 , , ( θ F ( U m ) (cid:3) ) m ∈ [0 , ).Now the claim follows by lemma 8.11. (cid:3) Connection between A and B Definition 10.1.
Let ( B ⊗K , γ ) be a G -algebra. Let E = B ⊗ e ii ∼ = ( B, β ) ⊆ B ⊗K be a G -invariant corner algebra. Then (( B ⊗ K ) E, γ ) is a G -Hilbert ( B, β )-modulewith all operations inherited from the G -algebra B ⊗ K . We define r : (cid:0) L B ⊗K ( B ⊗ K ) , Ad( γ ) (cid:1) → (cid:0) L B ( H B ) , Ad( Y ◦ γ ◦ Y − ) (cid:1) : r ( T )( ξ ) = Y ( T Y − ( ξ ))( T acts here by multiplication) where the G -Hilbert B -module isomorphism Y : ( B ⊗ K ) E → H B is the canonical map by regarding elements of the domain of Y as column vectors.The G -action on H B is Y ◦ γ ◦ Y − . Lemma 10.2. r and ζ : L B ( H B ) → L B ⊗K ( B ⊗ K ) : ζ ( T )( x ) = κ − ( T κ ( x )) ( T acts by multiplication) are inverse equivariant ∗ -homomorphisms to each other. Lemma 10.3.
Let θ be as in lemma 7.3 and µ θ = e e − : A → A Then A ( µ θ ) = [id A , ( A, γ ) , .Proof. Let A and A be the corner G -subalgebras of M ( A ). Then A ( µ θ ) = A ( e ) A ( e − ) = [( A A, θ ) , · [( AA , θ ) , A A, θ ) ⊗ A ( AA , θ ) ,
0] = [(
A, γ ) , (cid:3) Lemma 10.4.
Let the right part of the first line of the following diagram be doublesplit: B e / / (cid:15) (cid:15) B ⊗ K j / / ψ (cid:15) (cid:15) X f / / φ (cid:15) (cid:15) A u,h o o (cid:15) (cid:15) B / / B ⊗ K κ / / L B ( H B ) (cid:3) A / / A s ± o o Then A ( hµ ∆ u e − ) =[ uχr ⊕ hχr, ( H B , Y ◦ j − ◦ γ ◦ j ◦ Y − ) ⊕ ( H B , Y ◦ j − ◦ Γ ◦ j ◦ Y − ) , F ] where F is the flip, χ is like (2), r and Y are from definition 10.1, and (cid:18) γ Γ ′ Γ δ (cid:19) is the G -action on M ( X ) .Proof. Set I = B ⊗ K . For simpler notation we assume that I is embedded in X in the diagram. The G -action on M ( I ) is denoted in the same way as the one on M ( X ). We have two isomorphisms V : ( A ⊗ h X ⊗ χ I, α ⊗ Γ ⊗ γ ) → ( A ⊗ hχ I, α ⊗ Γ) : V ( a ⊗ x ⊗ i ) = a ⊗ xiW : ( A ⊗ h X ⊗ fuχ I, α ⊗ γ ⊗ γ ) → ( A ⊗ uχ I, α ⊗ γ ) : W ( a ⊗ x ⊗ i ) = a ⊗ u ( f ( x )) i of G -Hilbert I -modules.In the following computation F stands always for a flip operator (on possiblydifferent spaces). Notice that 0 F is a just a F -connection. Recall from [14, prop.9.(f)] the associativity of F -connections. We compute A ( hµ ∆ u ) = A ( h ) A ( µ ) A (∆ u ) = h ∗ ([id , ( X, Γ) , · [ f uχ ⊕ χ, ( I ⊕ I, γ ⊕ γ ) , F ]) SPECTS OF EQUIVARIANT KK -THEORY 19 = [id ⊕ id , A ⊗ h X ⊗ fuχ I ⊕ A ⊗ h X ⊗ χ I, F ]= [id ⊕ id , ( A ⊗ uχ I, α ⊗ γ ) ⊕ ( A ⊗ hχ I, α ⊗ Γ) , F ]= [ uχ ⊕ hχ, ( I ⊕ I, γ ⊗ Γ) , F ]Multiplying this with A ( e − ) = [( IE, γ ) , ∈ KK G (( I, γ ) , ( B, β )) this gives A ( hµ ∆ u e − ) = [ uχ ⊕ hχ, ( I, γ ) ⊗ ( I,γ ) ( IE, γ ) ⊕ ( I, Γ) ⊗ ( I,γ ) ( IE, γ ) , F ⊗ uχ ⊕ hχ, ( IE ⊕ IE, γ ⊕ Γ) , F ]= [ uχr ⊕ hχr, ( H B ⊕ H B , Y ◦ γ ◦ Y − ⊕ Y ◦ Γ ◦ Y − ) , F ]If we had allowed j to be general, then the G -actions on I would have been j − ◦ Γ ◦ j and j − ◦ γ ◦ j . (cid:3) Lemma 10.5.
