Cấu trúc các C* – đại số Connes liên kết với một lớp con các MD5 – nhóm

Tạp chí KHOA HỌC ĐHSP TP HCM Le Anh Vu et al. 
_____________________________________________________________________________________________________________ 
 15
THE STRUCTURE OF CONNES’ C* – ALGEBRAS 
ASSOCIATED TO A SUBCLASS OF MD5 – GROUPS 
LE ANH VU*, DUONG QUANG HOA** 
ABSTRACT 
The paper is a continuation of the authors’ works [18], [19]. In [18], we consider 
foliations formed by the maximal dimensional K-orbits (MD5-foliations) of connected 
MD5-groups that their Lie algebras have 4-dimensional commutative derived ideals and 
give a topological classification of the considered foliations. In [19], we study K-theory of 
the leaf space of some of these MD5-foliations, analytically describe and characterize the 
Connes’ C*-algebras of the considered foliations by the method of K-functors. In this 
paper, we consider the similar problem for all remains of these MD5-foliations. 
 Key words: Lie group, Lie algebra, MD5-group, MD5-algebra, K-orbit, Foliation, 
Measured foliation, C*-algebra, Connes’ C*-algebras associated to a measured foliation. 
TÓM TẮT 
Cấu trúc các C* – đại số Connes liên kết với một lớp con các MD5 – nhóm 
 Bài báo này là công trình tiếp nối hai bài báo [18], [19] của các tác giả. Trong [18], 
chúng tôi đã xét các phân lá tạo thành bởi các K – quỹ đạo chiều cực đại (các MD5 – phân 
lá) của các MD5 – nhóm liên thông mà các đại số Lie của chúng có ideal dẫn xuất giao 
hoán 4 chiều và đưa ra một phân loại tô pô tất cả các MD5 – phân lá được xét. Trong [19], 
chúng tôi đã nghiên cứu K – lý thuyết đối với không gian lá của một vài MD5 – phân lá 
trong số đó, mô tả giải tích đồng thời đặc trưng các C* – đại số của Connes liên kết với 
một số phân lá đó bằng phương pháp K – hàm tử. Trong bài này, chúng tôi xét bài toán 
tương tự đối với tất cả các MD5 – phân lá còn lại. 
 Từ khóa: Nhóm Lie, Đại số Lie, MD5-nhóm, MD5-đại số, K-quỹ đạo, Phân lá, Phân 
lá đo được, C*-đại số, C*-đại số Connes liên kết với một phân lá đo được. 
1. Introduction 
In the years of 1970s-1980s, the works of Diep [4], Rosenberg [10], Kasparov 
[7], Son and Viet [12],  showed that K-functors are well adapted to characterize a 
large class of group C*-algebras. In 1982, studying foliated manifolds, Connes [3] 
introduced the notion of C*-algebra associated to a measured foliation. Once again, the 
method of K-functors has been proved as very effective in describing the structure of 
Connes’ C*-algebras in the case of Reeb foliations (see Torpe [14]). 
* Department of Mathematics and Economic Statistics, University of Economics and Law, Vietnam 
National University, Ho Chi Minh City 
** Department of Mathematics and Infomatics, Ho Chi Minh City University of Education, Vietnam. 
Tạp chí KHOA HỌC ĐHSP TP HCM Số 27 năm 2011 
_____________________________________________________________________________________________________________ 
 16
Kirillov’s method of orbits (see [8, Section 15]) allows to find out the class of Lie 
