Non-commutative chern characters o f the c*-algebras o f the sphers
N g u v c n Q u o c Tho*
Ị'yeparỊm ení o f M athem aiics, Vinh Universilv, Ĩ 2 Le D uan, ih ĩh d lv , iĩcínưnĩ ReccÌNcd 8 Septem ber 2009
VNT.J Journal o f Scicnce, M ath em a tics - Physics 25 (2009) 24 9 -2 5 9
A b s t r a c t . We p ro p o se in this paper the construcion o f n o n-c om m utalive Chcrn characters
o f i h e 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 Hn a l c o m p u t a t i o n g i v e s u s 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 / ( 2 ) - 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 h ara cters o f ĩhe c * —algebras.
1. I n t r o d u c t i o n
For coinpacl Lie g roups the C h em character cfi : /v * ( G ) ® Q — ^ Ỉ I p Ị ị { C ; Q ) were construcled.
in [4 |-f5 ] wc com puted the non-conim utativc Clicrn characters o f com pact Lie goup c * —algebras and o f com pact quantum ụroups, which are also honiomorphisiiis from q u antum A '- g r o u p s into entire
c u r r c i u p e r i o d i c c y c l i c h o m o l o g y o f g r o u p
c
* - a l g e b r a s ( r e s p . , o f C * - a ! u c b r a q u a n t u m g r o u p s ) ,(7/(- : K J ( ' * ( C ) ) — - / / A \ ( C * ( G ) ) ) , (resp., chc* : A \ ( C ; ( C ) ) — ^ / / / s \ ( c ; ( ơ ) ) ) . We obtained viiiiO ilic icspuiKliii'4 ali;cl)iaic ihfiig '■ h * { ( J * ( U ) j — Ỉ Í J\ [L' * [Li) ))^ which coincidcs
w i t h t h e l - c d o s o v - C ' u n t / - Ọ u i l l c i i f o r m i i l a f o r C l i e r n c h a r a c t e r s [ 5 ] . W h e n A - C * { G ) w c f i r s t
com puted tlie / v - u r o u p s o f c * (G ) and the / /L \{ C ^ { G) ) ' T h ereafter w e com puted the Clicrn charactcr r ile - : h \ { C U ( > ) - Ỉ Ỉ E ^ { C * { G ) ) ) as an isomorphism m odulo torsions.
Usinir tiie results from [4 |-[5 ], in this paper vvc com pute llic non-com m ulative Chern characters
^ : A ^ ( / l ) — > / / / : > ( / ! ) , for two eases .4 = c * ( 5 “ )), tlic C * - a l u c b r a o f spheres and .4 “ the - a l g e b r a s o f quantum spheres. For com pact t»roups G -- (){ii f 1), the C h em character CỈI : K * { S " ) Q — » I ỉ p Ị ị { S ^ ^ \ Q) o f llic sphere 5 ” - 0 { n f i ) / 0 { 7 i ) is an isomorphism (sc. f 15]). In the paper, w e describe two Chern character hoiiiomorphism s
a n d
chc- : A',(C';(5")) — / / E . ( c ; ( 5 " ) ) .
Ỉ^ÍÌKÚI: ihonguNcnquoc Í/ g m a il.co m
219
250 N Q. Tho / V N V J o u rn a l o fS c ic n c e , S íaihcm ư íic s - f^hvsics 25 (2009) 2 4 9 -2 5 9
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 X s ’ o f p la ce o f
\ v X 5 " ) , from w hich w e d educe that c lic - is an isom orphism m odulo torsions.
