Finite and torsion KK-theories

FINITE AND TORSION K K -THEORIES HVEDRI INASSARIDZE AND T AMAZ KANDELAKI Abstract. W e dev elop a finite K K G -theory of C ∗ -algebras f ol lowing Arletta z- H.Inassaridze’s approac h to finite algebraic K -theory [1] . The Browder- Karoubi-Lambre’s theorem on the or ders of the elemen ts f or finite algebraic K -theory [ , ] is extended to finite K K G -theory . A new biv ariant theory , called to rsion K K -theory is defined as the direct l imit of finite K K -theories. Suc h biv ariant K -theory has almost all K K G -theory prop erties and one has the following exact sequenc e · · · → K K G n ( A, B ) → K K G n ( A, B ; Q ) → K K G n ( A, B ; T ) → · · · relating K K -theory , rational biv ariant K - theory and torsi on K K -theory . F or a giv en homology theory on the category of separable GC ∗ -algebras finite, rational and torsion homology theories are introduced and inv estigated. In particular, w e formulate finite, torsion and rational versions of Baum-Connes Conjecture. The lat er is equiv alent to the inv estigat ion of rational and q -finite analogues for Baum-Connes Conjecture f or al l prime q . Introduction In this pap er we provide a new biv ariant theory , which will b e called tor sion equiv a r iant K K G -theory . That is closely connected with the usual and rational versions of K K G -theories. By definition to rsion K K G -theory is a direct limit of K K G -theory with co efficients in Z q ( q -finite K K G -theory in our terminology), where q r uns ov er a ll natura l num bers ≥ 2. This new biv aria nt homolog y theory ha s all the prop erties of K K G -theory except of the existence of the iden tit y mo rphism. W e arrive to the fo llowing principle: some of problems that arise in usual K K G - theory may b e reduced to suitable problems in ra tional, finite and torsion K K G - theories. Namely , it will b e shown that Baum-Co nnes co njecture has analo gues in finite, tors io n and rational K K -theories and Baum-Connes assembly map is an isomorphism if and only if its rational and finite asse m bly maps a re iso morphisms for all prime q (Theor em 3.5). As a technical tool, we mainly w ork with homolo gy theo ries o n the categ ory of C ∗ -algebra s with a c tio n of a fix locally co mpact g roup G . In sectio ns 1 a nd 2 for a given homolo gy theory H torsion and q -finite ho mology theories H ( q ) are con- structed and their prop erties are in vestigated. Much of these pr op erties are k nown for experts in some concrete form, but we could not find suitable references for our purp oses. They a re redefined and reinv estigated here. F urthermore, in section 2 we define and inv estigate a new homology theory , so called torsio n homology theory . Esp ecially , w e mak e ac cent on the following tw osided long exa ct seq ue nc e of ab e lia n The authors were partially supp orted by GNSF-grant, INT AS-Caucasus gran t and V olkswa gen F oundation grant . 1 2 HVEDRI INASSARIDZE AND T AMAZ KANDELAKI groups · · · → H T n +1 ( A ) → H n ( A ) r − → H n ( A ) ⊗ Q → H T n ( A ) → H n − 1 ( A ) r − → · · · for any GC ∗ -algebra A whic h is used co ncretely for biv ariant K K -theor ies in the sequel section. In particular, based o n results o f these sections w e list pro pe r ties of torsion and finite biv ariant K K G -theories. Besides, there exists a long exact sequence, which is similar to the a b ove long e x act sequence: (0.1) · · · → K K G n +1 ( A, B ; Q ) → K K G n +1 ( A, B ; Q / Z ) → → K K G n ( A, B ) Rat n − − − → K K G n ( A, B ; Q ) → K K G n ( A, B ; Q / Z )) → · · · The similar result for K -theory of b o r nologica l a lgebras one can find in [5]. The ratio na l biv ar iant K K -theory and the torsion biv aria nt K K -theory hav e all the prop erties of usual biv ariant K K -theory . The only difference is that the torsion case hasn’t unital morphisms. Note that rational biv a riant K K -theor y used in this pap er differs from the s imilar one defined in [3]. In the next section 3 we study torsion and q - finite K K -theorie s , wher e the fi- nite K K -theo ry is redefined fo llowing Arlettaz- H.Inassaridze ’s a pproach to finite algebraic K -theory [1]. Sections 4 and 5 are devoted to the pro of of the follo wing Browder-Karo ubi-Lambre’ theor em for finite K K -theory (see Theore m 5.5): Let A a nd B be , resp ectively , s e parable and σ -unital C ∗ -algebra s, rea l or co m- plex; and G b e a metrizable