groups MD, for which the group C*-algebras can be characterized by means of suitable 
K- functors (see [5]). Moreover, for every MD-group G, the family of K- orbits of 
maximal dimension forms a measured foliation in terms of Connes (see [3, Section 2, 
5]). This foliation is called MD-foliation associated to G. Recall that an MD-group of 
dimension n (for short, an MDn-group), in terms of Diep, is an n-dimensional solvable 
real Lie group whose orbits in the co-adjoining representation (i.e., the K- 
representation) are the orbits of zero or maximal dimension. The Lie algebra of an 
MDn-group is called an MDn-algebra (see [5, Section 4.1]). 
Combining methods of Kirillov and Connes, the first author studied MD4-
foliations associated with all indecomposable connected MD4-groups in [16]. Recently, 
Vu and Shum [17] have classified, up to isomorphism, all the 5-dimensional MD-
algebras having commutative derived ideals. 
In [18], we have given a topological classification of MD5-foliations associated to 
the indecomposable connected and simply connected MD5-groups, such that MD5-
algebras of them have 4-dimensional commutative derived ideals. There are exactly 3 
topological types of considered MD5-foliations which are denoted by F1, F2, F3. All 
MD5-foliations of type F1 are the trivial fibrations with connected fibre on 3-
dimensional sphere S3, so Connes’ C*-algebras C*( F1) of them are isomorphic to the 
C*-algebra 3C S K following [3, Section 5], where K denotes the C*-algebra of 
compact operators on an (infinite dimensional separable) Hilbert space. 
In [19], we study K-theory of the leaf space and to characterize the structure of 
Connes’ C*-algebra C*(F2) of all MD5-foliations of type F2 by method of K-functors. 
The purpose of this paper is to study the similar problem for all MD5-foliations of type 
F3. Namely, we will express C*(F3) for all MD5-foliations of type F3 by a single 
extension of the form 
 0 3 00 * 0C X K C C Y K   F , 
then we will compute the invariant system of C*(F3) with respect to this extension. 
Note that if the given C*-algebra is isomorphic to the reduced crossed product of the 
form 0C V ⋊H , where H is a Lie group, then we can use the Thom-Connes 
isomorphism to compute the connecting map 0 1,   . 
2. The MD5-foliations of type F3 
Originally, we recall geometry of K-orbits of MD5-groups which associate with 
MD5-foliations of type F3 (see [17]). 
In this section, G will be always one of connected and simply connected MD5-
groups 5,4,14( , , ) G   which are studied in [17] and [18]. Then, the Lie algebra G of G 
will be the one of the Lie algebras 5,4,14 ( , , )   G (see [17] or [18]). Namely, G is the 
Tạp chí KHOA HỌC ĐHSP TP HCM Le Anh Vu et al. 
_____________________________________________________________________________________________________________ 
 17
Lie algebra generated by 1 2 3 4 5, , , ,X X X X X with 
  42 3 4 5: , . . . .X X X X        1G G G and 1 41Xad End Mat  G as 
follows 
1
cos sin 0 0
sin cos 0 0
: ; , , 0, 0, .
0 0
0 0
Xad
   