We now briefly review he structure o f the paper. In scclion 1, we com pute the C h e m ch ra c te r o f the c * - a l g e b r a s o f spheres. 'I'lie com putation o f C h e m character o f is based on two crucial points:
i) Because the sphere S " - ( ) { n f \ ) / 0 { 7 i ) is a hom ogeneous 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], w e have:
ii) U sing the stability property theorem and I I in [5], w e rcduce it to the com putation o f
c *
- a l g e b r a s o f subgroup0 { n )
in 0(n
I- 1) group.In section 2, w e com pute the Chern characlcr o f c * - a l g e b r a s o f q u antum spheres. For q u an tu m sphere 5 ^ , w e define the com pact quantum c * - a l g e b r a s w here t is a positive real n u m b e r Thcreafier, w e prove thát:
w here is the elemenlar>' algebra o f com pact 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 ) ,Sim ilar to Section 1, vve first com pute the h \ { C * { S ' ^ ) and Ị Ị E ^ { C * { S ' ' ) , and w e prove that
die- : /ú (c ;(5 ”)) — Iỉh\{c:isn)
is a isomorphism m odulo torsion.
N o tes o n N o ta t io n : For any com pact spacc -V, w e w rite /v*(A ’') for the Z / ( 2 ) - graded to p o logical A ^ -th e o ry o f X . We use S w a n ’s theorem to identify /v * (A ') w ith Z / { 2 ) - graded /v * (C (-Y )).
For any involution Banach algebra A , K ^ { A ) , H E ^ { A ) and I Ỉ P ^ { A ) are Z / ( 2 ) - 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.
I f T is a maximal torus o f a com pact 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 num bers and
c
the field o f com plex num bers, i ^ ( N ) the standard space o f square integrable sequences o f elem ents from A , and finally by C * { G ) w e denote the com pact q u an tu m algebras, C * { G ) the c * - a l g e b r a o f G.2. N o n -c o m m u la tiv e Chern characters o f c * algebras o f spheres.
In this section, we com pute non-comm utativc 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 r o p o s itio n 2.1. ( |5 |, Tlìcorciìì 2.6) Lei / / he a sep a ra b le liilh e r t sp a ce a n d B cm a rb itra ry B anach space. We have
K . ( t c ự ỉ ) ) ^ 7 v \ ( C ) ;
/ V \ ( Ỡ 0 / C ( / / ) ) ^ K . Ụ Ì )
/ / £ \ ( A C ( / / ) ) ^ / / £ . ( € ) ;
Ỉ I E . { B ® K ự I ) ) ^ HE. { D) ,
where K . ( H ) is the c lem e n ííỉỉy algebra o f com pact o p era to rs in a se p a ra b le ifijim te d im e m io n a l ỉỉỉlb e rt sp a c e II.
P r o p o s itio n 2.2. ([5], Theorem 3.1) L ei .4 be cm in vo lu tio n B anach algebra with unity. There is a ('hern c h a ra c te r h om o m o rp h ism
chc- ; h \ { A ) — ^ I I E. { A) .
P r o p o s itio n 2.3. (Ị5), rheoreni 3.2) L et G be an co m 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 ch a ra cter rh(^^ : / \ * ( C * ( G ) ) ^ / / E . ( C * ( G ) ) . is cm isoìnorpỉìism m odulo lorsiofis i.e.
