A Note on Kasparov Product and Duality

A NOTE ON KASP AR O V PR ODUCT AND DUALITY HYUN HO LEE Abstract. Using Pasc hk e-Higson dualit y , we can get a natural index pairing K i ( A ) × K i +1 ( D Φ ) → Z ( i = 0 , 1)(mo d2), where A is a separable C ∗ -algebra , and Φ is a r epresentation o f A on a separable infinite dimens io nal Hilbert space H . It is pr ov ed that this is a sp ecial case o f the Kaspar ov Pro duct. As a step, we show a pr o of of Bott-p erio dicity for KK- theory asserting that C 1 and S are K K -equiv alent us ing the o dd index pa ir ing. 1. Introduction In [Hig], Hig son show ed a conne ction b etw een K- t heory of the essen- tial commutan t of a C ∗ -algebra A and the extension group of A b y the C ∗ -algebra of the compact op erators o n a separable infinite dimensional Hilb ert space and defined index pairings (dualities) (1) K i ( A ) × K i +1 ( D Φ ) → Z for i = 0 , 1(mo d2) based on an earlier work of Pas c hke [P a]. In ligh t of this dualit y , K- theory of the essen tial commu tan t of a C ∗ -algebra A can b e view ed as K- ho mology of A (See Theorem 1.5 in [Hig]). Since Kasparov’s KK-theory generalizes b oth K-theory and K-homology , w e can expect that each index pair ing giv en by (1) is a Kasparo v pro duct. It is a goal of our pap er to show that P asc hk e-Higson duality can b e realized as a Kasparov product. F or this w e need to use Bott p erio dicity for KK- theory of the form KK( A, B ) ∼ = KK( S A, S B ) so that we give a pro of of Bott p erio dicity using the o dd pairing a s an application. 2. P aschke-Higson Duality and Index p airing In this section, w e review the P asc hke -Higson duality theory [Hig] more carefully fo r the con v enience of the reader. Throughout this article we denote b y H a separable infinite dimens ional Hilb ert space, b y B ( H ) the set of linear b ounded op era t o rs on H , b y Date : September 5, 2010. 2000 Ma thematics Subje ct Classific ation. Pr imary:46L 8 0;Secondar y :19K33,19K35. Key wor ds and phr ases. K K-theory , Kasparov P ro duct, Paschk e- Higson Dua lity . 1 2 HYUN HO LEE K ( H )(or just K ) the ideal of compact o p erators on H , and b y Q ( H )(or just Q ) the Calkin algebra. W e use the follo wing notation : if X and Y are op era t o rs in B ( H ) w e shall write X ∼ Y if X and Y differ by a compact op era t o r. Note that ev ery *- represen tation Φ of A on H induces a *-homomorphism ˙ Φ of A into the Calkin algebra. Definition 2.1. Let A b e a C ∗ -algebra. A *- represen tation Φ : A → B ( H ) is called admissible if it is non-degenerate a nd k er ( ˙ Φ) = 0. R emark 2.2 . If a * -represen tatio n is admissible, then it is necessarily faithful and its image contains no non-zero compact o p erators. Definition 2.3. Let Φ b e a *- represen tation of A on H . W e define the essential c ommutant o f Φ( A ) in B ( H ) as D Φ ( A ) = { T ∈ B ( H ) | [Φ( a ) , T ] ∼ 0 for all a ∈ A } Giv en t wo represen tations Φ 0 and Φ 1 on H 0 and H 1 resp ectiv ely , w e sa y they are appr oxi mately unitarily e quiva lent if there exists a sequence { U n } consisting of unitaries in B ( H 0 , H 1 ) suc h that for an y a ∈ A U n Φ 0 ( a ) U ∗ n ∼ Φ 1 ( a ) , k U n Φ 0 ( a ) U ∗ n − Φ 1 ( a ) k → 0 as n → ∞ . W e write Φ 0 ∼ u Φ 1 in this case. Theorem 2.4. (V oi culescu) L et A b e a sep ar able C ∗ -algebr a and Φ i : A → B ( H i ) ( i = 0 , 1) b e non-de gener a te *- r epr esentations. Then if k er ˙ Φ 0 ⊂ ker Φ 1 , then Φ 0 ⊕ Φ 1 ∼ u Φ 0 . Pr o of. See Corollary 1 in p343 of [Ar].  