Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 12, Number 2 (2016), pp. 1325-1335 © Research India Publications http://www.ripublication.com
On the generalized non-commutative sphere and their K-theory
Saleh Omran
Taif University, Faculty of Science, Taif , KSA South valley university, Faculty of science, Math. Dep. Qena , Egypt.
Abstract
In the work we follow the work of J.Cuntz, he associate a universal C* - algebra to every locally finite flag simplicial complex. In this article we define * nc a universal C -algebra S3 , associated to the 3 -dimensional noncommutative sphere. We analyze the topological information of the
noncommutative sphere by using it’s skeleton filtration Ik )( . We examine the nc K -theory of the quotient 3 /IS k , and I k for such kk 7, .
Keywords: Universal C* -algebra , Noncommutative sphere, K -theory of C* -algebra.
2000 AMS : 19 K 46.
1. Introduction Noncommutative 3 -sphere play an important role in the non commutative geometry which introduced by Allain Connes [4] .In noncommutative geometry, the set of points in the space are replaced by the set of continuous functions on the space. In fact noncommutative geometry change our point of view from the topological space itself to the functions on the space ( The algebra of functions on the space). Indeed, the main idea was noticed first by Gelfand-Niemark [8] they were give an important theorem which state that: any commutative C* -algebra is isomorphic of the continuous functions of a compact space, namely the space of characters of the algebra. More generally any C* -algebra ( commutative or not) is isomorphic to a closed subalgebra of the algebra of bounded operators on a Hilbert space. Our motivation is the problem of classification of noncommutative spheres by using the
1326 Saleh Omran
K -theory of C* -algebras associated these spheres.
2. Universal C * -algebras In this section we give some basic definitions and facts on C* -algebras and universal C* -algebras and their K -theory which we will use in this article for basic theory of C* -algebras, we refer to [15], [10]and [7].
Definition 2.1 A C* -algebra A is a complex Banach algebra with a conjugate-linear 2 involution : AA , such that ** xx xy **** =,=)(,=)( xxxxy 2 for all , yx in A . The C* -condition * = xxx implies that the involution is an isometry in the sense that * = xx for all x in A . A C* -algebra is called unital if it possesses a unit. It follows easily that 1=1 . Many C* -algebras can be constructed in the form of universal C* -algebras. In the following, we state the main definition of the universal C* -algebras.
Definition 2.2 Let A be a complex -algebra, such that for each Ax . Then we define the enveloping C* -algebra * AC )( to be the closure of the quotient /{ =| 0} xxA u with the induced norm. * AC )( has the following universal property: if : BA is a -homomorphism between C* -algebras A and B , then there exists a unique -homomorphism * )(:' BAC such that =' , where is the canonical -homomorphism from to * )( . Also, )( . In particular, if 0= this means that belongs A AC xx uu x u x to the kernel of .
Definition 2.3 Let and I be two index sets and let P be the -algebra of all * noncommutative polynomials in variables ii ,, Iixx , with involution * * * =)( * }|,{, }|{= 1 aa n n 1 ik i Iixxaaa . Let PR be a subset of P , and
J R be the ideal in P generated by the relations R . Then := /JPA R is called the * universal -algebra with generators and relations ii ,, Iixx relations . The enveloping C* -algebra of A is called the universal -algebra with generators and relations R .
3. K-theory groups The K -theory of C* -algebras generalizes and extends the classical topological K - theory which was introduced by Atiyah [1] to the noncommutative case. More
On the generalized non-commutative sphere and their K-theory 1327 precisely, topological K -theory of a compact space X is just the K -theory of the unital commutative C* -algebra XC )( . K -theory has found many applications in representation theory of groups, topology, geometry, index theory and many other subjects of mathematics and physics. The K - theory of C* -algebras plays a central role in what is called noncommutative geometry, which was pioneered by A. Connes. K -theory of C* -algebras is one of two powerful tools of this theory, the other one being cyclic homology. In noncommutative geometry the place of K theory of spaces , [1], [9], is taken by the K -theory of C* -algebras and Banach algebras. By now, the K -groups of many important classes of C* -algebras are known. So, the next important question is what K -theory can tell us about various kinds of C* - algebras respectively about noncommutative spaces. The K -theory of C* -algebras is defined by means of projections and unitaries. Here * we give the basic definitions of 0 AK )( and 1 AK )( groups of such unital C -algebra A , for more details we refer [2], [6], [7], [16] and [17].