r h r - : h \ { C * { G ) ) 0 C / / / i . ( C * ( G ) ) , vvhiJi CUN b e iiicntijicd wiih ific cIu.sMcai C hern charactcr
c h r - : A ' . ( C ( A t ) ) — / / / • : . ( C ( A ^ t ) ) , ỊỈìiỉỉ is a lso an ìsoỉììorphic m odulo torsion, i.e
rh : A \(A /'t)C s^C
Now, for 5 ” - (){ì L I l ) / 0 ( / i ) , where 0 { i i ) , 0 { n f 1) are the orthogonal matrix groups. We detDlc by T „ a fixed maximal torus o f 0 ( ; ỉ ) and A^Xn the norm alizer o f T „ in 0 ( n ) . Following [Proposition 1.2, there a natural C h e m character chc* : / Ú ( C ( 5 ' ' ) ) — ^ I Ỉ E ^ { C { S ^ ) ) , Now, we con 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 ] ] , Ặ T n ) ) . Pro ự We have
/ / / : . ( C ' ( 5 " ) ) - / / £ \ ( C ( 0 ( n - f 1 ) / 0 (h)))
/ / E . ( C * ( ơ ( n ) ) ® K l { L ^ { 0 { n + l ) / ơ ( r i ) ) ) )
N O. TĨU) / VNU J o u n u d o f S d c n c e , M a th e m a tic s - Physics 25 (2009) 24 9 -2 5 9 251
(in virtue of, llie K { ỉ ? { 0 { ì ì \ 1 ) / 0 ( r ỉ ) ) ) ) is a c * - a l g c l . r a co m p a c t operators in a separable 1 lilhcrt spacc I ? { ( ) { n + l ) / 0 ( n ) ) )
Ỉ I E , { C { ( ) { n ) ) ) (by I>n)Ị)().sition 1.1) - I Ỉ E . { C { M r ' ^ ) (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 ha 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 ( ( .Vt„ ) ) = / / / % ( C ( ( A 'tJ ) . W'c liave, hence
Ỉ Ỉ E , { C " { S n ) ~ / / £ ' * ( C ( A / ' t J ) = / / P . ( C ( ( A ' t J )
R e m a r k 1. Bccause ỈỊ]jỊị{AÍT ) is the dc Rham coliomology o f T n , invariant under the action o f the Weyl group \ \ \ following W atanabe [15], we have a canonical is o m o rp h ism ỉ ỉ p ] ị { T n ) ~ I Ỉ * { S O n ) l — A (X3,X7, ...,a-2z+3), w here X2x+ 3 = cr*(Pi) € / / ^ " ^ ^ ( 5 0 ( n ) ) and rr* i r { B S O { n ) , R ) — ► H * { S O { n ) , R ) for a com m utative ring /{ with a unit 1 G 1Ì, and Pi — t ị , l ị ) e / / + ( / i T n Z ) the P onlrjagin classes.
Thus, w e have
Proposition 2.5.
/v. ( C ( 5 " ) ) - /v*(7í tJ ) - Proof. \Vc have
/ Ú ( C ( 5 ” )) - h \ { C { ( ) { n i l ) / 0 { u ) ) )
h \ { C * { 0 { n ) ) ® i C { L \ 0 { n i l ) / 0 { n ) ) ) ) (see [10])
^ A ' , ( C * ( 0 (tí))) (by I ’r o p o s i t i o n 1.1)
~ A \ ( A /t ) (l->y L o i i i i n a 3 . 3 , f r o m [ 5 ] ) .
Thus, A \ ( C ( 5^ 0 ) = / w ( A t J .
R e m a r k 2. F ollow ing L em m a 4.2 from [5], w e have
/ v . ( A / t J = Ỉ C { S O { n \ l ) ) / T o r
= A ( / 3 ( A i ) , / J ( A „ _ 3 , £„4-1),
where (3 : R { S O { n ) ) — > /v ( 5 0 ( 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 [ 5 0 ( n ) , t / | = K ~ ^ { S O { n ) ) ,
w here i „ ; [/(71 + 1) — * Ư is the canonical one, U { n + 1) and Ư b y the n - t h and in fin ite unitary
groups respectively and € A '“ ^ (5 Ơ ( 7 1 + 1)). We have, finally A'*(C*(5")) s A (/3(Ai), ....,/y(A, ._3,en+i).
252 A '. ạ Thu / I 'M J J o u rn a l o / S c i e m v . Mưihcniưiics - P h y sic s 25 (2009) 249-259
M(ircovcr, the C hern chara ctcr OỈ S U ( n i 1) was compiilcd in [14], for all ì i ỳ \. Let us rccall the rc.suli. D efine a function
ộ : N x N x n — ^ z,
uivcii bv
k ^ ^
,1
N . ọ Tho / I'N U J o u r n a l of Scwnce, M a th e m a tic s - P hysics 25 (2009) 249-259 253
J'h eo reiii 2.6. Ỉ.CÍ T n h e a fix e d m axim al iorus o f 0 { n ) a m i T the fix e d m a xim a l 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
r h c - i f W ) - ^ ( ( - l ) ' - ' 2 / ( 2 i - l)!ự.(2,i f 1 , ẳ ; , 2 0 x 2 h i ( k = 1, n-1);
^ 1 ”
r h c . { € „ . , ^ ) - ^ ( ( - l ) ' “ ' 2 / ( 2 i - l ) ! ) ( ( ^ ^ 0 ( 2 u + 1 , A : , 2 0 x , , h .