Corollary 2.5. Assume Φ i : A → B ( H i ) ar e admissible r ep r esenta- tions for i = 0 , 1 . Then Φ 0 ∼ u Φ 1 . Pr o of. Admissib ilit y implies k er ˙ Φ i = 0. By symmetry , the result fol- lo ws.  Recall that an extension of a unital se parable C ∗ -algebra A is a unital *-monomorphism τ of A in to the Calkin a lg ebra. W e sa y τ is split if there is a non-degenerate *-represen tation ρ suc h that ˙ ρ = τ and se mi- split if there is a completely positive map ρ suc h that ˙ ρ = τ . In general, an extension τ of A b y B is a ∗ -homomorphism fro m A to Q ( B ) where Q ( B ) = M ( B ) /B and M ( B ) is the m ultiplier algebra o f B . W e sa y A N OTE ON KA SP AROV PR O DUCT AND DUALITY 3 τ is split or se mi-split if the lifting ρ is a *-homomorphism to M ( B ) or a completely p ositiv e map to M ( B ) respectiv ely . When B is stable, Ext ( A, B ) is the quotient of Hom( A, Q ( B )) by the equiv alence relation generated b y addition of trivial elemen ts a nd unitary equiv alence. Corollary 2.6. L et A b e a sep ar able unital C ∗ -algebr a. If τ is a unital inje ctive extension of A and if σ is a split unital extensi on of A , then τ ⊕ σ is unitarily e quivalent to τ . Pr o of. See p352-353 in [Ar ].  No w w e will pro v e that there is at least one admissible represe n t a tion of A . Prop osition 2.7. Ther e is a non-de gener ate *-r epr esentation Φ of for a sep ar able C ∗ -algebr a A such that k er ˙ Φ = 0 . Pr o of. Let π b e a faithful represen tatio n of A on H π . T ak e Φ to b e π ( ∞ ) = π ⊕ π ⊕ · · · and H = H ( ∞ ) π = H π ⊕ H π ⊕ · · · .  Prop osition 2.8. If Φ , Ψ ar e admi ssible r e pr esentations of A on H , D Φ ( A ) is isomorphic to D Ψ ( A ) . Pr o of. By Corollary 2.5, there is a unitar y U ∈ B ( H ) suc h that for a n y a ∈ A U Φ( a ) U ∗ ∼ Ψ( a ) . W e define a map θ on D Φ ( A ) by θ ( T ) = U T U ∗ for T ∈ D Φ ( A ). W e note that fo r an y a ∈ A Ψ( a ) θ ( T ) − θ ( T )Ψ( a ) = Ψ( a ) U T U ∗ − U T U ∗ Ψ( a ) ∼ U Φ T U ∗ − U T Φ U ∗ ∼ U (Φ T − T Φ) U ∗ ∼ 0 . Th us θ ( T ) ∈ D Ψ ( A ) a nd θ is an isomorphism onto D Ψ ( A ) since θ = Ad ( U ).  Definition 2.9. When Φ, Ψ are ad missible represen tations of A on H , D Φ ( A ) is isomorphic to D Ψ ( A ) b y Prop osition 2.8. Thus w e define D ( A ) = D Φ ( A ) as the dual algebra of A up to unitary equiv alence. If p is a pro j ection in D Φ ( A ), w e call it ample and can define an extension τ = τ Φ ,p : A p Φ( • ) p =Φ p − − − − − − → B ( p H ) π − → Q ( pH ) In g eneral, when A is se parable and B is stable, w e can express in v ert- ible elemen ts τ in Ext ( A, B ) a s pairs ( φ, p ), where φ is a repres en tatio n 4 HYUN HO LEE from A to M ( B ) and p is a