compact gr o up. Then, for all integer n , (1) q · K K G n ( A, B ; Z /q ) = 0 , if q − 2 is no t divided by 4; (2) 2 q · K K G n ( A, B ; Z /q ) = 0 , if 4 divides q − 2. It is clear tha t this result ho lds for non-unital rings to o. F or finite a lgebraic K -theory this theorem for n = 1 was pr ov ed algebraically b y Karoubi and Lambre [], and for n > 1 by Browder [2]. The key idea to carry out this pro blem is its reduction to the alg ebraic K -theory case. This is realized b y tw o steps. First we calculate finite top ologic a l K -theory of C ∗ -algebra s and additive C ∗ -categor ies by finite a lgebraic K -theory of rings. Then generalizing the main result of [8], finite biv ar iant K K G -theory is calculated by finite top o logical K - theory of the additive C ∗ -categor y o f F redholm mo dules. When G is a lo cally compact group, it is more complicated to get the s imilar result for finite G -equiv a riant biv aria nt K K -theory and we in tend to inv estigate this problem in a forthcoming paper . 1. On Finite homology theor y In this sectio n w e analyze some prop erties o f homo logy theory with coefficients in Z q which is sa id to b e q - finite homolo gy . There exist s ome different wa ys to construct for a given homology theory o n C ∗ -algebra s a corresp onding q - finite ho- mology theory; we c ho ose one o f them, suitable for our purp os es. Let S 1 be the unit cycle in the plane o f complex num bers with mo dule one. The map ˜ q : S 1 → S 1 , x 7→ x q , FINITE AND TORSION K K -THEORIES 3 q ≥ 2, q ∈ N , is called standar d q-th p ower map . Since 1 ∈ S 1 is in v aria nt relativ e to the map ˜ q , it can b e conside r ed as a map of p ointed spaces ˜ q : S 1 ∗ → S 1 ∗ , x 7→ x q , where ∗ = 1. These a r e basic q-th p ow er maps in algebr a and top olog y . Let C 0 ( S 1 ) be a C ∗ -algebra of contin uous complex (or real) functions on the unit cycle S 1 in the pla ne of complex num bers with module one v anishing a t 1. Then the map ˜ q : S 1 → S 1 , x 7→ x q , q ≥ 2, q ∈ N , induce s a ∗ -homomor phism ˆ q : C 0 ( S 1 ) → C 0 ( S 1 ) , f ( s ) 7→ f ( s q ) . Denote C ∗ -algebra C q as cone of the homomorphism ˆ q : C q = { ( x, f ) ∈ C 0 ( S 1 ) ⊕ C 0 ( S 1 ) ⊗ C [0; 1) | ˆ q ( x ) = f (0) } , The following lemma is one of the main prop erty of the degree map. T he idea of the pro of is tak en from [13]. Lemma 1.1. L et p q : C pq → C q b e a natur al map induc e d by a c ommutative C 0 ( S 1 ) p / / pq   C 0 ( S 1 ) q   C 0 ( S 1 ) = / / C 0 ( S 1 ) diagr am. T hen ther e is a natur al homomorphism ν p,q : C p q → C p , which is a homotopy e quivalenc e. Pr o of. A homomor phism ν p,q is induced by the commutativ e diagram C pq   p q / / C q   C 0 ( S 1 ) p / / C 0 ( S 1 ) . Cho ose a ho motopy H : [0 , 1] 2 × [0 , 1] → [0 , 1] 2 relative to L = [0 , 1] × { 1 } ∪ { 0 , 1 } × [0 , 1] such that H 0 = id and H 1 is the retraction of [0 , 1] 2 on L . Then, a homotop y inv erse to ν p,q is given b y χ : C p → C p q , χ ( a, b ) = ( a, b, ˜ b · H 1 ) F or b ∈ C 0 ( S 1 )[0 , 1 ) define ˜ b : L → C 0 ( S 1 ) by ˜ b ( s, 1 ) = (1 , t ) = 0 and ˜ b (0 , t ) = b ( t q ) , t, s ∈ [0 , 1]. W e hav e ν p,q · χ = C p . There is a homotop y betw een id C p q and χ · ν p,q which is given by a map G t ( a, b, c ) = ( a, b, c t ) where c t ( r , s ) = c ( H t ( r , s )).  4 HVEDRI INASSARIDZE AND T AMAZ KANDELAKI Lemma 1.2. (1) L et A b e a C ∗ -algebr a. Then the c ommutative diagr am A ⊗ C q / /   A ⊗ C 0 ( S 1 ) [0 , 1)   A ⊗ C 0 ( S 1 ) id A ⊗ ˆ q / / A ⊗ C 0 ( S 1 ) is a pul lb ack dia gr am. In p articular, A ⊗ C q ≃ C id A ⊗ ˆ q . (2) L et the diagr am A / /   B   C / / D b e a pul lb ack diagr am and ( X , x ) p ointe d c omp act sp ac e. Then the induc e d diagr am A ( X,x ) / /   B ( X,x )   C ( X,x ) / / D ( X,x ) is a pul lb ack dia gr am. Pr o of. (1). Let the diagram P / /   A ⊗ C 0 ( S 1 ) [0 , 1)   A ⊗ C 0 ( S 1 ) id A ⊗ ˆ q / / A ⊗ C 0 ( S 1 ) be a pullback diagram. Then P co nt ains the couple of functions ( f ( s ) , g ( s, t )), such that f ( s q ) = g ( s, 0), f ( 0) = 0, g (0 , t ) = 0; s ∈ S 1 t ∈ [0 , 1). Ther e fore the pair ( f ( s ) , g ( s, t )) defines a con tinu ous function on the co ne σ q of the degree map S 1 q − → S 1 with v alues in A . So, there is a homomo rphism P → A σ q ≃ A ⊗ C q (whic h is a morphism of suitable diag rams). Th us the diag ram is pullback and as a consequence we get the isomor phism A ⊗ C q ≃ C id A ⊗ ˆ q . (2) is trivia l.  