Definition 3.1 Let A be a unital C* -algebra. Two projections ,qp in A are called Murray-von Neumann equivalent, denoted ~ qp , if there exist some element Av such that p = vv* and = *vvq .
unitarily equivalent, denoted ~u qp , if there exist a unitary Au such that q = upu* .
homotopy equivalent, denoted ~ h qp if there exist a continuous path of projections in A , [0,1]:)( Atp , such that tqppp [0,1],=(1),=(0) .
It is not hard to prove that the relations above are equivalence relations. Now, let us denote by )( := AMAM )( , the union of the AM )( , n1 n n a 0 n n1 AMAMA )()( ... ( ), aAM , 00 and denote by AP )( the set of projections in AM )( . We have the following theorem.
Theorem 3.2 The equivalence relations, ,, ~~~ hu coincide on the set of projections
AP )( in AM )( .
Now, we describe the 0 AK )( -group.
* Definition 3.3 Let A be a unital C -algebra. For any APp )( denote by p][ the equivalence class of p with respect to the equivalence relation ~. The set
)( := {[ (|] APppAV )} with the addition given by qpqp ][=][][ , where
1328 Saleh Omran
p 0 qp := , is a semi-group. Actually AV )( is an abelian semi-group, since if 0 q 0 q APqp )(, , such that n APp )( and m APq )( , put v = mn AP )( , this p 0
* p 0 * q 0 gives vv = = qp and vv = = pq , so ~ pqqp . 0 q 0 p
We denote by 0 AK )( the Grothendieck group of AV )( . Recall that the Grothendieck group SG )( for some semi-group S is the universal abelian group generated by S .
The K0 group satisfies a limited set of properties( which characterize this groups on * * the category of all C -algebras. For a C -algebra, K0 satisfies the following short list of properties) homotopy invariance, half-exactness, continuity, stability, and
K0 C 0=)( .
On the other hand, 1 AK )( , it is described by unitaries in AM )( .
* Definition 3.4 Let A be a unital C -algebra. Denote by (( AMU )) is the group of unitaries in AM )( . Define 1 )( := (( ))/ 0 (( AMUAMUAK )) , where 0 (( AMU )) is the path component of the unit in AM )( . * Moreover, 1 1 AKAK )(: defines a functor from the category of C algebras to the category of abelian groups, with the same properties as the functor K0 but normalized in the sense that K1 C 0=)( and CK 01 .=(0,1))( * The following theorem states a deep relation between 0 AK )( and 1 AK )( for a C - algebra A , since the algebraic equivalence relations of projections in C* -algebras can be described in terms of homotopy equivalence, so one can find the following natural isomorphism .
Theorem 3.5 For any C* -algebra A and the suspension SA of A , there is a natural isomorphism 1 KAK 0 SA)()( .
There is a long exact sequence and a connecting map between K0 and K1 for each short exact sequence as follows. i For each exact sequence 0 BAJ 0 of C* -algebras, there is an exact sequence 1 1 1 0 0 0 ()()()()()( BKAKJKBKAKJK ). The connecting map is called the index map in K -theory. On the other hand Bott gives a very deep result in K -theory, Bott periodicity, it is very important to to determine the K -theory of C* -algebras related in an extension. The following theorem is given by Bott see [3].
On the generalized non-commutative sphere and their K-theory 1329
* Theorem 3.6 The map A 0 KAK 1 SA)()(: is an isomorphism for any C -algebra A . By using the connecting map and Bott periodicity, one can give a definition of the higher K group, Kn , for n 1, as follows.
* n Definition 3.7 For each integer n 2 and C -algebra A we set n )( := 0 ASKAK )( . By using the above theorems and definitions in this section we get the following very useful result
Proposition 3.8 For each short exact sequence 0 BAI 0
there is a six-term exact sequence
0 0 0 BKAKIK )()()(
1 1 1 ()()( IKAKBK ).
