DSpace at VNU: Non-commutative chern characters of the c -algebras of the sphers

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VNT.J Journal o f Scicnce, M athem atics - Physics 25 (2009) 249-259 Non-commutative chern characters o f the c*-algebras o f the sphers N gu vcn Quoc Tho* Ị'yeparỊmení o f M athem aiics, Vinh Universilv, Ĩ Le Duan, ih ĩh d lv, iĩcínưnĩ ReccÌNcd Septem ber 2009 A b s tra c t We p rop ose in this paper the construcion o f non-com m utalive Chcrn characters o f i he C ' —a l g e b r a s o f s p h e r e s a n d q u a n t u m s p h e r e s T h e Hnal c o m p u t a t i o n gi ve s us c i e a r r e l a t i o n With t he o r d i n a r y Z / ( ) - g r a d c d C h c r n c h r a c t e r s o f t o r s i o n o r t h e i r n o r m a l i z c r s Kc}^v()rks: C haracters o f ĩhe c * —algebras I n tr o d u c tio n For coinpacl Lie groups the C hem character cfi : /v * ( G ) ® Q — ^ Ỉ I p Ị ị { C ; Q ) were construcled in [4|-f5] wc com puted the non-conimutativc Clicrn characters o f compact Lie goup c * —algebras and o f compact quantum ụroups, which are also honiomorphisiiis from quantum A '- g r o u p s into entire currciu periodic cyclic h o m o lo g y o f group c * - a l g e b r a s (resp., o f C * - a ! u c b r a q u a n tu m groups), (7/(- : K J ( ' * ( C ) ) — - / / A \ ( C * ( G ) ) ) , (resp., chc* : A \ ( C ; ( C ) ) — ^ / / / s \ ( c ; ( ) ) ) We obtained viiiiO ilic icspuiKliii '4 ali;cl)iaic ihfiig '■ h * { ( J * ( U ) j — Ỉ Í J\ [L' * [Li) ))^ which coincidcs with the l-cdosov-C'unt/- Ọuillcii formiila for C l ie r n characters [5] W hen A - C*{G) wc first com puted tlie / v - u r o u p s o f c * (G ) and the / / L \{ C ^ {G) )' T hereafter we com puted the Clicrn charactcr r ile - : h \ { C U ( > ) - Ỉ Ỉ E ^ { C * { G ) ) ) as an isomorphism modulo torsions Usinir tiie results from [4|-[5], in this paper vvc com pute llic non-com m ulative Chern characters ^ : A ^ ( /l) the — > / / / : > ( / ! ) , for two eases = c * ( “ )), tlic C * - a l u c b r a o f spheres and “ - a l g e b r a s o f quantum spheres character CỈI : K * { S " ) For com pact t»roups G (){ii f 1), the C hem Q — » I ỉ pỊ ị { S^ ^ \ Q) o f llic sphere ” - { n f i ) / { i ) is an isomorphism (sc f 15]) In the paper, w e describe two Chern character hoiiiomorphisms and chc- : A',(C';(5")) — Ỉ^ÍÌKÚI: ihonguNcnquoc Í/ gm ail.co m 219 //E (c;(5")) N Q Tho / V N V Journal o fS cicn c e, Síaihcm ưíics - f^hvsics 25 (2009) 249 -2 250 Finally, \vc show that there is a coinmiitvative diagram A'.(C(AVJ) //E.