Recall that a family o f functor s H = { H n } n ∈ Z on the catego r y of (separable or σ -unital ) GC ∗ -algebra s (real or complex) [9] is sa id to b e homology theor y (cf. [3]): if (1) H n is a homo topy inv ariant functor for any n ∈ Z (2) for a ny ∗ -homomor phism ( G -equiv ariant ) of σ - unital algebras f : A → B there exists a natural t wosided long exact sequence of ab elia n groups: · · · → H n +1 ( B ) → H n ( C f ) → H n ( A ) → H n ( B ) → H n − 1 ( C f ) → · · · where C f is the co ne of f . FINITE AND TORSION K K -THEORIES 5 Definition 1 . 3. Under the q - finite homology of a homology H , q ≥ 2, we mean a family of functors H ( q ) = { H ( q ) n } n ∈ Z , where H ( q ) n = H n − 2 ( − ⊗ C q ) . Below w e list main prop erties o f the q -finite homology . Prop ositi o n 1. 4. L et H b e a homo lo gy t he ory on the c ate gory of C ∗ -algebr as. Then (1) H ( q ) is a homolo gy the ory; (2) ther e is a twoside d long exact s e quenc e of ab elia n gr oups · · · → H ( q ) n +1 ( A ) → H n ( A ) q · − → H n ( A ) → H ( q ) n ( A ) → H n − 1 ( A ) q · − → · · · (3) ther e is a twoside d long exact s e quenc e of ab elia n gr oups · · · → H ( p ) n +1 ( A ) → H n ( A ) ( q ) ˙ p − → H ( pq ) n ( A ) ´ q − → H ( p ) n ( A ) → H ( q ) n − 1 ( A ) ˙ p − → · · · (4) if homo lo gy H has an asso ciative pr o duct H n ( A ) ⊗ H m ( B ) → H n + m ( A ⊗ B ) then ther e is an asso ciative pr o duct H ( p ) n ( A ) ⊗ H ( q ) q ( B ) → H ( pq ) n + m − 2 ( A ⊗ B ) . Pr o of. The first a nd the second parts are immediate consequences of Lemma 1.2 and Definition 1 .3. The thir d is a trivia l consequence of the Pupp e’s exact sequence for the ho momorphism p q : A ⊗ C pq → A ⊗ C q , Lemma 1.4 and Lemma 1 .2. Finally we have (1.1) H ( p ) n ( A ) ⊗ H ( q ) m ( B ) = H n − 2 ( A ⊗ C p ) ⊗ H ( q ) m − 2 ( B ⊗ C q ) → → H n + m − 4 ( A ⊗ B ⊗ C p ⊗ C q ) → H n + m − 4 ( A ⊗ B ⊗ C pq ) → H ( pq ) n + m − 2 ( A ⊗ B ) where the pr o duct C p ⊗ C q → C pq is defined as follows. T he r e are na tural homo- morphisms ˘ q : C p → C pq , induced by the commutativ e diagram C p / / ˘ q   C 0 ( S 1 ) p / / =   C 0 ( S 1 ) q   C pq / / C 0 ( S 1 ) pq / / C 0 ( S 1 ) , and similar ly ˘ p : C q → C pq . Since all algebra s are n uclear (in C ∗ -algebra ic sense), these homomor phisms yield a homomorphism (pro duct) C p ⊗ C q → C pq which is asso ciative in the obvious s ense.  2. On Torsion Homology theor y Now we define a new homology theory using the fa mily o f q -finite homolo g y theories, q ≥ 2. Cons ider the or dered set N (2) = { q ∈ N | q ≥ 2 } , where q ≤ q ′ iff q divides q ′ . Note that if q ′ = q s , then there is a natura l tr ansformation of functors τ ( qq ′ ) n : H ( q ) n → H ( q ′ ) n 6 HVEDRI INASSARIDZE AND T AMAZ KANDELAKI induced by the homomorphism q s , C q / / q s   C 0 ( S 1 ) q / / =   C 0 ( S 1 ) s   C q ′ / / C 0 ( S 1 ) q ′ / / C 0 ( S 1 ) , where τ ( qq ′ ) n ( A ) : H ( q ) n ( A ) → H ( q ′ ) n ( A ) deno tes the homomorphism H n ( id A ⊗ q s ). Therefore one has an inductiv e system of ab elia n groups { H ( q ) n ( A ) , τ ( qq ′ ) n ( A ) } q ∈ N (2) for any GC ∗ -algebra A . Prop ositi o n 2. 1. L et H b e a homolo gy the ory and H T b e a family of functors define d by the e quality H T n ( A ) = lim − → q H ( q ) n ( A ) , n ∈ Z , q ≥ 2 , for any C ∗ -algebr a A . Then (1) H T is a homolo gy the ory H on the c ate gory GC ∗ -algebr as; (2) Ther e is a twoside d long exact se quenc e of ab elian gr oups · · · → H T n +1 ( A ) → H n ( A ) r − → H n ( A ) ⊗ Q → H T n ( A ) → H n − 1 ( A ) r − → · · · for any GC ∗ -algebr a A . (3) Ther e is a twoside d long exact se quenc e of ab elian gr oups · · · → H T n +1 ( A ) ˆ q − → H ( q ) n +1 ( A ) − → H T n ( A ) ` p − → H T n ( A ) ˆ q − → H ( q ) n ( A ) − → · · · for any GC ∗ -algebr a A . Pr o of. According to Pr op osition 1 .4 (1) the first part is an eas y consequence of the fact that the direc t limit preserves homotopy and ex c is ion pro p e rties. F or the second part cons ide r the comm utative dia gram · · · / / H ( q ) n +1 ( A ) / /   H n ( A ) q / / H n ( A ) / / q ′ q   H ( q ) n ( A ) / / τ ( qq ′ ) n ( A )   · · · · · · / / H ( q ′ ) n +1 ( A ) / / H n ( A ) q ′ / / H n ( A ) / / H ( q ′ ) n ( A ) / / · · · , where r ows are lo ng tw osided exact sequences. B y taking the dir ect limit of these long exact seque nc e s , one gets the following long tw osided exact sequence · · · / / H