Consider the last lemma. (i)
Then B ( A ( hµ ∆ u e − )) = hµ ∆ u e − provided δ = α ⊗ in corollary 7.5. (ii) The first line is the second line in GK of the diagram of the last lemma.That is, hµ ∆ u e − = s − µ ∆ s + e − .Proof. (i) If we put the computed KK -element of the last lemma into B , we getexactly the second line of the last lemma as follows:(ii) We complete the diagram of the last lemma by setting ψ = jφκ − and φ ( x ) = χr ( x ) ⊕ (cid:3) f ( x ) s − ( a ) = uχr ( a ) ⊕ (cid:3) as + ( a ) = hχr ( a ) ⊕ (cid:3) a We equip D := L B ( H B ) (cid:3) A with the G -action Ad( S ) (cid:3) α , where S is the obvious G -action notated in the cylce of the last lemma. We define the G -action of M ( D )to be θ F (cid:3) . The first line of the diagram is the second line by lemma 5.4 forΦ = φ ⊗
1. Thereby we note that χr ( x ) = Y ◦ j − ◦ m x ◦ j ◦ Y − where m x is multiplication with x , such that Y ◦ j − ◦ Γ g ◦ j ◦ Y − ◦ χr ( x ) ◦ Y ◦ j − ◦ γ g − ◦ j ◦ Y − = χr (Γ g ( x ))which shows equivariance of Φ in the lower left corner, which is sufficient by corollary7.4.For general δ we define the M -action of the second line of the diagram in suchway that the bijective map Φ is equivariant. (cid:3) Lemma 10.6.
A ◦ B = id .Proof.
We replace the first line of the diagram of lemma 10.4 by the diagram ofdefinition 9.3. So we have j = κ , and thus by lemma 10.2, χr = id.Let Ad( S ⊕ T ) (cid:3) ( α ⊗
1) be the M -action of the first line.Then the action on H B of the second line computes as follows: Note that Γ( x ) = T g ◦ x ◦ S g − .We take a column vector Y − ([ b i ] i ) ∈ ( B ⊗ K ) E , go with it into κ , and applythe action Γ there and see what happens:Γ g ( κ ( Y − ([ b i, ]))([ ξ j ]) = T g ([ b i, ] · S g − ([ ξ j ])) = T g ([ b i, β g − ( ξ )]) = T g ([ b i, ]) ξ
10 B. BURGSTALLER
Thus we get Γ g ( κ ( Y − ([ b i, ]))) = κ ( Y − ( T g [ b i, ]))or Y ◦ κ − ◦ Γ g ◦ κ ◦ Y − = T g Analogously, Y ◦ κ − ◦ γ g ◦ κ ◦ Y − = S g .If z = [ s − ⊕ s + , ( H B , S ⊕ V ) , H ] with H the flip, then T = V ⊕ S , and by lemma10.4 we have shown A ( B ( z )) = [ s − ⊕ ⊕ s + ⊕ , ( H B , S ⊕ V ⊕ V ⊕ S ) , F ] = z (cid:3) Preparation for pushout construction
We remark that the proofs given in this section also follow directly from knownresults in KK -theory and an application of lemma 10.5.(i). We still recall the proofsin the framework of GK -theory because it is an important technique and the proofis not so long.For two ∗ -homomorphisms s ± : A → L ( E ) and a Hilbert A -module F we write F ⊗ s ± E := F ⊗ s − E ⊕ F ⊗ s + E ∼ = F ⊗ s − ⊕ s + ( E ⊕ E ) Lemma 11.1.