 
 
 
We now recall the geometric description of the K-orbits of G in the dual space G* 
of G. Let * * * * *1 2 3 4 5, , , ,X X X X X be the basis in G* dual to the basis 1 2 3 4 5, , , ,X X X X X 
in G. Denote by F the K-orbit of G including , ,F i i     in 
* 5    G . 
- If 0i i    then F F (the 0-dimension orbit), 
- If 2 2 0i i    then F is the 2-dimension orbit as follows 
 ., . , . , , .ia e a iF x i e i e x a        
In [18], we show that, the family F of maximal-dimension K-orbits of G forms 
measure foliation in terms of Connes on the open sub-manifold 
  ** 2 2 2 2 4, , , , : 0V x y z t s y z t s   G . 
Furthermore, all the foliations      5,4,14 , ,, , , , 0, 0;V F , are 
topologically equivalent to each other and we denote them by F3 . So we only choose a 
“envoy” among them to describe the structure of C*(F3) by K-functors. In this case, we 
choose the foliation
5,4,14 0,1,
2
,V 
F . 
 In [18], we also describe the foliation 
5,4,14 0,1,
2
,V 
F by suitable action of 2 . 
Namely, we have the following assertion. 
Proposition 2.1. The foliation 
5,4,14 0,1,
2
,V 
F can be given by an action of the 
commutative Lie group 2 on the manifold V. 
Tạp chí KHOA HỌC ĐHSP TP HCM Số 27 năm 2011 
_____________________________________________________________________________________________________________ 
 18
Proof. One needs only to verify that the foliation 
5,4,14 0,1,
2
,V 
F is given by the action 
  2: V V of 2 on V as follows 
  r a x y iz t is, , , , : = ia iax r y iz e t is e, . , . , 
where        r a x y iz t is V2 4, and , , . Hereafter, for 
simply, we write F3 instead of 
5,4,14 0,1,
2
,V 
F . 
It is easy to see that the graph of F3 is identified with 2V  , so by [3, Section 5], 
it follows from Proposition 2.1 that 
Corollary 2.2. (Analytical description of C*(F3)) The Connes’ C*-algebra C*(F3) can 
be analytically described by the reduced crossed product of 0C V by 2 as follows 
C*(F3) 0C V ⋊ 2  . 
3. C*(F3) as a single extension 
3.1. Let 1 1, V W be the following sub-manifolds of V 
  *1 , , : 0V x y iz t is V t is     , 
  *1 1\ , , : 0W V V x y iz t is V t is    . 
It is easy to see that the action  in Proposition 2.1 preserves the subsets 1 1, V W . 
Let , i  be the inclusion and the restriction 
 0 1 0:i C V C V , 0 0 1: C V C W . 
where each function of 0 1C V is extended to the one of 0C V by taking the value of 
zero outside 1V . 
It is known a fact that , i  are - equivariant and the following sequence is 
equivariantly exact: 
(3.1) 0 1 0 0 10 0iC V C V C W    . 
3.2. Now we denote by 1 1 1 1, , ,V WF F restrictions of the foliations F3 on 1 1, V W , 
respectively. 
 Theorem 3.1. C*( F3) admits the following canonical extension 
 1 3 0 * 0iJ C B     

F , 
Tạp chí KHOA HỌC ĐHSP TP HCM Le Anh Vu et al. 
_____________________________________________________________________________________________________________ 
 19
where * 1 1 0 1,J C V C V F ⋊ 2 0C K     , 
 * 1 1 0 1,B C W C W F ⋊ 2 0C K    , 
 3 0*C C VF ⋊ 2  . 
and the homomorphism , i  is defined by 
  , , , , ,i f r s if r s f r s f r s   . 
Proof. Note that the graph of F3 is identified with 2V  , so by [3, section 5], we have: 
 * 1 1 0 1,J C V C V F ⋊ 2  , 
 * 1 1 0 1,B C W C W F ⋊ 2  . 
From -equivariantly exact sequence in 3.1 and by [2, Lemma 1.1] we obtain the 
single extension 1 . Furthermore, the foliations 1 1,V F and 1 1,W F can be come 
from the submersions 
*
'
: and
 , , ' , 'i i i
p V
x re r e re r 
      