n ( A ) ˆ q / / lim − → q H n ( A ) / / H T n ( A ) / / H n − 1 ( A ) / / · · · It is easy to chec k that the inductiv e sy s tem { H n ( A ) , q ′ q } is isomor phic to the inductive sy stem { H n ( A ) ⊗ Z { q } , q ′ q } , where Z { q } = Z for all q . Then lim − → q H n ( A ) ≃ H n ( A ) ⊗ lim − → q Z { q } ≃ H n ( A ) ⊗ Q , since one has the is omorphism lim − → q Z { q } ≃ Q defined by the map ( q , r ) 7→ r q . FINITE AND TORSION K K -THEORIES 7 F or (3) consider the comm utative diag ram · · · / / H ( q ) n +1 ( A ) / /   H ( p ) n ( A ) p / /   H ( pq ) n ( A ) / / q ′ q   H ( q ) n ( A ) / / / / · · · · · · / / H ( q ) n +1 ( A ) / / H ( p ′ ) n ( A ) q / / H ( pq ) n ( A ) / / H ( q ) n ( A ) / / · · · , where rows are long tw osided exa ct sequences given in P rop osition 1.4 (3). The direct limit o f these long exa c t seq ue nc e s with respect to p yields the required lo ng t wosided e x act sequence.  The homo logy theory H T is sa id to be the tors ion homology o f the homology H . Corollary 2.2. L et τ : H → ˜ H b e a natur al tra nsformation of homolo gy the ories. Then τ induc es natur al t r ansformatio ns τ ( q ) : H ( q ) → ˜ H ( q ) , τ T : H T → ˜ H T and τ Q : H ⊗ Q → ˜ H ⊗ Q . F u rthermor e the fol lowing c onditions ar e e qu ivalent. (1) τ ( A ) is an isomorphism for a C ∗ -Algebr a A . (2) τ T ( A ) and τ Q ( A ) is an isomorphism for a C ∗ -Algebr a A . (3) τ ( q ) ( A ) for al l prims q and τ Q ( A ) is an isomorphism for a C ∗ -Algebr a A . Pr o of. (1) ∼ = (2) is co nsequence o f the Fiv e Lemma and following commutativ e diagram of tw osided long exact sequence : .. / / H T n +1 ( A ) / / τ T ( A )   H n ( A ) / / τ ( A )   H n ( A ) ⊗ Q / / τ Q ( A )   H T n ( A ) / / τ T ( A )   .. .. / / ˜ H T n +1 ( A ) / / ˜ H n ( A ) / / ˜ H n ( A ) ⊗ Q / / ˜ H T n ( A ) / / .. (2) ∼ = (3) is a consequence of the Five Lemma and following comm utative dia- grams of the tw osided long exac t se q uence: .. / / H ( q ) n +1 ( A ) / / τ ( q )   H T n ( A ) / / τ T   H T n ( A ) / / τ T   H ( q ) n ( A ) / / τ ( q )   .. .. / / ˜ H ( q ) n +1 ( A ) / / ˜ H T n ( A ) / / ˜ H T n ( A ) / / ˜ H ( q ) n ( A ) / / .. and .. / / H ( q ) n +1 ( A ) / / τ ( q )   H ( p ) n ( A ) / / τ ( p )   H ( pq ) n ( A ) / / τ ( pq )   H ( q ) n ( A ) / / τ ( q )   . . . .. / / ˜ H ( q ) n +1 ( A ) / / ˜ H ( p ) n ( A ) / / ˜ H ( pq ) n ( A ) / / ˜ H ( q ) n ( A ) / / ..  3. Applica tions to K K -theor y 3.1. T orsion and fini te K K -theories . By considering K K G ( A, − ) as a homo l- ogy theory a nd acco rding to s ection 1 w e define finite, torsion and rational K K G - theories for all integer n as follows. 8 HVEDRI INASSARIDZE AND T AMAZ KANDELAKI Definition 3.1. (3.1) K K G n ( A, B ; Z q ) = K K G n − 2 ( A, B ⊗ C q ) . Definition 3.2. K K G n ( A, B , T ) = lim − → q K K G n ( A, B ; Z q ) . Definition 3.3. K K G n ( A, B ; Q ) = K K G n ( A, B ) ⊗ Q. Our definitions of finite and ratio na l K K G differ fro m the ex isiting definitions of finite and rational K K -theor ies ([3], 23.15.6-7 ). In effect, here we compare the t wo versions of the definitions of finite and r ational KK- theories. 1. Let N b e the smallest class of separ able C- a lgebras with the following pr op- erties: (N1) N contains field complex num ber s; (N2) N is closed under countable inductiv e limits; (N3) if 0 → A → D → B → 0 is an exa ct sequence, and tw o of them ar e in N, then s o is the third; (N4) N is closed under K K -eq uiv alence . Let D b e a C ∗ -algebra in N with K 0 ( D ) = Z p , K 1 ( D ) = 0. Define K K n ( A ; B ; Z p ) = K K n ( A ; B ⊗ D ) . As noted in ([3]), so defined K K S -gr oups are independent of the c hoice of D . Bellow we show that the ab ov e definition is eq uiv alent to our definition. One has K 0 ( C m ) = Z m and K 1 ( C m ) = 0. This is an easy consequence o f the Bott pe rio dicity theorem and the tw o-sided long exact sequence .... → K 2 ( C 0 ( S 1 )) → K 2 ( C 0 ( S 1 )) → K 2 ( C m ) → K 1 ( C 0 ( S 1 )) → ..., since K 1 ( C 0 ( S 1 )) = 0 . Therefore our definition of finite K K -theory agrees to its definition in the sense of [1] ta king into acc o unt the following isomor phism induced b y the Bott p erio dicity theorem: K K n ( A ; B ⊗ C m ( S 1 )) ≃ K K n − 2 ( A ; B ⊗ C m ( S 1 )) . 