(C(AVJ) (Similarly, for -4 = c * ( ” ), we have an analogous com m utalivc diagram with i r \v X X s ’ o f place o f " ) , from which we deduce that c lic - is an isomorphism modulo torsions We now briefly review he structure o f the paper In scclion 1, we com pute the C h e m chracter o f the c * - a l g e b r a s o f spheres 'I'lie computation o f C h em character o f is based on two crucial points: i) Because the sphere S " - ( ) { n f \ ) / { i ) is a homogeneous space and C " - a l g e b r a o f 9^ is (he transformation group c * - a l g e b r a , follwing J.Parker [10], we have: ii) U sing the stability property theorem and I I in [5], we rcduce it to the com putation o f c * - a lg e b r a s o f subgroup 0{ n) in (n I- 1) group In section 2, we com pute the Chern characlcr o f c * - a l g e b r a s o f quantum spheres For quantum sphere ^ , w e define the com pact quantum c * - a l g e b r a s where t is a positive real n u m b e r Thcreafier, we prove thát: where is the elemenlar>' algebra o f compact operators in a separable infinite dim ensional Hilbert space i and w is ihe Weyl o f a maximal tom s T „ in S O { n ) , Similar to Section 1, vve first com pute the h \ { C * { S ' ^ ) and Ị Ị E ^ { C * { S ' ' ) , and we prove that die- : /ú (c;(5 ”)) — Iỉh\{c:isn) is a isomorphism modulo torsion N otes on N o ta tio n : For any compact spacc -V, we write /v*(A ’') for the Z / ( ) - graded topo­ logical A ^-theo ry o f X We use S w a n ’s theorem to identify /v*(A ') with Z / { ) - graded /v*(C (-Y )) For any involution Banach algebra A , K ^ { A ) , H E ^ { A ) and I Ỉ P ^ { A ) are Z / ( ) - graded algebraic or topological A '- g r o u p s o f A , cnire cyclic lioinology, and periodic cyclic homology o f A , respectively If T is a maximal torus o f a compact group G , wilh the corresponding Wcyl group w , write C ( T ) for the algebra o f com plex valued functions on T We use the standard notation from the root theory such as p , for the positive highest weights, etc, We denote by A / t the norm alizer o f T in Ơ , by N the set o f natural num bers, R the fied o f real numbers and the standard c the field o f com plex num bers, i ^ ( N ) space o f square integrable sequences o f elem ents from A , and finally by C* { G ) we denote the com pact quantum algebras, C* { G ) the c * - a l g e b r a o f G N O TĨU) / VNU J o u n u d o f S d c n c e , M a them atics - Physics 25 (2009) 249-259 N o n -com m u lativ e Chern characters of c * 251 algebras o f spheres In this section, we com pute non-commutativc C h c m characters o f c * - a l g e b r a s o f spheres Let A be an involution [^anach alucbra We construct the non-com m utative Chern characters chc* ■ K ^ ( A ) — ►I I E J A ) s and show in [4] that ỉbr c * - a l g e b r a C * { G ) o f com pact Lie groups G , the Ciiern charactcr cìì('* is aii isomorphism P ro p o sitio n 2.1 ( |5 |, Tlìcorciìì 2.6) Lei / / he a sep a b le liilh e r t sp a ce a n d B cm arb itra ry Banach space where Wehave K (tcựỉ)) ^ 7v\(C); /V \(Ỡ /C (//)) ^ K Ụ Ì) //£ \(A C (//)) ^ //£ (€ ); ỈIE.