1=^1 1=1
Proof. By IVoposilion 1.5, w e have
A-.(C*(5")) ^ K , ( C { M r J ) =
and
/ / E . ( C * ( 5 ' ' ) ) ^ / / E . ( C ( ^ / J ) ^ H h n W r J (l>y P r o p o s i t i o n l.-l).
Now. consider the c o m m u ta tiv e diagram
K . { C " { S " ) ) Ị ỉ i : , { C ' { S ’'))
h - { M r „ ) ỉ ỉ h u i ^ - i n ) ) -
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
T hus, c/ic* : A ' , ( C * ( 5 " ) ) — > Ỉ I E . { C ' { S " ) ) is an isom orphic (Proposition 1.4 and 1.5), given by
chc-{P{\k)) = f ^ ( ( - l ) - ' 2 / ( 2i - I ) ! ) ộ ( 2a + 1 , ^ - , 20X 2,+1 ( k = l , n - 1 );
1 - 1
1:^1 i = l
where
H E , { C ^ S ' ‘)) 5^ A (j'3 ,X 7 ,...,X 2 ,4 3 ).
3. N o n -c o m m u ta tiv e C hern charactcrs o f c * - a l g e b r a s o f q u antu m 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'\
w e definec*
( 5 " ) , theC”
- a l g e b r a s o f compact quantum spheres as the C * - c o m p l e t i o n o f the ’ - a l g e b r a with respcct to the 6 ” - n o r m , whereis 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 254 N.Q. Tho / VN U J o u r n a l o f Science, M a th em a tic s - /Vỉv.v/c‘.v 25 (2009} 24 9 -2 5 9
w here fC{Hyj t) is the elem entary algebra o f compact operators in a separable infinite dimensional Hilbert space and Ỉ Ỉ ' is the Weyl group o f 5 " with respect to a m axim al loriis T .
A fter that,w e first com pute the / ^ - g r o u p s A ',( C * (S '* )) and the I I E ! , { C * { S ' ' ) ) , respectively.
Thereafter w e define the Chern character o f c * - a l g e b r a s q 'lanlum spheres, as a hom om orphism from A ' , ( Q ( 5 " ) ) to H E \ { C * { S ' ‘)), and wc prove that d i e - ■ A ' . ( C ; ( 5 ' " ) ) / / / Ỉ , ( Q ( . S ’'‘)) is an isomorphism m odulo lorsioii.
Let G be a com plex algebraic group with Lie algebra Q = L icG ' and £ is real number. £• :/ - 1 . D efin itio n 3.1. ([3], Definition 13.1). The q u a n tize d fu n c tio n a lg e b ra T e { G ) is the subal'^ehra o f the H o p f a lg eb ra d u a l to Ue{G), g en era te d b y the m a trix elem en ts o f the fin itc -d im o is io n u l U r(G )—m od ules o f type 1.
For com pact q uantum groups the unitar>' representations o f T ^ { G ) arc param etcri/cd b \ pairs (w, t), w here t is an elem ent o f a fixed maximal torus o f the com pact 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 1 w ith the lii«hcst wciuht A.
Then K ( A ) adm its a positive definite hcrmitian form' such that X V \ , V2) = lor i'll V\ , V<2 €. Vị : { ằ) , x € U{ G) . Let be an orthogonal basis for w eight space ft e . Then is an orthogonal basis for Ve(X). Let r ( x ) = v ị ) be the associated matrix elements o f ^^(À). T hen the matrix elements c ^ , A runs tlirongli p + , w h ile (/i, r ) and (í/, >s) runs independently through the index set o f a basis o f Ve(A) form a basis o f J^Ị^{G){SQC [3]).