pro j ection in M ( B ) whic h comm utes with φ ( A ) mo dulo B , i.e, τ ( · ) = q B ( pφ ( · ) p ) where q B = M ( B ) → Q ( B ). W e denote Ext − 1 ( A, B ) b y the group of in v ertible elemen ts in Ext ( A, B ) [Bl], [Kas]. T o define P asc hk e-Higson duality , w e need the following t wo t ec hnical lemmas. Lemma 2.10. L et A b e a unital C ∗ -algebr a. F or any α ∈ K 0 ( D ( A )) , ther e exists an ample pr oje ction p ∈ D ( A ) such that α = [ p ] 0 . Pr o of. Step1: By Corollary 2.6, there is a unitary u ∈ B ( p H ⊕ H , H ) suc h that Ad ( u )(Φ p ⊕ Φ)( a ) ∼ Φ( a ) for any a ∈ A if p is am- ple. Let U =  p 0  u . W e can easily c hec k that U ∈ M 2 ( D Φ ( A )) and U U ∗ =  p 0 0 0  , U ∗ U =  p 0 0 I  . Therefore w e hav e [ p ⊕ I ] 0 = [ p ⊕ 0] 0 . This implies [ p ] 0 + [ I ] 0 = [ p ] 0 . In particular, [ I ] 0 = 0. Similarly , w e can conclude ev ery t wo ample pro jections are Murra y-v on Neumann equiv alent. Step2: Note that p ⊕ 1 is alw a ys ample whether p is ample or not b e- cause (Φ ⊕ Φ) p ⊕ 1 ( a ) is nev er compact unless a is zero in A . Step3: An y elemen t in K 0 ( D Φ ( A )) can b e written as [ q ] 0 − [ I n ] 0 for some q ∈ M n ( D Φ ( A )). As w e observ ed in Step1, this is just [ q ] 0 . By Step2, w e may assume q is ample for Φ ( n ) = Φ ⊕ Φ ⊕ · · · ⊕ Φ | {z } n . No w if w e can sho w [ q ] 0 = [ p ] 0 for some ample p ∈ D Φ ( A ), w e are done. Since Φ ( n ) ∼ u Φ, there exists v : H n → H such that (1) v ∗ v = 1 B ( H n ) , v v ∗ = 1 B ( H ) (2) Ad( v )Φ ( n ) ( a ) − Φ( a ) ∈ K for an y a ∈ A . Then [ q ] 0 = [ v q v ∗ ] 0 . In addition, [ v q v ∗ , Φ( a )] ∼ v [ q , Φ ( n ) ] v ∗ ∼ 0 for ev ery a ∈ A . Th us v q v ∗ is ample and w e are done.  Lemma 2.11. L e t A b e as ab ove. F or any α ∈ K 1 ( D ( A )) , ther e exists an unitary u ∈ D ( A ) such that α = [ u ] 1 . Pr o of. Assume U ∈ M n ( D Φ ( A )) ≈ D φ ( n ) ( A ) is a unitar y whic h rep- resen ts α ∈ K 1 ( D Φ ( A )). Let V be ( n z }| { 1 , 0 , · · · , 0) T v where v is defined in Lemma 2.10 and S =  V 1 − V V ∗ 0 V ∗  ∈ M 2 n ( D Φ ( A )). It is easy to che c k that V U V ∗ + 1 − V V ∗ =  v U v ∗ 0 0 1  and S  U 0 0 1  S ∗ = A N OTE ON KA SP AROV PR O DUCT AND DUALITY 5  V U V ∗ + 1 − V V ∗ 0 0 1  . Therefore [ U ] 1 = [ v U v ∗ ] 1 and [ v U v ∗ , Φ( a )] ∼ v [ U, Φ n ( a )] v ∗ ∼ 0 for eve ry a ∈ A . Th us w e take u as v U v ∗ .  R emark 2.12 . A unital C ∗ -algebra A is said to hav e K 1 -surjectivit y if the natural map from U ( A ) /U 0 ( A ) to K 1 ( A ) is surjectiv e and is said to hav e (strong) K 0 -surjectivit y if the gr o up K 0 ( A ) is generated b y { [ p ] | p is a pro jection in A } . Thus Lemma 2.10 a nd Lemma 2.11 show D Φ ( A ) has (strong) K 0 -surjectivit y and K 1 -surjectivit y . No w w e are ready to define an index pairing b etw een K i ( A ) and K i +1 ( D Φ ( A )) for i = 0 , 1 . In the following t w o definitions, Ind will denote the classical F r e dhol m index . Giv en a pro jection p ∈ M k ( A ) and a unitary u ∈ D Φ ( A ), when Φ k is k-th amplification of Φ, the o p erator Φ k ( p ) u ( k ) Φ k ( p ) : Φ k ( p )( H k ) → Φ k ( p )( H k ) is essen tially a unita ry , and therefore F r e dholm . Definition 2.13. The (ev en) index pairing K 0 ( A ) × K 1 ( D Φ ( A )) → Z is defined b y ([ p ] , [ u ]) − − − → Ind  Φ k ( p ) u ( k ) Φ k ( p )  where p ∈ M k ( A ) and Φ k is k-th amplification of Φ. Similarly , give n a unitary v ∈ M k ( A ) and a pro jection p ∈ D Φ ( A ), the op erato r p ( k ) Φ k ( v ) p ( k ) − (1 − p ( k ) ) : H k → H k is essen tially a unita ry , and therefore F r e dholm . Definition 2.14. The (o dd) index pairing K 1 ( A ) × K 0 ( D Φ ( A )) → Z is defined by ([ v ] , [ p ]) − − − → Ind  p ( k ) Φ k ( v ) p ( k ) − (1 − p ( k ) )  where v ∈ M k ( A ) and Φ k is k-th amplification of Φ. 3. Kasp aro v Pro duct and Duality In this section, w e prov e our main results: Eac h index pairing is a sp ecial case of the Kasparov pro duct. Before do ing this, we recall some rudimen ts of KK- theory . W e only g iv e a brief description of elemen ts of the Kasparo v group KK( A, B ), but f o r the complete information w e refer the reader to the orig inal source [Kas] or to [JenThom] a nd [Bl] for detailed exp ositions. 6 HYUN HO LEE KK( A, B ) is describ ed in terms of triples ( E , φ, F ), whic h w e call cycles, where E is a coun tably generated graded Hilb ert B-mo dule, φ : A → L B ( E ) is a represen tation and F ∈ L B ( E ) is an elemen t o f degree 1 suc h that [ F , φ ( a )] , ( F 2 − 1) φ ( a ) , ( F ∗ − F ) φ ( a ) are in K B ( E ) for an y a ∈ A . W e remind the reader that the comm utator [ , ] is graded and φ preserv es the grading. W e shall denote E ( A, B ) b y the set of all cycles. A degenerate cycle is a triple ( E , φ, F ) suc h that [ F , φ ( a )] = ( F 2 − 1) φ ( a ) = ( F ∗ − F ) φ ( a ) = 0 for all a ∈ A . An op era- tor homotop y through cycle s is a homoto p y ( E , φ, F t ), where t → F t is norm contin uous. A theorem of Kasparov [Kas] sho ws that KK( A, B ) is the quotien t of E ( A, B ) b y the equiv alence relation generated by addition of degenerate cycles , unitary equiv alence and op erator homo- top y . A graded (maximal) tensor pro duct of tw o graded C ∗ -algebras C and D denoted b y C ˆ ⊗ D is the univ ersal env eloping C ∗ -algebra of t he algebraic tensor pro duct C ⊙ D with a new pro duct and in volution on C ⊙ D by ( c 1 ˆ ⊗ d 1 )( c 2 ˆ ⊗ d 2 ) = ( − 1) deg d 1 deg c 2 ( c 1 c 2 ˆ ⊗ d 1 d 2 ) ( c 1 ˆ ⊗ d 1 ) ∗ = ( − 1) deg c 1 deg d 1 ( c ∗ 1 ˆ ⊗ d ∗ 1 ) Let E = ( E , φ, F ) ∈ E ( A, B ). Then w e can form a Hilb ert B ˆ ⊗ C - mo dule E ˆ ⊗ C . In addition, φ ˆ ⊗ 1 : A ˆ ⊗ C → L B ˆ ⊗ C ( E ˆ ⊗ C ) is defined b y φ ˆ ⊗ 1( a ⊗ c ) = φ ( a ) ˆ ⊗ c. It follows that ( E ˆ ⊗ C , φ ˆ ⊗ 1 , F ˆ ⊗ 1) ∈ E ( A ˆ ⊗ C , B ˆ ⊗ C ). W e denote this map by τ C : KK( A, B ) → KK( A ˆ ⊗ C , B ˆ ⊗ C ). T h us the image of E is τ C ( E ). F or E ∈ E ( A, B ) and F ∈ E ( B , C ), t here exists a Kasparov pro duct of E b y F , whic h will b e denoted by E · F , in E ( A, C ) [Kas]. This pro d- uct is unique up to op erator homotopy so that w e define the pro duct KK( A, B ) × KK( B , C ) → KK( A, C ) using the same notation. F or t he remaining, A and B will denote (ungraded) separable C ∗ - algebras. Prop osition 3.1. K K ( S, B ) = K 1 ( B ) w her e S = { f ∈ C ( T ) | f (1 ) = 0 } . Pr o of. It is w ell-kno wn. W e just note that an y unitary in K 1 ( B ) can b e lifted to φ ∈ K K ( S, B ) where φ : S → B is determined by sending z − 1 to u − 1.  