{B®KựI)) ^ HE { D) , K ( H) is the clem en ííỉỉy algebra o f com pact o p era to rs in a sep a b le ifijim te d im e m io n a l ỉỉỉlb e rt sp a ce II P ro p o sitio n 2.2 ([5], Theorem 3.1) Lei be cm involution B anach algebra with unity There is a ('hern ch a cter hom om orphism chc- ; h \ { A ) — ^ IIE { A) P ro p o sitio n 2.3 (Ị5), rheoreni 3.2) Let G be an com pact Stroup a m i T a fix e d ìììaxim al torus o f (Ĩ \viih Weyl i r A ' t / T Then the Chern character rh(^^ : / \ * ( C * ( G ) ) ^ isoìnorpỉìism m odulo lorsiofis i.e rhr- : h \ { C * { G ) ) C / / E ( C * ( G ) ) is cm ///i.(C * (G )), vvhiJi CUN be iiicntijicd wiih ific cIu.sMcai Chern charactcr c h r- : A ' ( C ( A t ) ) — / / / • : ( C ( A ^ t) ) , ỊỈìiỉỉ is also an ìsoỉììorphic m odulo torsion, i.e rh : A \(A /'t)Cs^C Now, for ” - (){ìL I l ) / ( / i ) , where { i i ) , { n f 1) are the orthogonal matrix groups We detDlc by T „ a fixed maximal torus o f ( ; ỉ ) and A^Xn the normalizer o f T „ in ( n ) Following [Proposition 1.2, there a natural C h em character chc* : / Ú ( C ( ' ' ) ) — ^ I Ỉ E ^ { C { S ^ ) ) , Now, we pute first k \ { C ( S ' ' ) ) and then H E ^ { C { S ' ^ ) ) o f C * - a l g c b r a o f the sphere i*ruposition 2.4 H E , ( C { S ’')) H]],ẶTn)) Pro ự We have ///:.( C '( " ) ) - / / £ \ ( C ( ( n - f ) / ( h ))) //E (C * (ơ (n )) ® Kl{L^{0{n + l)/ơ (ri)))) 252 A ' Thu / I 'M J Jou rna l o / S c i e m v Mưihcniưiics - P hysics 25 (2009) 249-259 (in virtue of, llie K { ỉ ? { { ì ì \ ) / ( r ỉ ) ) ) ) is a c * - a lg c l.r a com pact operators in a separable 1lilhcrt spacc I ? { ( ) { n + l ) / ( n ) ) ) ỈIE,{C{(){n))) (by I>n)Ị)().sition 1.1) IỈE.{C{Mr'^) - (s e e Ịõ Ị) Thus, we have H E , { C ^ { S ’‘)) ~ / / £ ' * ( C ( V t J ) A p a r t f r om t t, b e c a u s e C ( A ' t ) is tlicn c o m i m i t a t i v c c * —a l i ỉ c b r a b y a C i i n t z - Q u i l l e n ’s r e s u l t [I], \vc have an isomotpliisni //n (C ((A 'T „ ))S ^ //ô » (A /'T j) Moreover, by a result o f Klialkhali [8],[9], we have ///> (C ((.V t „ )) = / / / % ( C ( ( A ' t J ) W'c liave, hence ỈỈE,{C"{Sn) ~ // £ '* ( C ( A /'t J ) = / / P ( C ( ( A ' t J ) R e m a r k Bccause ỈỊ]jỊị{AÍT ) is the dc Rham coliomology o f T n , invariant under the action o f the Weyl group \ \ \ following Watanabe [15], we have a canonical isom orphism ỉ ỉ p ] ị { T n ) ~ I Ỉ * { S O n ) l — A (X3 ,X , ,a- z+3 ), where X2 x+ = cr*(Pi) € / / ^ " ^ ^ ( ( n ) ) and rr* i r { B S O { n ) , R) — ► H * { S O { n ) , R ) for a com m utative ring /{ with a unit G 1Ì, and Pi — t ị , l ị ) e / / + (/iT n Z ) the P onlrjagin classes Thus, we have Proposition 2.5 / v ( C ( " ) ) - / v *(7í t J ) Proof \Vc have / Ú ( C ( ” )) h\{C {(){nil)/0{u))) - h\{C*{0{n))® iC{L\0{n i l)/0{n)))) ^ A ', ( C * ( ( tí))) ~ A \(A /t ) (see [10]) (by I ’r o p o s iti o n 1.1) (l->y L o i i i i n a , f r o m [5]) Thus, A \ ( C ( 5^0 ) = / w ( A