2. The rational K K -theory is defined in ([1 ], 23.1 5.6) by the following manner. Let D be a C ∗ -algebra in N with K 0 ( D ) = Q, K 1 ( D ) = 0. Define K K n ( A ; B ; Q ) = K K n ( A ; B ⊗ D ) . In genera l K K n ( A ; B ; Q ) 6 = K K n ( A ; B ) ⊗ Q ([3], 23.15.6 ). F or exa mple, K K ( D ; C ; Q ) = Q and K K ( D ; C ) ⊗ Q = 0 . This means that our rational K K -theory differs fro m that of [3]. According to results of the previous section one has the following pr op erties of q -finite and to rsion K K -theo ries. FINITE AND TORSION K K -THEORIES 9 (1) The groups K K G ( A, B ; Z q ) hav e Bott p erio dicity proper ty and s atisfy the excision prop er ty r elative to b o th arguments. (2) there is a natura l t wosided exact sequence: (3.2) · · · → K K G n ( A, B ) q × − − → K K G n ( A, B ) → K K G n ( A, B ; Z q ) → → K K G n − 1 ( A, B ) q × − − → K K G n − 1 ( A, B ) → · · · (3) there is a natura l t wosided exact sequence: (3.3) ´ q − → K K G n ( A, B , Z pq ) ` p − → K K G n ( A, B ; Z q ) → → K K G n − 1 ( A, B , Z p ) ´ q − → K K G n − 1 ( A, B , Z pq ) → (4) There is a as so ciative pr o duct (3.4) K K G n ( A, B ; Z p ) ⊗ K K G m ( A, B ; Z q ) → K K G n + m − 2 ( A, B ; Z pq ) (5) there is a natura l t wosided exact sequence: (3.5) · · · → K K G n ( A, B , T ) ` q − → K K G n ( A, B ; T ) ˘ p − → K K G n ( A, B , Z q ) → K K G n − 1 ( A, B , T ) → . . . (6) there is a natura l t wosided exact sequence: (3.6) · · · → K K G n ( A, B ) r − → K K G n ( A, B ; Q ) ˘ t − → K K G n ( A, B , T ) → K K G n − 1 ( A, B , ) → . . . In addition there is an asso cia tive pro duct K K G n ( A, B ; Q ) ⊗ K K G m ( B , C ; Q ) → K K G n + m ( B , C ; Q ) T ensor pro duct is considere d o ver ring of in tegers. The pr o duct is a comp os ition o f the isomor phism: (3.7) ( K K G n ( A, B ) ⊗ Q ) ⊗ ( K K G n ( B , C ) ⊗ Q ) ∼ = ∼ = ( K K G n ( X ; A, B ) ⊗ K K G n ( X ; B , C )) ⊗ ( Q ⊗ Q ) , which is the c o mp o sition of the twisting and asso ciativity iso morphisms of tensor pro duct, and a ho momorphism (3.8) ( K K G n ( A, B ) ⊗ K K G n ( B , C )) ⊗ ( Q ⊗ Q ) − → K K G n ( A, C ) ⊗ Q defined by a map ( f ⊗ r ) ⊗ ( f ′ ⊗ r ′ ) 7→ ( f · f ) ′ ⊗ rr ′ , wher e f · f is K asparov pro duct of f and f ′ . Thu s we can form an additiv e category K K G Q , where GC ∗ -algebra s are ob jects and the gro up of morphisms from A to B is given b y the equa lity (3.9) K K G n ( A, B ; Q ) = K K G n ( X ; A, B ) ⊗ Q . There is a natural additive functor Rat : K K G − → K K G Q which is identit y on ob jects, and on morphisms is defined by the map f 7→ f ⊗ 1. It is clear tha t R at is a n a dditive functor. 10 HVEDRI INASSARIDZE AND T AMAZ KANDELAKI The result b elow says that K K G Q is a biv ariant theory on the category of sepa- rable GC ∗ -algebra and it is said to b e the ra tional K K G -theory . Theorem 3 .4. The additive c ate gory K K G Q is a bivariant the ory on the c ate gory of sep ar able GC ∗ -algebr as, i.e . has al l fundamental pr op erties of usu al bivariant K K -t he ory. Besides, K K G T is a bimo dule on the c ate gory K K G such t hat it is c ohomolo gic al functor r elative the first ar gument and homolo gic al functor re lative to t he se c ond ar gument satisfying the Bott p erio dicity pr op erty. Pr o of. This is easy consequence of the fac t that Q is a fla t Z -module and the tensor pro duct on a fla t mo dule preserves exactness.  3.2. A l o ok at Baum-Conne s Conjecture. In the form ulation of B aum-Connes Conjecture a crucial role play the groups K top n ( G, A ), so called the topolo gical K - theory of G with coefficients in A , and the homomorphism µ A : K top n ( G, A ) → K n ( G ⋉ r A ) , which is called the Baum- C o nnes asse m bly map. The Ba um-Connes Conjectur e for G with co efficients in A asse r ts that this map is an isomorphism. Note that K top n ( G, − ) and K n ( G ⋉ r − ) are homology theories in the s ense that we hav e defined in the first section (cf. [10]). There fo re w e hav e ra tional, torsion and finite versions of Baum-Connes Conjecture: • (Rational version) the assembly map µ A ⊗ id Q : K top n ( G, A ) ⊗ Q → K n ( G ⋉ r A ) ⊗ Q is an iso mo rphism; • (Finite version) the q -finite a ssembly map µ ( q ) A : K top n ( G, A ; Z q ) → K n ( G ⋉ r A ; Z q ) ⊗ is an iso mo rphism; • (T o rsion version) the torsion as sembly map µ ( q ) A : K top n ( G, A ; T )) → K n ( G ⋉ r A ; T )) is an iso mo rphism. According to Cor ollary 2.2, we ha ve the follo wing theorem Theorem 3.5. The fol lowing Conje ctur es ar e e quivalent. (1) Baum-Connes Conje ctur e; (2) Baum-Connes r ational and torsion Conje ctur es; (3) Baum-Connes r ational and q -finite Conje ctur es for al l primes. 