Prop osition 3.2. KK( C , C ) ∼ = Z . A N OTE ON KA SP AROV PR O DUCT AND DUALITY 7 Pr o of. Ev ery elemen t in KK( C , C ) can b e represen ted by a triple  ˆ H ,  φ 0 0 φ  ,  0 u ∗ u 0  whic h satisfies [ u, p ] ∼ 0, ( u − u ∗ ) p ∼ 0 , and ( u 2 − 1) p ∼ 0 for φ (1) = p . Here ˆ H means H ⊕ H with the standard eve n grading. Suc h a triple is called a standard triple. Then a standard triple is mapp ed to Ind( pup ) = Ind( φ (1 ) uφ (1)). And this map is a group isomorphism ( See Example 17.3 .4 in [Bl]).  Recall that C 1 is the Clifford algebra of R , whic h is isomorphic to C ⊕ C with the odd grading that transp oses the tw o copies of C . Th us B ˆ ⊗ C 1 is the C ∗ -algebra B ⊕ B with the o dd gr a ding. Definition 3.3. KK 1 ( A, B ) = K K ( A, B ˆ ⊗ C 1 ) . Theorem 3.4 (Kasparo v) . Ext − 1 ( A, B ) ∼ = K K 1 ( A, B ) . Pr o of. See [Kas ]. W e note that an in vertible extension τ in Ext − 1 ( A, B ), whic h correspo nds to a pair ( φ , p ), is mapp ed to a cyc le ( H B ⊕ H B , φ ⊕ φ, 2 p − 1 ⊕ 1 − 2 p ) where H B ⊕ H B is gra ded by 1 ⊕ ( − 1) whic h w e call the standard ev en grading.  Lemma 3.5. L et φ : A → B ( H 1 ⊕ H 2 ) b e a *-r epr esen tation. Write φ ( a ) =  φ 11 ( a ) φ 12 ( a ) φ 21 ( a ) φ 22 ( a )  . Supp ose φ 11 is a *-homomorphism mo d- ulo K ( H 1 ) , i.e., ˙ φ 11 is a * -homomorphism . Then φ 12 ( a ) , φ 21 ( a ) ar e c omp acts for any a ∈ A and ˙ φ 22 is a *-homomorphis m. Pr o of. Using φ ( aa ∗ ) = φ ( a ) φ ( a ∗ ) under the decomposition of φ o n H 1 ⊕ H 2 and the fact φ 11 is *-homomorphism mo dulo K ( H 1 ), w e hav e φ 12 ( a ) φ 12 ( a ∗ ) is compact. Th us φ ( a ) is compact for an y a ∈ A . Sim- ilarly , using φ ( a ∗ a ) = φ ( a ) ∗ φ ( a ), w e ha v e φ 21 ( a ) is compact for an y a ∈ A . It follo ws that φ 22 is *- homomorphism mo dulo K ( H 2 ).  The following theorem due to Higson [Hig ] is imp ortant to us so that w e giv e a complete pro of. Prop osition 3.6. K 0 ( D ( A )) ∼ = Ext − 1 ( A, C ) . Pr o of. In ligh t of Lemma 2.10 , we define the map from K 0 ( D ( A )) to Ext − 1 ( A , C ) b y [ p ] 0 → [ τ Φ ,p ] . 8 HYUN HO LEE where Φ is an admissible represen tation of A on H and p ∈ D Φ ( A ). When [ p ] 0 = [ q ] 0 , as we hav e seen in the pro of of Lemma 2.1 0, p and q are Murra y-v o n Neumann equiv alen t in D Φ ( A ) so that the par- tial isometry implemen ting this equiv alence induces the equiv alence b et w een τ Φ ,p and τ Φ ,q . Con verse ly , unitary equiv a lence b etw een τ Φ ,p and τ Φ ,q induces Murra y-v on Neumann equiv alence b etw een p and q eviden tly . F rom Φ ⊕ Φ ∼ u Φ, w e get a unitary u ∈ B ( H ⊕ H, H ) whic h in- duces a natural isomorphism Ad ( u ) : M 2 ( D Φ ( A )) → D Φ ( A ). Note that π ◦ Ad ( u ) = Ad ( u ) ◦ ( π ⊗ id 2 ). Since [ p ] 0 + [ q ] 0 = [ p ⊕ q ] 0 and p ⊕ q ∈ D Φ ⊕ Φ ( A ), [ p ] 0 + [ q ] 0 is mapp ed to [ π ◦ Ad ( u ) ◦ (Φ ⊕ Φ) p ⊕ q ] = [ Ad ( u ) ◦ (( π ⊗ id 2 ) ◦ (Φ ⊕ Φ) p ⊕ q ] whic h is indeed [ τ Φ ,p ] + [ τ Φ ,q ]. So far w e ha v e sho wn the map is a monomorphism. It remains to sho w the map is on to. Supp ose ρ is an in v ertible extension. Then ρ is a semi-split extension of A with a completely p ositiv e lif ting ψ : A → B ( H ). By the Stinespring’s dilation theorem, there is a non-degenerate *-represen tation φ : A → B ( H 0 ) and an isometry V : H → H 0 suc h that ψ ( a ) = V ∗ φ ( a ) V for all a ∈ A . It follo ws that k er( ˙ φ ) = 0 and φ is an admis sible repre sen tatio n. If w e set P 1 = V V ∗ and P 2 = 1 − P 1 , then H 0 = P 1 ( H 0 ) ⊕ P 2 ( H 0 ) = H 1 ⊕ H 2 . If w e dec omp ose φ on H 0 = H 1 ⊕ H 2 and write φ ( a ) =  φ 11 ( a ) φ 12 ( a ) φ 21 ( a ) φ 22 ( a )  , w e hav e V ψ ( a ) V ∗ = V V ∗ φ ( a ) V V ∗ = P 1 φ ( a ) P 1 = φ 11 ( a ). Since ˙ ψ = ρ is a (injective ) *-homomorphism, we can conclude φ 11 is a injectiv e *-homomorphism mo dulo compact. By Lemma 3.5, φ 12 ( a ), φ 21 ( a ) are compacts for a ∈ A a nd φ 22 is *a - ho momorphism mo dulo compact. This implies that [ P 1 , φ ] ∈ K . Th us ˙ φ 11 is τ φ,P 1 . Viewing V : H → P 1 ( H 1 ) as a unitary , w e can also see that ρ is unitarily equiv alent to ˙ φ 11 . Th us we finish the pro of.  Theorem 3.7 (O dd case) . The mapping K 1 ( B ) × K 0 ( D Φ ( B )) → Z is the Kasp ar ov pr o duct K K ( S, B ) × K K 1 ( B , C ) → Z . Pr o of. Without lo ss of generalit y , w e ma y assume tha t an ele men t in K 1 ( B ) is represen ted b y a unitary v in B. Supp ose an elemen t in K 0 ( D Φ ( B )) is represen ted b y a pro jection p in D Φ ( B ). As w e ha ve noted, [ v ] 1 ∈ K 1 ( B ) is mapp ed to ψ ∈ K K ( S, B ) where ψ : S → B is determined b y sending z − 1 to v − 1. On the other hand, [ p ] is mapp ed to [ τ Φ ,p ] b y Propo sition 3.6 . Using the is omorphism Ext ( B , C ) → K K 1 ( B , C ) b y Theorem 3.4, the image of τ Φ ,p is a cycle F = ( H ⊕ H , Φ ⊕ Φ , T ⊕ − T ) where T = 2 p − 1. A N OTE ON KA SP AROV PR O DUCT AND DUALITY 9 Then the Kasparo v pro duct ψ b y [ F ] is ψ · [ F ] = [( H ⊕ H , (Φ ◦ ψ ) ⊕ ( Φ ◦ ψ ) , T ⊕ − T )] . Note that (Φ ◦ ψ )( z − 1) = Φ( v − 1) = Φ ( v ) − 1. Under the iden t ificatio n of K K 1 ( S, C ) with K 1 ( Q ( H )), ψ · [ F ] is mapp ed to p Φ( v ) p − (1 − p ) b y Prop osition 1 7 .5.7 in [Bl]. Using the index map ∂ 1 : K 1 ( Q ( H )) → K 0 ( K ) = Z in K- theory , w e kno w ∂ 1 ( p Φ( v ) p − (1 − p ) ) = Ind( p Φ( v ) p − (1 − p )).  It is said that an elemen t x ∈ KK( A, B ) is a KK-equiv alence if t here is a y ∈ KK( B , A ) suc h that x · y = 1 A and y · x = 1 B . A and B are K K -equiv alent if there is a KK-equiv alence in KK( A, B ). The follo wing coro llary , whic h is originally due to Kasparo v, establishes the KK-equiv alence of C 1 and S . Lemma 3.8 (Morita inv ariance) . L et M n denote the n × n matrix algebr a of C . Then KK( A, B ) = KK( A ⊗ M n , B ⊗ M m ) for any p ositive inte gers n, m . Lemma 3.9. KK( A ˆ ⊗ C 1 , B ˆ ⊗ C 1 ) ∼ = K K ( A, B ) Pr o of. Note that τ C 1 ◦ τ C 1 = τ M 2 . Th us Morita in v aria nce implies that τ C 1 has the inv erse as τ − 1 M 2 ◦ τ C 1 .  