Let the first line of the following diagram be given. Then it can becompleted to this diagram B e / / (cid:15) (cid:15) B ⊗ K / / (cid:15) (cid:15) L ( E ⊕ H B ) (cid:3) A / / φ (cid:15) (cid:15) A s ± ⊕ (cid:3) o o (cid:15) (cid:15) B / / B ⊗ K / / L ( ˜ A ⊗ s ± E ⊕ H B ) (cid:3) A / / A u ± o o B [0 , / / O O (cid:15) (cid:15) B [0 , ⊗ K / / O O (cid:15) (cid:15) L (cid:16) Z ⊗ y ± E [0 , ⊕ H B [0 , (cid:17) (cid:3) A O O / / b (cid:15) (cid:15) b O O A t ± o o (cid:15) (cid:15) O O B / / B ⊗ K / / L ( A ⊗ s ± E ⊕ H B ) (cid:3) A / / A v ± o o such that the first line is the last line in GK, i.e. ( s + ⊕ (cid:3) µ ∆ s − ⊕ (cid:3) e − = v + µ ∆ v − e − . Thereby v − ( a ) = a ⊗ ⊕ ⊕ (cid:3) av + ( a ) = U ( H ) ◦ (0 ⊕ a ⊗ ⊕ ◦ U ( H ) ∗ (cid:3) a where H is a F -connection on A ⊗ s ( E ⊕ E ) ∼ = A ⊗ s ± E for the flip operator F on E ⊕ E .We assume that the G -action on M ( L ( E ⊕ H B ) (cid:3) A ) of the given first line of thediagram is Ad( S ⊕ R, T ⊕ R ) (cid:3) ( α ⊗ . The G -action on M := L ( A ⊗ s ± E ⊕ H B ) (cid:3) A of the last line of the diagram then is Ad( α ⊗ S ⊕ α ⊗ T ⊕ R ) (cid:3) α and on M ( M ) it is θ U ( H ) (cid:3) . SPECTS OF EQUIVARIANT KK -THEORY 21 Proof.
Let C ([0 , G -action. Let ( ˜ A, ˜ α ) be the unitization of( A, α ), see [2, def. 3.3] for inverse semigroups and [13] for groupoids. Set Z = { f ∈ ˜ A [0 , | f (1) ∈ A } ⊆ ( ˜ A, ˜ α ) ⊗ C ([0 , E [0 ,
1] :=
E ⊗ C ([0 , s + , ˜ s − : ˜ A → L ( E ) be the natural extensions of s ± , and set y ± : ˜ A [0 , → L ( E [0 , y ± ( x ⊗ f ) = ˜ s ± ( x ) ⊗ f Let F be the flip operator on F := E [0 , ⊕ E [0 , G = Z ⊗ y ± E [0 , ∼ = Z ⊗ y − ⊕ y + F =: H . The G -action on G is(˜ α ⊗ ⊗ ( S ⊗ ⊕ (˜ α ⊗ ⊗ ( T ⊗ G -action on H by the last isomorphism.Choose a F -connection V ∈ L B ( H ) on H by [12, lemma 2.7].Since a ∈ K A ( Z ) ⊗ ⊆ L B ( H ) for all a ∈ A , by [12, lemma 2.6], [ a, V ] ∈ K B ( H ).Similarly, by [3, Lemma 10] ag ( V ) − ag (1) V ∈ K B ( H )for all a ∈ K A ( Z ) ⊗ z = [id , H , V ] ∈ KK G ( A, B ). Go with z into lemma 9.5 andcreate the third line of the diagram with it. That is we set t − ( a ) = ( a ⊗ ⊕ ⊕ (cid:3) at + ( a ) = U ( V )(0 ⊕ a ⊗ ⊕ U ( V ) ∗ (cid:3) a The M -action of the third line of the diagram is θ U ( H ) (cid:3) .The second and the fourth line of the diagram are the evaluations of the thirdline at time zero and one as written in lemma 8.10.By lemma 9.2.(iii) we have U ( V ) ⊗ φ t U ( H ) for H := V ⊗ φ