4. Remarks on finite algebraic and topol ogical K -theories W e beg in with some preliminary definitions and pro p erties. In [2], Br owder has defined algebr a ic K - theory of an unital ring with co efficien ts in Z /q , q ≥ 2 as follows: K n ( R ; Z /q ) = π n ( B GL ( R ) + ; Z /q ) by using so called homotopy groups with co efficients in Z /q . R emark . B ellow ”Algebraic K -theory of an unital r ing with co efficients in Z /q ” will be replaced by ” q - finite algebr aic K -theory of an unital r ing”. FINITE AND TORSION K K -THEORIES 11 F o r our purp o s es we use equiv alent definition used in [1]: K a n +1 ( R ; Z /q ) = π n ( F q ( B GL ( R ) + )) . Here, in genera l, F q ( X ) is defined a s the homotopy fiber of the q -p ow er map of a lo op space X = Ω Y (se e [1 ]). There exists similar in terpretatio n for q -finite topo logical K -theor y of C ∗ -al ge- bras. If A is an unital C ∗ -algebra . Then GL ( A ) has the standard topo logy induced by the no rm in A . Denote this top olo gical g roup b y GL t ( A ). It is known that GL t ( A ) a nd Ω B ( GL t ( A )) are homoto py equiv alen t spa ces. Therefore to p o logical K -gro ups ma y be defined equiv ale ntly b y the equality K t n ( A ) = π n ( B ( GL t ( A ))) , n ≥ 1 . Therefore one can define the q -finite top olo gical K -theory as follows: K t n +1 ( R ; Z /q ) = π n ( F q ( B ( GL t ( R )))) . W e hav e na tur al, up to homoto py , maps B ( GL ( A )) + → B ( GL t ( A )) and F q B ( GL ( A )) + → F q B ( GL t ( A )) . Therefore we ha v e natural homomor phisms α n : K a n ( A ) → K t n ( A ) and α n,q : K a n ( A, Z /q ) → K t n ( A, Z /q ) , n ≥ 1 , q ≥ 2 . Prop ositi o n 4.1. L et A b e a C ∗ -algebr a and K b e a C ∗ -algebr a of c omp act op er- ators on a sep ar able Hilb ert sp ac e. Then the natur al homomorphisms ε − 1 α n,q : K a n ( A ⊗ K , Z /q ) → K t n ( A, Z /q ) n ≥ 1 , q ≥ 2 , ar e isomorphisms, wher e ε : K t n ( A ; Z q ) ∼ = − → K t n ( A ⊗ K ; Z q ) is the isomorphism of stability for t he finite top olo gic al K -the ory of C ∗ -algebr as. Pr o of. It is enough to show that the homomorphism α n,q : K a n ( A ⊗ K ; Z q ) → K t n ( A ⊗ K ; Z q ) is an isomorphis m. T o this end consider the follo wing commut ative diagram · · · / / K a n +1 ( A ⊗ K ; Z q ) / / α n +1 ,q   K a n ( A ⊗ K ) × q / / α n   K a n ( A ⊗ K ) / / α n   · · · · · · / / K t n +1 ( A ⊗ K ; Z q ) / / K t n ( A ⊗ K ) × q / / K t n ( A ⊗ K ) / / · · · . Since the natural homomorphis ms α n : K a n ( A ⊗ K ) → K t n ( A ⊗ K ) are iso mor- phisms for any in teger n [12], then b y the Five Lemma the homomorphism K a n ( A ⊗ K ; Z q ) → K t n ( A ⊗ K ; Z q ) is an isomorphis m too for all n ≥ 2.  12 HVEDRI INASSARIDZE AND T AMAZ KANDELAKI 5. Bro wder-Kar oubi-lambre ’ s theorem for finite K K -theor y One has the follo wing in terpretatio n of the q -finite topolog ical K -theory . Prop ositi o n 5.1. Ther e ar e a natur al isomorphisms K t n ( A ; Z /q ) ∼ = K t n − 2 ( A ⊗ C q ) , for al l n ≥ 1 and q ≥ 2 . Pr o of. Since cla s sifying s pace co nstruction has functorial pr op erty , according to the functorial prop er t y of the functor B ( GL t ( − )) and the co mm utative diagram A ⊗ C q / /   A ⊗ C 0 ( S 1 ) ⊗ C [0; 1)   A ⊗ C 0 ( S 1 ) / / A ⊗ C 0 ( S 1 ) , one gets the comm utative diag rams B ( GL t ( A ⊗ C q )) / /   B ( GL ( A ⊗ C 0 ( S 1 ) ⊗ C [0; 1)))   B ( GL t ( A ⊗ C 0 ( S 1 ))) / / B ( GL ( A ⊗ C 0 ( S 1 ))) and F q (Ω B ( GL t ( A ))) / /   Ω B ( GL t ( A )) [0 , 1)   Ω B ( GL t ( A ) q / / Ω B ( GL t ( A )) Since the second diag ram is universal, there exists a natur al map χ : B ( GL t ( A ⊗ C q )) → F q (Ω B ( GL t ( A ))) . Therefore one has a na tur al homomorphis m π n χ : π n ( B ( GL t ( A ⊗ C q ))) → π n (Ω F q ( B ( GL t ( A )))) Thu s there is a natural homomorphism χ n : K t n ( A ⊗ C q ) → K t n +2 ( A, Z /q ) . Now, co nsider the following commut ative diagram · · · / / K t n ( A ⊗ C q ) / / χ n   K t n ( A ⊗ C 0 ( S 1 )) × q / / =   K t n ( A ⊗ C 0 ( S 1 )) / / =   · · · · · · / / K t n +2 ( A, Z /q ) / / K t n +1 ( A ) × q / / K t n +1 ( A ) / / · · · According to the Fiv e Lemma, one concludes that χ n are isomor phisms, n ≥ 1.  