t J R e m a r k Following Lem m a 4.2 from [5], w e have /v (A /tJ where (3 : R { S O { n ) ) — > /v = ỈC{SO{n \ l))/Tor = A ( / ( A i ) , / J ( A „ _ , £„4-1), ( ( n ) ) be the homomorphism o f Abelian groups assigning to each rep­ resentation p : S O { n ) — » U { n - \ - 1) the homotopic class ị3{p) = [inp] G [ ( n ) , t / | = K ~ ^ { S O { n ) ) , where i „ ; [/(71 + 1) — * Ư is the canonical one, U { n + 1) and Ư b y the n - groups respectively and € A '“ ^ (5 Ơ (7 + 1)) We have, finally A'*(C*(5")) s A (/3(Ai), ,/y(A,._3,en+i) t h and in fin ite unitary N ọ Tho / I'N U J o u r n a l of Scwnce, M ath em a tics - Physics 25 (2009) 249-259 253 M(ircovcr, the Chern charactcr OỈ S U ( n i 1) was compiilcd in [14], for all ì i ỳ \ Let us rccall the rc.suli Define a function ộ : N x N x n — ^z, uivcii bv k ^ ^ ,1 J'heoreiii 2.6 Ỉ.CÍ T n he a fix e d maximal iorus o f { n ) a m i T the fix e d m axim al torus of S ( ) [ n ) , with Wcyl ị:,n)ups i r A t / T , th e C hern characier o f C"‘[S'^) rhc^ : h \ { c * { s n ) //£' *(C *(5”)) I.s Ufi ỉsíììtiorphism, ịỉìven by rhc-ifW ) - ^ ( ( - l ) ' - ' / ( i - l)!ự.(2,i f , ẳ ; , x h i rhc.{€„.,^) - ^ ” ^ ( ( - l ) ' “ ' / ( i - l ) ! ) ( ( ^ ^ ( u + ,A :,2 x ,,h 1=^1 ( k = 1, n-1); 1=1 Proof By IVoposilion 1.5, we have A-.(C*(5")) ^ K , ( C { M r J ) = and //E ( C * ( '') ) ^ / / E ( C ( ^ / J ) ^ H h n W r J (l>y P r o p o s i tio n l.-l) Now consider the com m u tative diagram K.{C"{S")) Ị ỉ i : , { C ' { S ’')) h -{M r„ ) ỉỉhui^-in))- Moreover, by the results o f W atanabe [15], the Chcrn cliaractcr c/i : A '* ( A t „ ) © C — ' ỉỉ])ii{ -^ T „ )) is an isom orphism Thus, c/ic* : A ', ( C * ( " ) ) — > Ỉ I E { C ' { S " ) ) is an isomorphic (Proposition 1.4 and 1.5), given by chc-{P{\k)) = f ^ ( ( - l ) - ' 2/ ( 2i - I ) ! ) ộ ( 2a + 1, ^ - , 20X 2,+1 ( k = l , 1-1 1:^1 i= l n - 1); N.Q Tho / VNU Jo u rn a l o f Science, M athem atics - /Vỉv.v/c‘.v 25 (2009} 249-259 254 where H E , { C ^ S ' ‘)) 5^ A (j'3,X 7, ,X 2,4 3) N on-com m utative C hern charactcrs o f c * - a l g e b r a s o f quantum spheres In this section, w e at first recall definition and main properties o f com pact quantum spheres and their representations M ore precisely, for S'\ we define c* ( " ) , the spheres as the C * - c o m p le ti o n o f the ’ - a l g e b r a C” - a lg e b r a s o f compact quantum with respcct to the ” - n o r m , where is the quantized H o p f subalgebra o f the Mopf algebra, dual to the quantized universal enveloping algebra Ư{Ợ), generated by matrix elements o f the U{C/) modules o f type l ( s e e [3]) We prove that where fC{Hyj t) is the elementary algebra o f compact operators in a separable infinite dimensional Hilbert space and Ỉ Ỉ ' is the Weyl group o f " with respect to a maxim al loriis T A fter that,we first com pute the / ^ - g r o u p s