1. Note that weget v ± as claimed.Completely analogously are u ± defined for H ′ := V ⊗ φ
1. Note that H ′ is a F ′ -connection on (cid:0) ˜ A ⊗ s − ⊕ s + ( E ⊕ E ) , ˜ α ⊗ ( S ⊕ T ) (cid:1) ∼ = ( E ⊕ E , S ⊕ T )where F ′ is the flip on E ⊕ E . Recall that there is a operator homotopy (id , E ⊕E , H ′ ) ∼ (id , E ⊕ E , F ′ ). By lemma 9.2.(ii) we can replace U ( H ′ ) by F ′ in thedefinition of u + . Under identification of the last isomorphism, define φ ( x (cid:3) a ) = x ⊕ (cid:3) a where 0 is the operator on E ∼ = ˜ A ⊗ s + E . Verify the identity in GK of line one andtwo of the diagram with lemma 5.4. (cid:3) Lemma 11.2.
The last lemma still holds true if we choose any F -connection H .Proof. Let H ∈ L ( F ) be any F -connection. Consider the unit 1 A := [ id A , A, ∈ KK G ( A, A ). Set H := A ⊗ s + ⊕ s − ( B ⊕ B ). Then [ id A , H , H ] is a Kasparov product1 A ⊗ A z because [ a, H ] ∈ K ( H ) for all a ∈ A by [14, prop. 9.(e)], H is a F -connection, and a [0 , H ] a ∗ = 0 ≥ K ( H ), and so [12, def. 2.10] applies. Hence by the uniqueness of the Kasparov product, ( id A ⊕ , H , H ) and ( id A ⊕ , H , H ) are operator homotopic by Skandalis [14, 12.(a)] for any two F -connections H and H .Skandalis’ proof is non-equivariant, but it works also equivariant, see for example[3] for inverse semigroups G .Hence the definition in GK of v + µ in lemma 11.1 does not depend on H as onecan connect different choices by a homotopy in the sense of lemma 9.6. (cid:3) Fusion with a synthetical split
The following proposition shows that a composition of a double split exact se-quence with a synthetical split yields a double split exact sequence again.
Proposition 12.1.
Let the first line of the diagram of lemma 11.1 be given, andconsider its fourth line, (equivalently) rewritten down in the first line of the nextdiagram (without e − ).Let the right column of the next diagram be a given split exact sequence. Thenwe can complete these data to the following diagram B ⊗ K / / (cid:15) (cid:15) L B (cid:16)(cid:0) A ⊗ s − E ⊕ A ⊗ s + E (cid:1) ⊕ H B (cid:17) (cid:3) A / / φ (cid:15) (cid:15) A c ± o o j (cid:15) (cid:15) B ⊗ K / / L B (cid:16)(cid:0) A ⊗ s − E ⊕ A ⊗ s + E (cid:1) ⊕ H B (cid:17) (cid:3) X / / X t ± o o g (cid:15) (cid:15) Q u O O such that ∆ u (( s + ⊕ (cid:3) µ ∆ s − ⊕ (cid:3) e − = ∆ u c + µ ∆ c − e − = t + µ ∆ t − e − Proof.