N o w ver>' irreducible ‘ - re p re s e n ta tio n o f T e { S L2{C.) is equivalent to a representation bclonginụ to one o f the following tw o families, each o f which is param eterized bv 5 ' {/ G c \ | / | = 1}
i) the family o f one-dim cnsional representations Ti ii) the family 7T( o f representations in £^(N )(sec |3J).
Moreover, there exists a surjective homom orphism J^e{G) — f ( S /> j( C ) ) induccd h\ llic natural inclusion S L2C G and by com posing the representation 7T_1 o f J ^ ^ { S L2'C) with this homom orphic, w e obtain a representation o f J -\{G ) >n £^(N) denoted bv TTs , w here Si appears ir. the reduced decom position UJ = ,s,,, s ,2, 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
Then representation Tị is onc-dim cnsional and is o f the form Tt{Ci^s4L.r) ổr„,<5^,,^exp(27Tv/^/i(x)), if / ( 'x ị ) ( 2 7 T \ / ^ / í ( x ) G T , for X e L i e T ( s c c [3]).
P r o p o s i tio n 3.1. ([3], 13.1.7). Every’ irreducible u n ita ry representation o f J~e{G) on a sep ara ble ỉĩilh c r t .space is the c o m p le tio u o j a unita riza b le highest w eight reprt\se?ìíaíion. Moreover, two such rep resen ta tio n are e q u iv a le n t i f a n d only i f they have the s a m e h ig h est
P r o p o s i tio n 3.2. ([3],1 3 .1 .9 ) L et UỈ == b e a red u ced d eco m p o sitio n o f an elemeniuj o f the H eyl g ro u p U ' o f G . Then
u The ỉỉilh c r t s p a c e te m o r p ro d u ct f)^ I ~ lĨỊị Tĩs s ...®7ĨS irreducible rưpre.scniaíiotì o f Tẹ{ G) w hich is a sso cia ted to the S ch u b ert c e ll s.^\
ĩi) I p to eq u iva len ce, the representation I does not d ep en d on the choice o f (he reduced d e c i . m p o s i t i o n o f u j \
Hi) Every’ irred u cib le *-r e p r e s e n ta iio n o f J^e{G) is e q u iva len t to so m e
The sp h ere ca n b e rea lized as the o rb it u nder th e a ctio n o f the co m p a ct group S U { n - \- 1) of the hiị:^hest w eight v e c to r V o in iis ĩuỉiural (?i ~f \ ) - d i m e m i o n a l representation V o f S U { n -f 1).
// /r.si 0 < 7', 5 < n , a re th e m a trix entries o f V , the a lg e b ra o f fu n c tio n s on the orbit is g en era te d by (he en tries in the "first c o lu m n ” tsQ a n d th eir com plex coiijugates. In fa c t,
C [/o o . w here " is ih e fo llo w in g eqiùvaỉetĩce relation
N.Q. Tho / V N U J o u r n a l of Science. M a th e m a tic s - P hysics 25 (2009) 2 49-259 255
n
^sO^sO = 1*
5= 0
P r o p o s itio n 3.3. ([3], 13.2.6). The * —siruciure on H o p f a lg e b ra / * t ( S L2( C ) ) , is given by
w here 7V.S is the m a trix o h ta ifw d b y removing:, the row a n d the co lu m n fr o m T .
D efin itio n 3.2. (Ị3J,13.2.7). The -s u b a lg e b r a o f ^£(5 L „4 -1 (C ) ) g e n e ra te d b y he elem ents iso ỉỳ ~ 0, is c a lle d the q u a n tize d a lg e b r j o f fu n c tio n s on (he sp h ere 5 ^ , a n d is den o ted hy It is a q u a n tu m s L r i \ - \ ( C ) —space.