A ',(C * (S '* )) and the I I E ! , { C * { S ' ' ) ) , respectively Thereafter we define the Chern character o f c * - a l g e b r a s q'lanlum spheres, as a homom orphism from A ' , ( Q ( " ) ) to H E \ { C * { S ' ‘)), and wc prove that d i e - ■ A ' ( C ; ( '" ) ) isomorphism modulo lorsioii / / / Ỉ , ( Q ( S ’'‘)) is an Let G be a com plex algebraic group with Lie algebra Q = L icG ' and £ is real number £• :/ - D efin ition 3.1 ([3], Definition 13.1) The qu a n tized fu n ctio n alg eb T e { G ) is the subal'^ehra o f the H o p f algebra d u a l to Ue{G), g en era ted by the m a trix elem ents o f the fin itc -d im o isio n u l U r(G )—m odules o f type For compact quantum groups the unitar>' representations o f T ^ { G ) arc param etcri/cd b \ pairs (w, t), where t is an elem ent o f a fixed maximal torus o f the compact real (brin o f c and u.' is a element o f the Weyl group w o f T in G Let A G be the irreducible i/ £ ( i / ) - m o d u l e o f type w ith the lii«hcst wciuht A Then K ( A ) admits a positive definite hcrmitian form' such that X V \ , V ) = V\ , V n £^(N) denoted bv TTs , where Si appears ir the reduced decom position UJ = ,s,,, s, , M o r e precisely, TTa : T e { G ) — » £(ế'^(N )) is o f class CCR(see [11]),i.e its image is dense in the ideal con.pact operators £ ( i ^ ( N ) ) t N.Q Tho / V N U Jo u r n a l of Science M a th em atics - Physics 25 (2009) 249-259 255 Then representation Tị is onc-dimcnsional and is o f the form Tt{Ci^s4L.r) ổr„,() ị ^ r íí n, Oĩì íh c l ĩ i l h c r í s p a c e íc ìĩs o r p r o d u c t / -^('^1) ■ / c give by P r Á - l ) { ( ' k , ® : í I V, Ạ r (1 - ‘ ẽ:‘ ♦‘ +' > ) s = a H + ® f'A-,,2 ® if s < r ® ® PAv i f s- - r if ,s > r The representation fh)Ị is eqiiivalenỉ /() the restriction o f th e representation Ti o f !F ^(S Ln-\\\ C )) (cf.2.3); a n d or ì' > (ì, f)j-1 is equivaìent to ihe restriction o f TTs^ y %- From Theorem 2.6, we have kcr/)^,/ -= {()}, (*',i)eiV'x7- i.e the representation ©;^eu I'f is faithful and f if u; (’ (liin () t ' ' \ We recall now ihc definition o f compact quanlum o f spheres D efinition 3.3 *—algebra c* — algebra The c * —algebraic com pact quantum sp h ere c * ( “ ) is he c * —com plciion oj the with respect to the C * ~ n o rm ll/ll-sup i|M/)ll, p where p runs through the*— representations o f (cf Theorem 6) a n d ihe norm on the riglii- hcmd sid e is (he operator It sufTcies to show that ll/ll is finite for all / € for it is d e a r that ||.