Set F ′ := A ⊗ s − E ⊕ A ⊗ s + E and F := F ′ ⊕ F ′ with the imagined grading − ⊕ +. We regard F ⊕ H B as havingfive summands (the four A ⊗ s ± E and one H B ).We define the Kasparov cycle z = [ s − ⊕ s + , E ⊕ E , T ] ∈ KK G ( A, B ) , where E ⊕ E has the obvious grading and T is the flip operator. As in definition4.1.(iii) we set w = [ guχ ⊕ χ, A ⊕ A, V ] ∈ KK G ( X, A ) , where A ⊕ A has the obvious grading and V is the flip operator.We form the Kasparov product w ⊗ A z and obtain a cycle w ⊗ A z = [ v − ⊕ v + , H , F ] ∈ KK G ( X, B ) , and using a canonical isomorphism, which we are going to sloppily use as an iden-tification, F ∼ = ( A ⊕ A ) ⊗ s − ⊕ s + ( E ⊕ E ) =: H SPECTS OF EQUIVARIANT KK -THEORY 23 (via ⊕ ( i,j )=(0 , , (1 , , (1 , , (0 , a i ⊗ ξ j ( a ⊕ a ) ⊗ ( ξ ⊕ ξ )) we have v − ( x ) = (cid:0) χ ( x ) ⊗ (cid:1) ⊕ (cid:0) guχ ( x ) ⊗ (cid:1) ⊕ ⊕ v + ( x ) = 0 ⊕ ⊕ (cid:0) guχ ( x ) ⊗ (cid:1) ⊕ (cid:0) χ ( x ) ⊗ (cid:1) for all x ∈ X (right hand side are operators on F ). We set c − ( a ) = ( a ⊗ ⊕ ⊕ ⊕ ⊕ (cid:3) ac + ( a ) = U ( F ) ◦ (cid:0) ⊕ ⊕ ⊕ ( a ⊗ ⊕ (cid:1) ◦ U ( F ) ∗ (cid:3) at − ( x ) = v − ( x ) ⊕ (cid:3) xt + ( x ) = U ( F ) ◦ (cid:0) v + ( x ) ⊕ (cid:1) ◦ U ( F ) ∗ (cid:3) x By lemma 9.5, the second line of the diagram is double split with M -action θ U ( F ) (cid:3) .Recall that F is a T -connection on H . Applying this to Lemma 11.1 for A ⊕ A instead of A , E ⊕ E instead of E , and s + ⊕ s + ⊕ (cid:3) , s − ⊕ s − ⊕ (cid:3) s ± ⊕ (cid:3)
1, together with lemma 11.2, and using lemma 7.1 two times, first in thefirst line of the diagram of lemma 11.1 and second in the fourth line, for A ⊕ A instead of A , X = A , φ : A → A ⊕ A the injection onto the first corrdiante, andusing a similar assertion as in lemma 8.9 that we may deliberately add summandswithout changing anything, we obtain that c + µ ∆ c − e − = ( s + ⊕ (cid:3) µ ∆ s − ⊕ (cid:3) e − Also note that we have shown that the first line of the diagram is double splitwith M -action θ U ( F ) (cid:3) .Consider the (null) cycle σ := [ guv − ⊕ guv + , H , V ⊗ ∈ KK G ( X, B )(null because [ guv − ⊕ guv + , V ⊗
1] = 0).Recall that x [ V ⊗ , F ] x ∗ ≥ K ( F )for all x ∈ X , and in particular for all x = gu ( x ), by the definition of the Kasparovproduct [12, def. 2.10.(c)].Hence, by [14, lemma 11] (or [3, lemma12] in the inverse semigroup equivariantsetting), applied to σ and σ := ( gu ) ∗ ( w ⊗ A z ) = [ guv − ⊕ guv + , H , F ] ∈ KK G ( X, B ) ,σ is operator homotopic to σ .Therefore, gut + µ = gut ′ + µ by lemma 9.6, where t ′ + ( x ) := U ( V ⊗ ◦ (cid:0) v + ( x ) ⊕ (cid:1) ◦ U ( V ⊗ ∗ (cid:3) x = ( V ⊗ ◦ v + ( x ) ◦ ( V ⊗ ∗ ⊕ (cid:3) x in GK by lemma 9.2.(ii), where we have identified F ∼ = H for simplicity. Hence gut + µ ∆ t − = gut − µ ∆ t − = gut − ∆ t − = 0where we have achieved µ = 1 by a rotation homotopy.We define φ ( x (cid:3) a ) = x (cid:3) j ( a ) We verify all conditions of lemma 5.4 for the first line and second line of thediagram and for Φ = φ ⊗
1. We then obtain jt + µ ∆ t − = c + µ ∆ c − . Thus we get t + µ ∆ t − = 1 X t + µ ∆ t − = (∆ u j + gu ) t + µ ∆ t − = ∆ u c + µ ∆ c − (cid:3) Fusion with the inverse of a corner embedding
In the following lemma we turn the composition of a double split exact sequencewith the inverse of a corner embedding to another double split exact sequence.