*:
 , i
q W
x re r 
    
Hence, by a result of [3, p.562], we get 
 * 1 1 0 1,J C V C V F ⋊ 2 0C K     , 
 * 1 1 0 1,B C W C W F ⋊ 2 0C K    . 
4. Computing the invariant system of 3*C F 
Definition 4.1. The set of element 1 corresponding to the single extension 1 in 
the Kasparov group Ext ,B J is called the system of invariant of 3*C F and denoted 
by Index 3*C F . 
Remark 4.2. Index 3*C F determines the so-called table type of 3*C F in the set 
of all single extension 
0 0J E B    . 
The main result of the paper is the following 
Theorem 4.3. Index 3 1*C  F , where 
 1 0,1 in the group , , ,Ext B J Hom Hom     . 
Tạp chí KHOA HỌC ĐHSP TP HCM Số 27 năm 2011 
_____________________________________________________________________________________________________________ 
 20
To prove this theorem, we need some lemmas as follows 
Lemma 4.4. Set 2 1 10 and I C S A C S  . 
The following diagram is commutative 
 3 1... ...j j j jK I K C S K A K I      
 0 1 0 0 1 1 0 1... ...j j j jK C V K C V K C W K C V      
where 2 is the Bott isomorphism, / 2j   . 
Proof. Let 
 2 1 30:k C S C S   , 3 1:v C S C S . 
be the inclusion and restriction defined similarly as in 3.1. 
One gets the exact sequence 
 30 0k vI C S A    . 
Note that 
 2 10 1 0 0 0C V C C S C I         
 3 30 0 0C V C S C C S       
 10 1 0 0C W C C S C A        
So, the extension (3.1) can be identified to the following one 
 30 0 00 0Id k Id vC I C C S C A             . 
So, the assertion of lemma is derived from the naturalness of Bott isomorphism. 
Remark 4.5. 
i) 2 1 10 0 , / 2j jK C S K C S j       . 
ii) 3 , / 2jK C S j    . 
iii) 10K C S   is generated by  0 2 1  , 11K C S   is generated 
by  1 2 Id  (where 1 is a unit element in 1C S ; , / 2j j   , is the Thom-Connes 
isomorphism; Id is the identity of 1S ). 
Proof of Theorem 4.3. Recall that the extension 1 in theorem 3.1 gives the rise to a 
six-term exact sequence 
2
2
2
2
Tạp chí KHOA HỌC ĐHSP TP HCM Le Anh Vu et al. 
_____________________________________________________________________________________________________________ 
 21
0 1 
 0 0 3 0*K J K C K B  F 
(4.1) 
 1 1 3 1*K B K C K J F 
By [11, Theorem 4.14], the isomorphism 
 0 1 1 0, , ,Ext B J Hom K B K J Hom K B K J   
associates the invariant 1 ,Ext B J to the pair 
 0 1 0 1 1 0, , ,Hom K B K J Hom K B K J    . 
Since the Thom-Connes isomorphism commutes with K-theoretical exact sequence 
(see [14, Lemma 3.4.3]), we have the following commutative diagram / 2j   : 
 3 1... * ...j j j jK J K C K B K J      F 
In view of Lemma 4.4, the following diagram is commutative 
 0 1 0 0 1 1 0 1... ...j j j jK C V K C V K C W K C V      
Consequently, instead of computing the pair 0 1,  from the direct sum 
 0 1 1 0, ,Hom K B K J Hom K B K J  , it is sufficient to compute the pair 
 0 1 0 1 1 0, , ,Hom K A K I Hom K A K I    . In other words, the six-
term exact sequence (4.1) can be identified with the following one 
 2 1 3 10 0 0 0K C S K C S K C S    
 (4.2) 
 1 3 2 11 1 1 0K C S K C S K C S   
By remark 3.5, this sequence becomes 
      
(4.3) 0 
     
By the exactness, the sequence (4.3) will be the one of the following ones 
0 1 
2 2 2 2 
j j j 1j 
 0 1 0 0 1 1 0 1... ...j j j jK C V K C V K C W K C V      
 3 1... ...j j j jK I K C S K A K I      
1 
Tạp chí KHOA HỌC ĐHSP TP HCM Số 27 năm 2011 
_____________________________________________________________________________________________________________ 
 22
1 1 0 0 
1 0 0 1 
1 1 0 0 
0 1      
0 1     
or 1 0      
1 0     
Now we choose 11ia e GL C S , 1b a . Then 
 0 1200
i
i
e
a b GL C S
e
 
. 
Let u = 1 2 1 2 1 2 1, , , cos cos cos ,cos cos sin ,cos sin ,sinu x y z t u        
1
1
0 32 2
2
2 2
. .cos sin
sin . .cos
ii
ii
e e
GL C S
e e
 
 
 
  
is a pre-image of a b . So, 
1
1
1 2 2
2 2
cos sin
sin cos
ii
ii
e e
u
e e
 
 
 
 
. 
Let 1 1
1 0
0
0 0
q I  
. We get 
1
1
2
1 2 12 2 2
2 02
2 2 2
cos cos sin
cos sin sin
ii
ii
e e
p uqu P C S
e e
 
 
  