|| is a C * - ii o r n i, i-C- ll/-/*ll — \ \ f\\ ^‘ We now prove that ibllowing result abo ut he structure o f compact quantum C * - a l g c b r a o f sphere S ' \ I 'h e o r c m 3,5 ỉilíh Pìion as above, vt't' have c ; ( " ) ^ c ( ') 0 r where C ( * ) is (he alịiehra o f com plex valued co n tin u o u s fim c tio tis on a n d K \ ỉ l ) ideal o f com paci operators in a separable m ih e r t space // Proof Let UJ — be a rcduccd decom position o f the elem ent Ú,’ G i r into a product o f reflections Then by Proposition 2.6, for r > o, the representation t is equivalent to the restriction Í 7T.,^ ® ®7T5^ ® T t, w here 1ĨSX is the composition o f the hom om orph ism o f T'e(G) onto / ’ê ( /.2 ( C ) ) and the representation 7T_1 o f J^ị:{SL ‘2{C)) in the iỉilb c rt sp ace and the family o f onc- dimensional representations T(, given by 7Ị(a) - T; (0) - 7Ị(c) - 0, ĩ(ư ) - where t G 5^ and a j j , c , d are give by: A lgebraJT £(5L 2(C )) is generated by the matrix elements o f type [ ỵc ^ uy Hence, by construction the representation T , : C ; ( S ' ‘) ►c ; ( S L , ( ) Í = TTg ’ *Ỉ ®7ĨS £ (ế'^(N )S ^) ® 7i- T hus, vvc have *k N ọ Tho / V N U Jo u r n a l o fS c ic n c c M ath em atics - Physics 25 (2009) 249-259 257 Now, TTs is CCR (see, 111Ị) and so, w e have 7rs,(C'*(5") ~ /C(//^,() M oreover Tt{C*{S'^)) —c I Icncc /-,(C (^ " )) = (7T,._ ' v 7T,.^0 ; ) ( Q ( S " ) = rr,., (c;(S")) ® s 7t,./(c;(S'‘)) T((c;(5 '*)) I C{ I L, t ) , - where n ^ j = //ô, â H s, ® c p ^ A C : { S ’')) = K ụ U t ) T hus, 1Icncc, r(D /*0 Now, recall a result o f s Sakai from [11]: Let -4 be a com m utative c * - a l g e b r a and B be a Then, c* -a lg e b B ) ~ ,4 Cv /i, where ÍÌ is the spectrum spacc o f A A pplying this result, lor B — AC(7 /^ / ) ^ /C and A — C ( U ' x ^ ) be a com mutative c * - a l g e b r a * I'hus we liave c Now, vve first com pute the A \ ( C * ( ” )) and llic I I E ^ { C * { S ' ' ) ) o f c * - a l g e b r a o f quantum sphere 5'^ P ro p o sitio n 3.6 / / A \ ( c ; ( S ' - ) ) ^ / / ; „ , ( U ' x ố ’') Proof Wc have Ỉ Ỉ E A C : ( S ’'}) r = // E ,( C ( ') © = / / £ ( C ( S ’' ) © / / E ( — H E { C { \ V X ‘) / C ^ / / / T ( C ( i r X 5*)) ! C{n^^t )dt ) r IC{ỈUt)dt)) (b y P r o p o s i tio n 1.1) Since C (U '' X s ' ) is a com m utative ’ - a l g e b r a , by F’roposition 1.5 §1, we have / / E ( C ; ( S " ) ) ^ H E , { C { \ V X ‘ )) ^ H h n i ^ V X 5*))' Proposition 3.7 K,{C*{S’')) ^ IC(\V X 5^) N.Q TỊìo / VN U Jo u rn a l o f Science M ưíhcmưỉics - Physics 25 (2009) 249-259 258 Proof Wc have A\(c;(S'‘)) = A\(C(S‘)Q riC{U^^t)df) U' ' ^ ) dt ) ) ^ K,{C{\V x S ^ ) @I C ^ A '.( C ( ir X S ') ) (1)V P r o p o s i t i o n 1.1) In result o f Proposition 1.5, §1, we have h \iC { \V X ') ) K { \ v X ‘ ) Theorem 3.8, IFiih rwiaiion above, the Chern chanicỉer o f C * —algebra o f quantum sphere chc- : A \ ( C (S") — ỉỉi:.{C c (S") ( ”)) is an isom orphism Proof By Proposition 2.9 and 2.10, we have H E , { C ; { S ' ^ ) ) ^ H E , { C { \ V X ‘ )) ^ i / ; j / i ( i r X ' ) ) , / : ( c ; ( " ) ) ^ A '.( C ( ir X ‘ )) ^ I C { \ V X * )) Now, consider the com mutative diagram /c * (c ;(5 " )) ỈỈE.