Lemma 13.1.
Let the first line of the diagram of lemma 11.1 be given, and considerits fourth line, partially written down in the first line of the next diagram.Let ≤ n ≤ ∞ and set M ∞ ( A ) := A ⊗ K . Let f be a corner embedding.Then we can complete these data to the following diagram B ⊗ K κ / / l (cid:15) (cid:15) L B (cid:16) A ⊗ s ± B ⊕ H B (cid:17) (cid:3) A / / φ (cid:15) (cid:15) A v ± o o f (cid:15) (cid:15) B ⊗ K ⊗ M n / / L B (cid:16) M n ( A ) E ⊗ s ± E ⊕ H nB (cid:17) (cid:3) M n ( A ) / / M n ( A ) t ± o o such that f − s + µ ∆ s − = f − v + µ ∆ v − = t + µ ∆ t − .Proof. Let E = f ( A ) be the corner algebra (the upper left corner say). The map l is the equivariant corner embedding.Recall that the G -action on F := A ⊗ s ± E ⊕ H B is the canonical one inducedby the actions of ( A, α ), ( E , S ) and ( H B , R ). The G -action on M ( L B ( F ) (cid:3) A ) is θ U ( H ) (cid:3) .Denote the G -action on M n ( A ) by γ .Since M n ( A ) E ∼ = A n (column vectors) as non-equivariant Hilbert A -modules, byjust reordering summands we may consider the canonical isomorphism Y : M n ( A ) E ⊗ s ± E ⊕ H nB =: G → ( A ⊗ s ± E ⊕ H B ) n of non-equivariant Hilbert B -modules. Recalling U ( H ) from lemma 11.1, put V = Y − ◦ U ( H ) n ◦ Y. We define the G -action on the domain of Y to be γ ⊗ S ⊕ γ ⊗ T ⊕ R n . The G -action on M ( L B ( G ) (cid:3) M n ( A )) is θ V (cid:3) .We set t − ( x ) = ( x ⊗ ⊕ ⊕ n (cid:3) xt + ( x ) = V ◦ (0 ⊕ ( x ⊗ ⊕ n (cid:1) ◦ V ∗ (cid:3) x Note that t + ( x ) − t − ( x ) ∈ K ( G ) by lemma 11.1 (observe it first for a matrix x with a single non-zero entry to get essentially v + ( y ) − v − ( y ) ∈ K ( F )).Observe that L ( G ) ∼ = M n ( L ( F )) and this is the equivariant “corner embedding”: φ ( T (cid:3) a ) = Y − ◦ ( T ⊕ n − ) ◦ Y (cid:3) f ( a )Notice that Φ = φ ⊗ M is equivariant, as for example, by observing the lowerleft corner of the range of Φ, φ (cid:16) U ( H ) ◦ g (cid:0) U ( H ) ∗ ◦ T (cid:1) (cid:3) g ( a ) (cid:17) = V ◦ g (cid:0) V ∗ ◦ φ ( T (cid:3) a ) (cid:1) SPECTS OF EQUIVARIANT KK -THEORY 25 for g ∈ G and T ∈ L ( F ).It is now straightforward to check the claim with lemma 5.4. (cid:3) The standard form
Recall that G is a locally compact (not necessarily Hausdorff) groupoid or acountable inverse semigroup, and that all C ∗ -algebras are separable. Lemma 14.1.