  
 . 
Then rank (p) = 1. So       2 11 1 0 00a p I K C S  . Therefore, K-
theoretical exact sequence associate to 1 is 
 0 1      
0 1     
The proof is completed. 
Tạp chí KHOA HỌC ĐHSP TP HCM Le Anh Vu et al. 
_____________________________________________________________________________________________________________ 
 23
REFERENCES 
1. Brown L. G.; Douglas R. G., Fillmore P. A. (1977), “Extension of C*-algebra and K-
homology”, Ann. of Math, 105, pp. 265 – 324. 
2. Connes A. (1981), “An Analogue of the Thom Isomorphism for Crossed Products of 
a C*–algebra by an Action of  ”, Adv. In Math., 39, pp. 31 – 55. 
3. Connes A. (1982), “A Survey of Foliations and Operator Algebras”, Proc. Sympos. 
Pure Mathematics, 38, pp. 521 – 628. 
4. Diep D. N. (1975), “Structure of the group C*-algebra of the group of affine 
transformations of the line”, Funktsional. Anal. I Prilozhen, 9, pp. 63 – 64 (in 
Russian). 
5. Diep D. N. (1999), Method of Non-commutative Geometry for Group C*-algebras. 
Research Notes in Mathematics Series, vol. 416. Cambridge: Chapman and Hall-
CRC Press. 
6. Gelfand I., Naimark A. (1943), “On the imbedding of normed rings into the ring of 
operators in Hilbert space”, Mat. sb., 12, pp. 197 – 213 (in Russian). 
7. Kasparov G. G. (1981), “The operator K-functor and extensions of C*-algebras”, 
Math. USSR Izvestija, 16(3), pp. 513 – 572. 
8. Kirillov A. A. (1976), Elements of the Theory of Representations, Springer – Verlag 
Pub., Berlin – Heidenberg – New York. 
9. Rordam, M., Larsen F., Laustsen N. (2000), An Introduction to K –Theory for C*–
Algebras, Cambridge University Press, United Kingdom. 
10. Rosenberg J. (1976), “The C*-algebras of some real p-adic solvable groups”, Pacific 
J. Math., 65(1), pp. 175 – 192. 
11. Rosenberg J. (1982), “Homological invariants of extension of C*-algebras”, Proc. 
Sympos. Pure Math., 38, AMS Providence R.I., pp. 35 – 75. 
12. Son V. M. ; Viet H. H. (1984), “Sur la structure des C*-algebres d’une classe de 
groupes de Lie”, J. Operator Theory, 11, pp. 77 – 90. 
13. Taylor J. L., Banach Algebras and Topology (1975), Academic Press in Algebras 
and Analysis, New York, pp 118–186. 
14. Torpe A. M. (1985), “K-theory for the Leaf Space of Foliations by Reeb 
Component”, J. Func. Anal., 61, pp. 15-71. 
15. Vu L. A. (1990), "On the structure of the *C –Algebra of the Foliation formed by the 
K –Orbits of maximal dimension of the Real Diamond Group", Journal of Operator 
theory, 24, pp. 227–238. 
16. Vu L. A. (1990), The foliation formed by the K – orbits of Maximal Dimension of the 
MD4-group, PhD Thesis , Ha Noi (in Vietnamese). 
(Continued page 53) 
Tạp chí KHOA HỌC ĐHSP TP HCM Số 27 năm 2011 
_____________________________________________________________________________________________________________ 
 24
17. Vu L. A., Shum K. P. (2008), “Classification of 5-dimensional MD-algebra having 
commutative derived ideals”, Advances in Algebra and Combinatorics, Singapore: 
World Scientific co, pp. 353-371. 
18. Vu L. A., Hoa D. Q. (2009), “The topology of foliations formed by the generic K-
orbits of a subclass of the indecomposable MD5-groups”, Science in China, series A: 
Mathematics, 52 (2), pp. 351-360. 
19. Vu L. A.; Hoa D. Q. (2010), “K-theory of the leaf space of foliations formed by the 
generic K-orbits of some indecomposable MD5-groups”, Vietnam Journal of 
Mathematics, 38 (2), pp. 249 – 259.