Let H : C ∗ → Ab b e a functor, wher e C ∗ is the categ o ry o f unital C ∗ - algebr as and their homo mo rphisms (non-unital). Then (1) if the inclusion in the upp er left corner A ֒ → M n ( A ) induces is omorphism H ( A ) ∼ = H ( M n ( A )), H is said to b e matrix in v a r iant functor. FINITE AND TORSION K K -THEORIES 13 (2) if H comm utes with direct system of C ∗ -algebra s, H is sa id to be contin u- ous. F o r a given matrix in v ar ia nt and contin uous functor H ther e ex ists a n extension H of it on the categ ory of small a dditive C ∗ -categor ies Add C ∗ such that the following diagram C ∗ pro j f / / H ! ! C C C C C C C C Add C ∗ H z z v v v v v v v v v Ab commutes, where proj f is a functor whic h sends unital C ∗ -algebra A to the additive C ∗ -categor y of finitely genera ted pro jective A -mo dules. The functor H is defined by the follo wing manner (cf.[7], [8]). First note that the functor H is a inner in v ar iant funct or (see Lemma 2.6.12 in [6]). Let A be an a dditiv e C ∗ -categor y . Set L ( a ) = hom A ( a, a ), a ∈ ob A . Let us write a ≤ a ′ if there is an isometry v : a → a ′ in A , i.e. v ∗ v = id a . The r e la tion ” a ≤ a ” makes the set of ob jects into a directed set. An y iso metry v : a → a ′ in A defines a ∗ -homomor phis m of C ∗ -algebra s Ad( v ) : L ( a ) → L ( a ′ ) by the rule x 7→ v xv ∗ . Using tech nics from [7], one has the following. Let v 1 : a → a ′ and v 2 : a → a ′ be t wo isometr ies in A . Then the homomorphisms Ad ∗ v 1 , Ad ∗ v 2 : H ( L ( a )) → H ( L ( a ′ )) are equa l. Indeed, let u =  0 1 1 0  be the unitar y element in a n unital C ∗ -algebra M 2 ( L ( a ′ )). Since H is a ma trix inv ariant functor, it is inner invariant functor too (see Lemma 2.6.1 2 in [6]), i.e. the homomorphis m H ( ad ( u )) is the identit y map. Therefore, the maps x 7→  x 0 0 0  and x 7→  0 0 0 x  sending L ( a ′ ) in to M 2 ( L A ( I )( a ′ )), induces the s ame is omorphisms after applying the functor H . It is clear that the ho momorphism ν aa ′ ∗ = H ( ν aa ′ ) is not dep ending on the choice of an is o metry ν aa ′ : a → a ′ . Therefor e one has a direct sys tem { H ( L ( a )) , ν aa ′ ∗ ) } a,a ′ ∈ obA of ab elian g r oups. Definition 5.2. Let A be a n additive small C ∗ -categor y . Then by definition H ( A ) = lim − → H ( L ( a )) . So defined functor makes co mmutative the ab ov e diagram. That follows from the matrix inv ariant a nd contin uous proper ties of H and is a simple exercise (see [8]). Since the functors K t ( − ; Z /q ) hav e the ab ov e mentioned pro pe r ties, one can define the q -finite topolo g ical K -theory for an additiv e C ∗ -categor y A b y setting K t n ( A ; Z /q ) = lim − → K t n ( L ( a ); Z /q ) . This definition is in acco rdance with other definitions o f q -finite top o logical K - theories b ecause of the matrix in v ar iant and c o ntin uo us prop er ties. Therefore we 14 HVEDRI INASSARIDZE AND T AMAZ KANDELAKI get a gener alization of B rowder-k ar oubi-Lambre ’s theorem for small additiv e C ∗ - categorie s. Prop ositi o n 5.3. L et A b e a smal l additive C ∗ -c ate gory. Then, for al l n ∈ Z , (1) q · K t n ( A , Z /q ) = 0 , if q − 2 is not divide d by 4 ; (2) 2 q · K t n ( A , Z /q ) = 0 , if 4 divides q − 2 . Pr o of. It is conseq uence of Prop os ition 4.1.  The next step is to giv e an in terpretation of q - finite K K G -theory as topolo g ical K -theory of the additive C ∗ -categor y Re p G ( A, B ). Suc h an in terpretatio n exists for K K G -theory , where G is a compact metrizable gr oup [8]. Theorem 5.4. L et A and B b e, r esp e ctively, sep ar able and σ -unital G − C ∗ - algebr as, r e al or c omplex; and G b e metrizable c omp act gr oup. Then, for al l int e ger n and q ≥ 2 , ther e exists a natur al isomorphi sms K K G n ( A, B ; Z q ) ∼ = K t n +1 (Rep( A, B ); Z /q ) , When G is loca lly compact group, the pro of is more complicated and this ca se will be in vestigated in the further pap er. First we r ecall the definition of the C ∗ -categor y R ep ( A, B ). This categ ory w as constructed in [8]. Let H G ( B ) b e the additive C ∗ -categor y of c o untably generated right Hilbert B - mo dules equipped with a B -linear, norm-contin uous G -action over a fixed compact second countable group G [9]. Note that the compact group ac ts o n the morphis ms by the following rule: for f : E → E ′ the morphism g f : E → E ′ is defined b y the formula ( g f )( x ) = g ( f ( g − 1 ( x ))). The c a tegory H G ( B ) contains the class of compact B -homomo rphisms [9 ]. De- note it by K G ( B ). Known prop erties of compa ct B -homomorphisms imply that K G ( B ) is a