Lemma 3.10. KK( A ˆ ⊗ C 1 , B ) ∼ = KK 1 ( A, B ) . Pr o of. Recall that K K 1 ( A, B ) = K K ( A, B ˆ ⊗ C 1 ). Th us KK( A ˆ ⊗ C 1 , B ) ∼ = K K ( A ⊗ C 1 ˆ ⊗ C 1 , B ˆ ⊗ C 1 ) ∼ = K K ( A ⊗ M 2 , B ˆ ⊗ C 1 ) ∼ = K K ( A, B ˆ ⊗ C 1 ) ∼ = K K 1 ( A, B ) .  Theorem 3.11 (Kasparo v) . L et x ∈ KK( C 1 , S ) ∼ = KK 1 ( C , S ) ∼ = Ext ( C , S ) b e r epr esente d by the extension 0 → S → C → C → 0 wher e C is the c one and y ∈ KK( S, C 1 ) ∼ = KK 1 ( S, C ) ∼ = Ext ( S, C ) b e r epr esente d by the extension 0 → K → C ∗ ( v − 1) → S → 0 wher e v is a c o-isometry of the F r e dho lm index 1 (e. g., the adjoint of the unilater al shift on H ). Then the Kasp ar ov pr o duct x · y = 1 C 1 . 10 HYUN HO LEE Pr o of. Note that x corresp onds t o the unitary t → e 2 π it in K 1 ( S ) b y the Bro wn’s Unive rsal Co efficien t Theorem [Br]. Also, the Busb y in v ariant of 0 → K → C ∗ ( v − 1) → S → 0 is the extension τ : S → Q whic h sends e 2 π it − 1 to π ( v ) − 1. Th us τ corresponds to a pa ir (Φ , 1) suc h that Φ( e 2 π it ) = v . If w e apply Theorem 3.7 to the pair ([ e 2 π it ] , [1]) ∈ K 1 ( S ) × K 0 ( D ( S )), x · y is identifie d with Ind( v ) = 1. Since KK( C 1 , C 1 ) ∼ = KK( C , C ) ∼ = Z b y Prop osition 3.2, w e conclude x · y = 1 C 1 .  Com bining the ab o v e f a ct with a ring theoretic argument (See [Bl] for details), we can pro v e that y · x = 1 S . Th us K K -equiv alence of S induces the fo llowing corollary which will b e used later. Corollary 3.12. τ S : KK( A, B ) → KK( S A, S B ) is an isom orphism. Prop osition 3.13. K K ( A, C ) ∼ = K 1 ( D Φ ) wher e Φ i s an admiss ible r epr esentation of uni tal sep a r able C ∗ -algebr a A on a s ep ar able Hilb ert sp ac e H . Pr o of. Using Lemma 2.11, we define a map f r o m K 1 ( D Φ ) to K K ( A, C ) b y [ u ] 1 →  ˆ H ,  Φ 0 0 Φ  ,  0 u ∗ u 0  where ˆ H is H ⊕ H with the standard ev en gr a ding. Indeed, this con- struction giv es rise t o a well-define d group homomorphism. If [ u ] = [ v ], then u ⊕ 1 is homoto pic to v ⊕ 1. Th us  ˆ H ⊕ ˆ H ,  Φ ⊕ Φ 0 0 Φ ⊕ Φ  ,  0 u ∗ ⊕ 1 u ⊕ 1 0  =  ˆ H ,  Φ 0 0 Φ  ,  0 u ∗ u 0  ⊕  ˆ H ,  Φ 0 0 Φ  ,  0 1 1 0  is op erator ho - motopic to  ˆ H ,  Φ 0 0 Φ  ,  0 v ∗ v 0  ⊕  ˆ H ,  Φ 0 0 Φ  ,  0 1 1 0  =  ˆ H ⊕ ˆ H ,  Φ ⊕ Φ 0 0 Φ ⊕ Φ  ,  0 v ∗ ⊕ 1 v ⊕ 1 0  . Similarly , it can b e sho wn that the map is a group homomor phism. If w e sho w that it is surjectiv e, w e a r e done. W e will use Higson’s idea in p354 [Hig]. Let α ∈ K K ( A, C ) b e represen ted b y  H 0 ⊕ H 1 ,  φ 0 0 0 φ 1  ,  0 u ∗ u 0  where u is a unita ry in B ( H 0 , H 1 ). Let Ψ = · · · ⊕ φ 0 ⊕ φ 0 ⊕ φ 1 ⊕ φ 1 ⊕ · · · and H = · · · ⊕ H 0 ⊕ H 0 ⊕ H 1 ⊕ H 1 · · · ⊕ . W e consider a degenerate cycle  H ⊕ H ,  Ψ 0 0 Ψ  ,  0 I I 0  A N OTE ON KA SP AROV PR O DUCT AND DUALITY 1 1 Then  H 0 ⊕ H 1 ,  φ 0 0 0 φ 1  ,  0 u ∗ u 0  ⊕  H ⊕ H ,  Ψ 0 0 Ψ  ,  0 I I 0  is unitarily equiv alent to  H ⊕ H ,  Ψ 0 0 Ψ  ,  0 F ∗ F 0  where F =   I 0 0 0 u 0 0 0 I   ◦ shifting