Let two diagrams as in (11) be given, say for actions
S, T andhomomorphisms s ± , t ± . (i) Then we can sum up these diagrams to (12) B / / B ⊗ K / / L B (cid:0) ( H B , S ) ⊕ ( H B , T ) (cid:1) (cid:3) A / / A s − ⊕ t − ,s + ⊕ t + o o and this corresponds to the sum of the associated elements in GK , i.e. s + µ ∆ s − e − + t + µ ∆ t − e − = ( s + ⊕ t + ) µ ∆ s − ⊕ t − e − The M -action is Ad( V ⊕ W ) (cid:3) ( α ⊗ for the two M -actions Ad( V ) (cid:3) ( α ⊗ and Ad( W ) (cid:3) ( α ⊗ of the given diagrams.(Note that we used the wrong but suggestive notation s − ⊕ t − := x ⊕ y (cid:3) for s − = x (cid:3) , t − = y (cid:3) .)Proof. We drop the proof. One does this as in [5, lemma 3.6]. (cid:3)
Corollary 14.2.
Consider the diagram (11). Make its ‘negative’ diagram wherewe exchange t − and t + and transform the M -action under coordinate flip. Thenits associated element in GK is the negative, that is, − t + µ ∆ t − e − = t − µ ∆ t + e − Proof.
Considering a sum as in the last lemma we have( t + ⊕ t − ) µ ∆ t − ⊕ t + = ( t − ⊕ t + )∆ t − ⊕ t + = 0by the rotation homotopy V s ( t − ⊕ t + ) V − s , where V s = (cid:16) cos s sin s − sin s cos s (cid:17) ⊕ ∈L B ( E ) ⊕ ˜ A for s ∈ [0 , π/
2] and where we define the G -action on M ( X [0 , X = L B ( E ⊕ E ) by ( θ V s ) s ∈ [0 ,π/ so that we can apply lemma 8.11. (cid:3) Theorem 14.3.
Every morphism in GK can be represented as t + µ ∆ t − e − (calledstandard form) for a diagram as in (11).Proof. Write id A = id A ee − for the corner embedding e : A → A ⊗ K . By applyinglemma 8.1 and lemma 10.5.(ii) (or lemma 8.3) to id A e , we can present the identityhomomorphism id A in the claimed form.Assume by induction on n ≥ w = w n . . . w of length n in GK allows such a presentation as claimed. If v is a homomorphism then vw allows alsosuch a presentation by lemma 8.2. Similarly we use proposition 12.1 if v = ∆ u , andlemma 13.1 if v is the inverse of a corner embedding to complete the induction stepfor vw . For sum of words we apply lemma 14.1 (or [5, lemma 3.6] before induction)and corollary 14.2. (cid:3) Corollary 14.4.
The assignment B is multiplicative, so is a functor. Proof.
By lemma 10.6 we have B ( zw ) = B (cid:0) A ( B ( z )) · A ( B ( w )) (cid:1) = B (cid:0) A (cid:0) B ( z ) · B ( w ) (cid:1)(cid:1) for composable morphisms z, w ∈ KK G . Since by theorem 14.3 we can bring B ( z ) · B ( w ) to standard form, by lemma 10.5 this is B ( z ) · B ( w ).For the unit 1 A = [0 + id A ⊕ , ( H A , R ) , ∈ KK G ( A, A ) we get B (1 A ) = id A by lemma 8.1. (cid:3) Corollary 14.5.
B ◦ A = id .Proof.
We bring a morphism in GK to standard form by theorem 14.3 and thenapply lemma 10.5. (cid:3) Corollary 14.6.
The functors A : GK → KK G and B : KK G → GK are isomor-phims of categories and inverses to each other. In particular, GK ∼ = KK G .Proof. By lemma 10.5 and corollary 14.5. (cid:3)
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