{C:{S^^)) K J C ( \ V y s^)) Í Ỉ E J C ( \ V X S' )) A '*(U ’ X ‘) ^ M o re o v e r, follvving W a ta n a b c [ ] , the e ll : A ' * ( i r X ‘ ) :-; C — » X * ) is an is o m o r p h is m Thus, chc* : h \ { C *£ (5'*) — » H E ^ { C (5'” )) 'S an isomorphism A c k n o w le d g m e n t T he author would like to thank Professor Do Ngoc D iep for his guidance anti encouragement during this paper References |1] |2J [3] [4| |5 | [6| J Cunlz, Hntice cvclic cohoniolog\ of Banach algebra and characl^T of “ Suiĩimablo i-rcdhoni tìKxlulcs K-'rhcai'y., (1998) 519 J Cunl/, I) Quillen, The X coinplcx Í IỈ1C unuvcrsai cxtci.iions ỉ^irpm nt i\ỉatfi Inst Vni U tidtlbtg, (1993) V Chari, A PrcsslcN A guide to quantum (j7X}ups, Cai.ibriJgc Uni Press, (1995) D.N Diep, A.o Kuku N.Q Tho, Non-coininulativc Chcm ciiaractcr Í compact Lie group c*-algeb ras K- 'rỉieory 17(2) (1999) 195 O.N Dicp, A.o Kuku N.Q Tho, Non-comniutative Chcm characlcr of compact quantum group, K- Thcor'y, 17(2) (1999), 178 D.N Dicp, N.v Thu lloniotopy ivariancc of entire cumt cyclic homology, Vielĩiarn J o f Math, 25(3) (1997) 21! N Ọ Tho / V N U J o u rn a l o f Science, M aihcm atics - Physics 25 (2009) 49-259 |7| |8| 110| |H{ [I2| |13| 114| I li>| 259 Ị) N, Dicp, N.v Thu I jitirc homology of non-commulativc dc.Rhani curcnts, ỈC T P , ỊC /9 /2 I4 , (1996), 23pp; to appirar in ỉ^iiblicaíioĩi o f C FC A, H anoi C ity V ietnam N ational Iniversily {\991) M Klialkhali On the inlive c>c!ic coh(>!iiolog\ of Hanach algebras: Moita invariancc M athem atisches Inst Uni Ilv.idclbcry., 54 (1992) 24 M Khalkhali On ihe intivc c>c!ic cohomology of iianach algebras: II Honiotopy invariancc, M a t h e T n a t i s c h e s Inst I 11 Ucidelbf rg 54 (1992) 18 J i*:.ctkcr Translormation group C’*~a!gcbra: A scicctivc sur\c\ C ontem porary M athem atics., Volume 167 (1994) S akai, ( " —algebras and w * —algebras, Sprincr-Vcrlag Berlin Heidelberg New York, (1971) N A' Thu, Morila invarianc’c of entire curronl cyclic homology, V ietnam J M ath., 26:2 (1998) 177 NA' ĩhu, líxaciness ol cnũrc current cyclic honiloiỉỴ A cta M ath Vic-lnamica (to appear) T Waianabc C'hem charactcr iin compact l.ic groups of low rank Osuka J M ath., (1985) 463 'I' Waianahc, On the Cilcm characicr of s\mmctric space related o S U (n ), J Math K yoto Uni, 34 (1994) 149  ... this section, we com pute non-commutativc C h c m characters o f c * - a l g e b r a s o f spheres Let A be an involution [^anach alucbra We construct the non-com m utative Chern characters chc*... curronl cyclic homology, V ietnam J M ath., 26:2 (1998) 177 NA' ĩhu, líxaciness ol cnũrc current cyclic honiloiỉỴ A cta M ath Vic-lnamica (to appear) T Waianabc C' hem charactcr iin compact l.ic groups... a l of Scwnce, M ath em a tics - Physics 25 (2009) 249-259 253 M(ircovcr, the Chern charactcr OỈ S U ( n i 1) was compiilcd in [14], for all ì i ỳ Let us rccall the rc.suli Define a function