We set -- /.,0 from now on. Using Proposition 2.4, it is easy to see that the following relations hold in ^ , ( 6 ’” );
Z s Z r i f r < s
if r ^ s
+ ( e - 2 + 1) = 0
H e n c e , h a s [ C P ) as its defining relations. T he construction o f irrcducible " -r e p re s e n ta tio n o f is given by.
T h e o r e m 3.4. ([3],13.2.9). Every^ irreducible *-r e p r e s e n ta tio n o f is equivalen t exactly to one o f the fo llo w ing:
i) the o n e -d im e n sio m il representation p o t t e g iv en b y P o t i ^ o ) ~ P o t ( K ) — 0 i f r > 0,
I
ii) ĩh c r c p ĩ v s c n ĩ d í ĩ ũ ì ĩ />() ị 1 ^ r í í n , / c Oĩì í h c l ĩ i l h c r í s p a c e í c ì ĩ s o r p r o d u c t / -^('^1) ■
g ive by
P r Á - l ) { ( ' k , ® . . . : í =
Ạ
.V, (1 - ♦ ‘ + ' > ) 0 ... s a H +
1® f'A-,,2 ® if .s < r
r ‘ẽ:‘ ® . . . ® PAv i f .s- - r
0 if ,s > r.
The representation fh)Ị is eqiiivalenỉ /() the restriction o f th e rep resen ta tio n Ti o f !F ^(S Ln- \ \ \ C. ) ) (cf.2.3); a n d o r ì' > (ì, f)j-1 is equivaìent to ihe restriction o f TTs^ y. %-
From Theorem 2.6, w e have
kcr/)^,/ -= {()}, (*',i)eiV'x7-
i.e. the representation ©;^eu I'f is faithful and
f 1 if u; -- (’
(liin (). t ' ' \
We recall now ihc definition o f com pact quanlum o f sp h eres
c*
— algebra.D efin itio n 3.3. The c * —a lg eb ra ic co m pact quantum sp h e re c * ( 5 “ ) is he c * —co m p lciio n oj the
*—algebra with respect to the C * ~ n o rm
ll/ll-sup i|M/)ll,
p
where p runs through the*— representations o f (cf.. T heorem 2 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 * - i i o r n i , i-C- ll/-/* ll — \ \ f \ \ ^‘ We now prove that ibllowing result ab o u t he structure o f com pact quantum C * - a l g c b r a o f sphere S ' \
I 'h e o r c m 3,5. ỉilíh Pìoíaíion as above, vt't' have
25() A-.Ọ Tho / \ 'NU J o u rn a l o f Science. Miỉíhernưtics - Ị^hysics 25 (2009) 249-259
c ; ( 5 " ) ^ c ( 5 ' ) 0 0 r
where C ( 5 * ) 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 ) id ea l o f com paci operato rs 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 rep resentation t is equivalent to the restriction 0Í 7T.,^ ®....®7T5^ ® T t, w here 1ĨSX is the com position o f the h o m o m o rp h is m o f T'e(G ) onto / ’ê ( 5 / . 2 ( C ) ) and the representation 7T_1 o f J^ị:{SL ‘2{C)) in the iỉilb c rt sp a c e and the family o f onc- dim ensional representations T(, given by
7 Ị ( a ) - T; (0) - 7Ị(c) - 0, 7 ĩ( ư ) -
where t G 5^ and a j j , c , d are give by: A lgebraJT £ (5 L 2 (C )) is generated by the matrix elements o f type [ ^ . H ence, by construction the representation Í = TTg
0
.... ®7ĨS ® 7i- T h u s, vvc haveỵ c u y ’ * Ỉ * k
7 T , . : C ; ( S ' ‘) --- ► c ; ( S L , ( 0 ) £ (ế'^ (N )S ^ ).
Now,
TTs
is C C R (see, 111Ị) and so, w e have7rs,(C'*(5") ~ /C(//^,().
M oreoverTt{C*{S'^)) — c .
I Icncc.