Here is the index map and is given by p])([ [= exp ix)](2 , where p][ is a class of projections in 0 BK )( and x is a selfadjoint lift of p to A .
4. Commutative 3-sphere and their K-theory Define the commutative 3 -sphere by 3 4 2 2 2 2 = {( 3210 |),,, 0 1 1 xxxxxxxxS 3 1}.= We denote by SC 3 )( the algebra of all complex-valued continuous functions on S 3 with pointwise addition and multiplication. The algebra SC 3 )( with involution defined by * xfxf )(=)( for each ( 3 ), XxSf , and with the norm = {| )( |, Sxxfsupf 3} is a commutative C* -algebra. In the language of Universal C* -algebra , this algebra is isomorphic to the universal C* -algebra generated by self- adjoint elements ,,, xxxx with relations 2 1=,= where 3310 ijji ixxxxx i , ji {0,1,2,3}. S 3 is homeomorphic to the one-point compactification of R 3 , so we have an ~ 3 3 isomorphism 0 RCSC )()( . Since the real line is homeomorphic to the open 3 3 interval, 0 RC )( isomorphic to SC and consequently CS is isomorphic to 0 RC )( . 3 3 By Bott periodicity, we obtain 00 (( RCK )) 0= and 01 (( )) = ZRCK Now, consider the split exact sequence
1330 Saleh Omran
~ 3 3 0 0 ()(0 ) CRCRC 0, and apply the split exactness of K groups. Then we obtain ~ 3 3 3 0 (( )) 00 (( )) 00 (( )) = ZZRCKRCKSCK and 3 3 1 (( )) 01 (( )) ZRCKSCK .= 3 3 3 0 (( SCK )) is generated by the unit of SC )( and 1 (( SCK )) is generated by the 22 unitary matrix with entries viewed as the coordinate functions on S 3 .
5. Noncommutative sphere and their K-theory Noncommutative circle and their K -theory was introduced in [11]. The author in [13], and [14] analyze the topological information of the noncommutative circle and 2 -dimensional sphere by using it’s skeleton filtration.
Definition 5.1 A simplicial complex consists of a set of vertices V and a set of non-empty subsets of V , the simplexes in , such that:
• If Vs , then s .}{ • If F and FE then E .
Definition 5.2 A simplicial complex is called flag or full, if it is determined by its
1-simplexes in the sense that { 0 ,..., n ssss ji },{} for all <0 nji . Cuntz in [5] associate a universal C* -algebra to every locally finite flag simplicial complex as follows.
Definition 5.3 Let be a locally finite flag complex. Denote by V the set of its f * vertices. Define C as the universal C -algebra with positive generators s , Vsh , satisfying the relations
tts ,= Vthhh Vs and
ts tsforhh .},{0= f Denote by I k the ideal in C generated by products containing at least n 1-different f generators. The filtration Ik )( of C is called the skeleton filtration. 1100 33 Let we consider the flag complex 3 with vertices ,,,{ ,..., VVVVVV }, S and the condition that exactly the edges ii },{ iVV {0,1,2,3} do not belong to 3 3 , the geometric realization of 3 is the 3 sphere S . We consider the universal S S
On the generalized non-commutative sphere and their K-theory 1331
* C -algebra with 8 positive generators ,hh where V i V i , ii := ,,,{ 1100 ,..., VVVVVVVVV 33 }, 3 S and satisfying the relations
= 1, hhhh = 0. i i i i i V i V V V We denote it by C f .. 3 S * * Let xxxxC 3210 ),,,( denote the unital universal C -algebra with self-adjoint generators ,,, satisfying the relations that 2 =1, xxxxx for all xxxx 3210 i i ijji , ji {0,1,2,3}. The abelianization of this C* -algebra is isomorphic to the algebra of continuous functions on the 3 -sphere S 3 as shown in [5]. Consider the following the positive and negative parts xi )( and xi )( of (xi ). We have 2 2 2 22 i i xxx i and xx ii =)()()()(=)( 0. and consider :
hhx . i V i V i and
xh ,)( and xh .)( V i i V i i * Then we get a new positive generators for xxxxC 3210 ),,,( and satisfy
hh =1,and = 0, ihh {0,1,2,3}. i i i i i V i V V V Therefore the algebra * xxxxC ),,,( is exactly isomorphism the algebra C f . 3210 3 S For simplicity we denote C f by S nc. 3 3 S nc The K theory of the algebra S3 is given by the following proposition.