C ∗ -ideal [4] in H G ( B ). Ob jects of the c a tegory Rep ( A, B ) ar e pairs of the form ( E , ϕ ), where E is an ob ject in H G ( B ) and ϕ : A → L ( E ) is an equiv a riant ∗ -homomo rphism. A morphism f : ( E , φ ) → ( E ′ , φ ′ ) is a G -inv ar iant mor phis m f : E → E ′ in H G ( B ) such tha t f φ ( a ) − φ ′ ( a ) f ∈ K G ( E , E ′ ) for all a ∈ A . The structure of a C ∗ -categor y is inherited from H G ( B ). It is easy to see that R ep ( A, B ) is an additiv e C ∗ -categor y , not idempotent-complete. Now, w e are re ady to construct our main C ∗ -categor y , tha t is Rep( A, B ). Its ob jects ar e triples ( E , φ, p ), wher e ( E , φ ) is an ob ject and p : ( E , φ ) → ( E , φ ) is a morphism in Rep ( A, B ) such that p ∗ = p and p 2 = p . A morphism f : ( E , φ, p ) → ( E ′ , φ ′ , p ′ ) is a morphis m f : ( E , φ ) → ( E ′ , φ ′ ) in Rep ( A, B ) such that f p = p ′ f = f . In detail, f must satisfy (5.1) f φ ( a ) − φ ′ ( a ) f ∈ K ( E , F ) and f p = p ′ f = f . So, by definition Rep( A, B ) = ^ Rep ( A, B ) . The structure of a C ∗ -categor y on Rep( A, B ) comes fro m the corresp onding struc- ture on Rep ( A, B ). FINITE AND TORSION K K -THEORIES 15 Pr o of. ( of the the or em 5.5) The following isomorphisms θ a n : K a n (Rep( A ; B )) ≃ K K G n − 1 ( A ; B ) , and θ t n : K t n (Rep( A ; B )) ≃ K K G n − 1 ( A ; B ) , was pro ved in [8]. Accor ding to the definition of the finite K K G -groups a nd these isomorphisms, in par ticular, we have the follo wing result for finite K K G -theory: Let A and B be , resp ectively , separ able and σ -unital G − C ∗ -algebra s. Then (5.2) K K G n ( A, B ; Z q ) ∼ = K t n − 1 (Rep( A ; B ⊗ C q )) ∼ = K a n − 1 (Rep( A ; B ⊗ C q )) . Therefore it is enough to show that K t n +1 (Rep( A, B ); Z /q ) ∼ = K t n − 1 (Rep( A ; B ⊗ C q )) . Note that K t n − 1 (Rep( A, B ⊗ C q ) ∼ = lim − → a ∈ Rep( A,B ⊗ C q ) K t n − 1 ( L ( a )) and K t n +1 (Rep( A, B ; Z q )) = lim − → b ∈ ob Rep( A,B ) K t n − 1 ( L ( b ) ⊗ C q ) . So it is enough to compare the right-hand sides. Consider Rep( A, B ) ⊗ C q as the C ∗ -tensor pr o duct of C ∗ -categor oids in the se ns e of [8] (or as non- unital C ∗ -categor ies in the sense of [11]). There is a natural (non-unital) functor ν : R ep ( A, B ) ⊗ C q → Rep( A, B ⊗ C q ) defined by maps: (1) b = ( ϕ, E , p ) 7→ ϕ ⊗ id C q , E ⊗ C q , p ⊗ id C q ) = a b on ob jects; (2) f 7→ f ⊗ id C q on mor phis ms. One has induced mor phism of direct sy s tems of ab elian g roups { ν a } : { K t n ( L ( a ) ⊗ C q ) } → { K t n ( L ( b ) } , where ν a : K t n ( L ( a ) ⊗ C q ) → K t n ( L ( a b ) is induced by ν . Therefore one ha s a natural homomorphism ¯ ν n : K t n +1 (Rep( A, B ); Z /q ) → K t n − 1 (Rep( A ; B ⊗ C q )) . Then comparing the tw o tw osided exact sequences · · · / / K t n +1 (Rep( A ; B ) , Z q ) / / ¯ ν n   K t n (Rep( A ; B )) × q / / =   K t n (Rep( A ; B )) / / =   · · · · · · / / K t n − 1 (Rep( A ; B ⊗ C q )) / / K t n (Rep( A ; B )) × q / / K t n (Rep( A ; B )) / / · · · one concludes that ¯ ν is an isomorphism.  Now,w e sho w the Browder-Ka roubi-Lambre’s theorem for finite K K G -theory . Theorem 5.5. L et A and B b e, r esp e ctively, sep ar able and σ -unital G − C ∗ - algebr as, r e al or c omplex; and G b e metrizable c omp act gr oup. Then, for al l n ∈ Z , (1) q · K K G n ( A, B ; Z q ) = 0 , if q − 2 is not div ide d by 4 ; (2) 2 q · K K G n ( A, B ; Z q ) = 0 , i f 4 divid es q − 2 . 16 HVEDRI INASSARIDZE AND T AMAZ KANDELAKI Pr o of. F ollows from Pr op ositions 4.1, 5.1 and 5.3, from Theorem 5 .4 and from the Browder-Karo ubi-Lambre’s theo r em for algebr aic K-theory .  References [1] A r lettaz D. Inassaridze Finite K -theory spaces M ath. Pro c. Camb. Phil. Soc. (2005 ), 139. 261-286. [2] Br owd er W. Algebraic K -theory with coefficients Z /p , in Geomet ric Applications of Homo- top y Theory I, Lect. Notes in M ath. 657 (Springer, 1978), 40-84. [3] Bl ac k adar B. K - the ory for Op er ator Algebr as , M.S.R.I. Publ. 5, Springer-V erl ag, (19 86). [4] P . Chez, R. Lima and J. Roberts, W ∗ -c ate gories , Pacific J. Math. 1 20 (1985) , No 1, 79-109. 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T op olo g ic al invariant for Gene r alize d O p e r atorial A lgebr as Dissertation, Heidel- berg (1 987). H. Ina ssaridze , T.Kandelaki:, Tbilisi Centre for Ma thema tical Sciences, A. Raz- madze Mathema tical Institute, M. Alexidze Str. 1, 380093 Tbilisi, Georgia