/ - , ( C ( ^ " ) ) = (7T,._ 'v . . . 0 7T,.^0 7 ; ) ( Q ( S " )
= rr,., (c;(S")) ®... s
7t,./(c;(S'‘)) 0 T((c;(
5'*))
- I C{ I L , t ) , where n ^ . j = //«,
0
.... © H s, ® c .T h u s, p ^ A C : { S ’')) = K . ụ U t ) . 1 Icncc,
N . ọ Tho / V N U J o u r n a l o f S c i c n c c . M a th e m a tic s - P hysics 25 (2009) 249-259 257
/*0 r(D
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
c*
-a lg e b r a . Then, B ) ~ ,4 Cv / i , w here ÍÌ is the spectrum spacc o f A .A pp ly in g this result, lor B — AC(7 /^ / ) ^ /C and A — C ( U ' x 5 ^ ) be a com m utative c * - a l g e b r a .
*
I'hus. w e liave
c
Now, vve first com pute the A \ ( C * ( 5 ” )) 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 s itio n 3.6.
/ / A \ ( c ; ( S ' - ) ) ^ / / ; „ , ( U ' x ố ’') . Proof. Wc have
Ỉ Ỉ E A C : ( S ’'}) = / / E , ( C ( 5 ' ) © 0 r ! C{n^ ^t ) dt )
= / / £ . ( C ( S ’' ) © / / E . ( 0 r I C { Ỉ U t ) d t ) )
— H E . { C { \ V X 5 ‘) 0 / C (b y P r o p o s i t i o n 1.1)
^ / / / T . ( C ( i r X 5*)).
Since C (U '' X
s ' )
is a com m u tativ e ’ - a l g e b r a , by F’roposition 1.5 §1, w e have / / E . ( C ; ( S " ) ) ^ H E , { C { \ V X 5 ‘ )) ^ H h n i ^ V X 5*))'.P roposition 3.7.
K,{C*{S’')) ^ IC(\V
X 5^).Proof. Wc have
A\(c;(S'‘)) = A\(C(S‘)Q
0 r i C { U ^ ^ t ) d f ) U' ' ^258 N.Q. TỊìo / V N U J o u r n a l o f Science. M ưíhcm ưỉics - P hysics 25 (2 0 0 9 ) 249-259
) d t ) )
^ K , { C { \ V x S ^ ) @ I C (1)V P r o p o s i t i o n 1.1)
^ A ' . ( C ( i r X S ' ) ) . In result o f Proposition 1.5, §1, w e have
h \ i C { \ V X 5 ' ) ) K . { \ v X 5 ‘ ).
T heorem 3.8, IFiih rwiaiion above, the Chern chanicỉer o f C * —algebra o f quantum sphere
c
( S " ) chc- : A \ ( C (S") — ỉ ỉ i : . { C ( 5 ”))is an isom orphism .
Proof. By Proposition 2.9 and 2.10, w e have
H E , { C ; { S ' ^ ) ) ^ H E , { C { \ V X 5 ‘ )) ^ i / ; j / i ( i r X 5 ' ) ) , / : . ( c ; ( 5 " ) ) ^ A ' . ( C ( i r X 5 ‘ )) ^ I C { \ V X 5 * ) ) . Now, consider the com m utative diagram
/ c * ( c ; ( 5 " ) ) Ỉ Ỉ E . { C : { S ^ ^ ) )
K J C ( \ V y s ^ ) ) Í Ỉ E J C ( \ V X S ' ) )
A ' * ( U ’ X 5 ‘ ) ^
M o r e o v e r , follvving W a t a n a b c [ 1 5 ] , th e e ll : A ' * ( i r X 5 ‘ ) :-; C — » X 5 * ) is an i s o m o r p h i s m . Thus, chc* : h \ { C *£ (5'*) — » H E ^ { C (5'” )) 'S an isom orphism .
A c k n o w le d g m e n t. T h e author w ould like to thank Professor Do Ngoc D ie p for his guidance anti encouragem ent during this paper.
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