nc 1 Proposition 5.4 The evaluation map ev : 3 CS at the vertex V , which maps the generator h to 1 and all the other generators to 0 , induces an isomorphism in 1 V K theory.
Proof. nc nc Define to be the homomorphism from S3 to S3 that maps the generator h V i V i to 1 and the other generators to zero. It is clear that is the composition of the V i j i nc evaluation map for the vertex V and the natural inclusion map SC 3 . Denote by nc id the identity homomorphism of S3 . Our purpose is to show that
1332 Saleh Omran
K K id )(=)( for a fixed vertex V 1 . * 1 * V nc nc Consider the homomorphisms and from S3 to SM 32 )( in the form
id 0 0 0 = and = . 0 1 0 1
Here 1 , 0 and 1 are defined as follows :
1 =)( 1 iforhhhh 0,1,=0=)(, V i V i V i V i
1 hh ss ,=)( for all other generators hs . =)( hhhh =)(0=)(, hh , for all other generators 0 0 0 0 0 0 0 ss V V V V and =)( hhhh =)(0=)(, hh for all other generators . 1 1 11 1 1 1 ss V V V V One has the following homotopies nc nc • is homotopic to , by using the homomorphisms t from S3 to SM 32 )( mapping ,hh to * )(,)( RhRRhR * and all other generators h 00 t 0 tt 0 t s VV V V
to hs )( where Rt are rotation matrices, t /2][0, . Clearly, we have 0 =
and /2 = . • is homotopic to , using the homomorphism which maps h to 1 0 0 V V )( thht )1(1 , h to 0 , h to hht )( , h to 0 and all 00 0 1 11 1 V V V V VV V
the other generators hs to ths , t [0,1]. • is homotopic to , using the homomorphism, which maps h to 0 0 0 V V )( thht )1(1 , h to 0 and all the other generators h to th . 00 0 s s V V V • is homotopic to , using the homomorphism, which maps h to 1 1 1 V V )( thht )1(1 , h to 0 and all the other generators h to th . 11 1 s s VV V From the above homotopies KK id K id )(=)(=)( * * *1 0 V and K ( K (=) ). * 0 1 * V V Since KK )(=)( , this implies that K K id )(=)( , and in the other side * * * 1 * V K K ev jK )()(=)( . This means that K ev)( is the inverse of jK )( and this * 1 * * * * V nc prove that SK 3* )( and * CK )( are isomorphic. nc Next we study the K -theory of the skeleton filtration for S3 . We study the K -
On the generalized non-commutative sphere and their K-theory 1333
nc theory of the quotient 3 /IS k for all k . We find that the K -theory of the quotients nc 3 /IS k is related to the K -theory of I k as follows.
Theorem 5.5 For each k 7 we have isomorphisms nc 30 k 1 k )(=)/( ZIKISK and nc 31 k 0 (=)/( IKISK k ). Proof. Consider the skeleton filtration nc = 2103 ... IIIIS 7. The following short sequence i nc nc 0 k 3 3 ISSI k 0/ nc nc is exact. 33 /: ISS k is the quotient map. Hence, by the six term exact sequence in K theory, we get i* * nc nc 0 IK k )( SK 30 30 ISK k )/()( nc nc 31 k SKISK 31 )()/( 1(IK k ). nc Since we have 3* KSK * C)()( by 5.4, the above exact cyclic sequence becomes
i* * nc 0 IK k )( 30 ISKZ k )/( nc 31 ISK k 0)/( 1 (IK k ). nc nc Where * maps the class [1] in SK 30 )( to the class [1] in 30 ISK k )/( . So * nc nc 30 ISK k )/( contains a copy of Z and 30 ISKZ k )/( is an embedding.