to the righ t, i.e., F sends ( · · · , η 1 , η 0 , ξ 0 , ξ 1 , · · · ) to ( · · · , η 2 , η 1 , uη 0 , ξ 0 , · · · ) . Again b y adding a degenerate cycle  ˆ H ,  Φ 0 0 Φ  ,  0 I I 0  , w e get α =  H ⊕ H,  Ψ ⊕ Φ 0 0 Ψ ⊕ Φ  ,  0 F ∗ ⊕ I F ⊕ I 0  Since Φ is a dmissible, w e can ha v e a unitary U ∈ B ( H ⊕ H , H ) suc h that Ω = Ad ( U ) ◦ (Ψ ⊕ Φ) ∼ Φ. =  ˆ H , Ω ⊕ Ω ,  0 Ad ( U )( F ∗ ⊕ I ) Ad ( U )( F ⊕ I ) 0  By Lemma 4.1 .1 0. in [JenThom] =  ˆ H , Φ ⊕ Φ ,  0 Ad ( U )( F ∗ ⊕ I ) Ad ( U )( F ⊕ I ) 0  It is not hard to c heck tha t Ad ( U )( F ⊕ I ) ∈ D Φ ( A ) and α is the image of it. Th us we finish the pr o of.  Theorem 3.14 ( Ev en case) . The mapping K 0 ( A ) × K 1 ( D Φ ) → Z is the Kasp ar ov pr o duct K K ( S, S A ) × K K ( S A, S ) → Z . Pr o of. Without loss of generalit y , w e ma y assume p is the elemen t o f A . (If necessary , consider C k ⊗ A .) Using the Bott map in K-theory , p gets mapp ed to f p ( z ) ∈ K 1 ( S A ). Then, as w e ha v e no t ed in Prop osition 3.1, f p ( z ) is lifted to Ψ as an elemen t of K K ( S, S A ) where Ψ is the *-homomorphism from S to S A that is determined b y sending z − 1 to ( z − 1) p . On the other hand, [ u ] ∈ K 1 ( D Φ ( A )) is mapp ed to [ E ] in K K ( A, C ) where E =  ˆ H ,  Φ 0 0 Φ  ,  0 u ∗ u 0  b y Prop osition 3.1 3. Using natural isomorphism τ S : K K ( A, C ) → K K ( S A, S ), w e can think of a Ka sparo v pro duct Ψ b y [ τ S ( E )]. Using eleme n t a ry functorial prop erties, w e can c hec k Ψ · [ τ S ( E )] is equal 12 HYUN HO LEE to h ( ˆ H ⊗ S, ((Φ ⊕ Φ) ⊗ 1) ◦ ψ , G ⊗ 1 ) i denoting  0 u ∗ u 0  b y G . Since τ S : K K ( C , C ) → K K ( S, S ) is an isomorphism, there is a map ρ : C → B ( H ) suc h that τ S (( ˆ H , ρ ⊕ ρ, G ) ) = ( ˆ H ⊗ S, ((Φ ⊕ Φ) ⊗ 1) ◦ ψ , G ⊗ 1) . This implies that ( ρ ⊕ ρ ) ⊗ 1 = ((Φ ⊕ Φ) ⊗ 1) ◦ ψ . Th us ( ρ ⊗ 1)( z − 1) = ( z − 1 ) ρ (1 ) = (Φ ⊗ 1)(Ψ( z − 1)) b y equalit y (3) = (Φ ⊗ 1)(( z − 1) p ) = ( z − 1)Φ( p ) . Consequen tly , ρ (1) = Φ( p ). Then  ˆ H , ρ ⊕ ρ, G  is mapp ed to Ind ( Φ( p ) u Φ( p )) by Prop osition 3.2.  Reference s [Ar] W. Arveson Notes on ext ensions of C ∗ -algebr as Duke Math. Jour nal V ol44 no2 329-3 55 (1977) [Br] L . G. Brown The un iversal c o efficient the or em for Ext and quasidiagonali ty pp60-64 in O pe rator algebras and group representations, I. Monogr aphs of Stud. Math. 1 7, P itman, Boston, Mas s. 19 84 [Bl] B. B lack ada r K-the ory for Op er ator Algebr as MSRI P ublicatons V ol.5 Second Edition Cambridge Universit y Pres s 1998 [Hig] N. Higs o n C ∗ -Algebr a Extension The ory and Duality J. F unc. Anal. 129(19 95), 349-3 6 3. [JenThom] K . N. Jensen, K. Thomsen Element s of KK- the ory Birkh¨ auser, B o ston, 1991 MR94b:190 08 [Kas] G. G. Ka sparov The Op er ator K functor and ex tensions of C ∗ -algebr as Mathe. USSR-Izv 16(19 81) 513-5 72[Englis h T anslatio n] [Pa] W. P aschek e K-the ory for c ommu tants i n the Calkin algebr a Pacific J . Math. 95 (1981) 4 27-43 7 Dep ar tment o f Ma thema tical Sciences S eoul Na tio nal University Seoul, South K orea 151-747 E-mail ad dr ess : hyu n.lee7 425@g mail.com