Consequently i* 0= and we have an isomorphism nc 31 k 0 IKISK k )(=)/( and a short exact sequence nc 0 30 k 1 IKISKZ k .0)()/( In the remaining part of the proof we show that this short exact sequence splits. Let nc 3 /: k CIS be the evaluation map. nc Consider the natural map ev : 3 CS which sends h 1 to 1 and all other generators V to 0 . Then we get the commutative diagram nc nc 3 3 /ISS k ev C
1334 Saleh Omran
Since K* is a functor, we obtain the following corresponding commutative diagram in K -theory, * nc nc SK 3* 3* ISK k )/()(
ev* *
* (CK ), nc By proposition 5.4 the K groups of S3 and C coincide, and the map nc ev 3** * CKSK )()(: is equivalent to the identity. So the composition ** is equivalent to the identity and thus the sequence (5) splits. Hence we obtain the desired isomorphism nc 30 k 1 k ZIKISK .)(=)/(
Now, we apply the above theorem k 1= , to obtain IK 10 )( and IK 10 )( .
nc nc 8 nc Lemma 5.6 Let 3 /IS k as in the above, then 130 =)/( ZISK and ISK 131 0=)/( .
Proof. nc Let hi denote the image of a generator hi for /IS 03 . One has the following relations : i =1, ji = 0, jihhh . i nc For every hi in /IS 13 we have 2 ii hhhh ii .=)(= i nc Hence /IS 13 is generated by 8 different orthogonal projections and therefore nc 8 nc 8 nc / 13 CIS . And therefore 130 =)/( ZISK , ISK 131 0=)/( . Apply theorem 2.5 7 and the lemma 2.6 above we get. 11 =)( ZIK and IK 10 0=)( .
Acknowledgements The author wish to thank the Taif university for the fund of the project number 1-436- 4322. And thank . Mohamed A. El-Sayed for his help and assist during the work in the project.
References
[1] M. F. Atiyah. K-theory. W. A. Benjamin, New York (1967). [2] B. Blackadar. K-theory for operator Algebras. Springer, 1986 [3] R. Bott. The stable homotopy of the classical groups., Ann. Math. 70(1959), 313-337.
On the generalized non-commutative sphere and their K-theory 1335
[4] A. Connes. Noncommutative geometry, Academic Press, San Diago, CA, 1994 [5] J. Cuntz. Non-commutative simplicial complexes and Baum-Connes- conjecture. GAFA, Geom. func. anal. Vol. 12(2002) 307-329. [6] J. Cuntz. A survey of some aspects of noncommutative geometry. Jahresber. d. dt. Math.-Verein.,95:60-84,1993. [7] K. R. Davidson. C* -algebras by Example., Fields Institute monographs, 1996. [8] I. Gelfand and M. Naimark. On the embedding of normed rings into the ring of operators in Hilbert space., Math. Sb. 12(1943), 197-213. [9] M.Karoubi. K-theory: An introduction. Springer Verlag, New York, 1978 [10] G. J. Murphy. C* -algebras and Operator Theory. Academic Press, 1990. [11] G. Nagy. On the K -theory of the noncommutative circle. J. Operator Theory,31(1994), 303-309. [13] Saleh. Omran, C* -algebras associated with higher-dimensional noncommutative simplicial complexes and their K -theory, Münster: Univ. Münster-Germany, (Dissertation 2005). [13] Saleh. Omran, C* -algebras associated noncommutative circle and their K - theory , The Australian Journal of Mathematical Analysis and Applications,Volume 10, Issue 1, Article 7, pp. 1-8, 2013. [14] Saleh Omran, Certain higher-dimensional simplicial complexes and universal C*-algebras, SpringerPlus 2014, 3:258 [15] G. K. Pedersen. C* -algebras and their automorphism groups. Academic Press,London, 1979. [16] M. Rørdam, F. Larsen, N. Laustsen. An introduction to K-theory for C*- algebras. London Mathematical Society Student Text 49 [17] N. E. Wegge-Olsen. K-theory and C*-algebras Oxford University Press, New York, 1993
1336 Saleh Omran