0771 the structure of connes’ c – algebras associated to a subclass of md5 – groups

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[.]

Le Anh Vu et al Tạp chí KHOA HỌC ĐHSP TP HCM _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 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 MD 5-foliations, analytically describe and characterize the Connes’ C*-algebras of the considered foliations by the method of Kfunctors 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 C* – đại số Connes liên kết với lớp MD5 – nhóm Bài báo cơng trình tiếp nối hai báo [18], [19] tác giả Trong [18], xét phân tạo thành K – quỹ đạo chiều cực đại (các MD5 – phân lá) MD5 – nhóm liên thơng mà đại số Lie chúng có ideal dẫn xuất giao hoán chiều đưa phân loại tô pô tất MD5 – phân xét Trong [19], nghiên cứu K – lý thuyết không gian vài MD5 – phân số đó, mơ tả giải tích đồng thời đặc trưng C* – đại số Connes liên kết với số phân phương pháp K – hàm tử Trong này, chúng tơi xét tốn tương tự tất MD5 – phân lại Từ khóa: Nhóm Lie, Đại số Lie, MD5-nhóm, MD5-đại số, K-quỹ đạo, Phân lá, Phân đo được, C*-đại số, C*-đại số Connes liên kết với phân đo 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 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 Krepresentation) are the orbits of zero or maximal dimension The Lie algebra of an MDngroup is called an MDn-algebra (see [5, Section 4.1]) Combining methods of Kirillov and Connes, the first author studied MD4foliations associated with all indecomposable connected MD 4-groups in [16] Recently, Vu and Shum [17] have classified, up to isomorphism, all the 5-dimensional MDalgebras having commutative derived ideals In [18], we have given a topological classification of MD 5-foliations associated to the indecomposable connected and simply connected MD5-groups, such that MD5algebras of them have 4-dimensional commutative derived ideals There are exactly topological types of considered MD 5-foliations which are denoted by F1, F2, F3 All MD5foliations of type F1 are the trivial fibrations with connected fibre on 3dimensional sphere S3, so Connes’ C*-algebras C*( F1) of them are isomorphic to the C*-algebra C ( 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 →C0 ( X ) ⊗ K →C *( F3 ) →C0 ( Y ) ⊗ K →0 , 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 C0 ( V ) ⋊H , where H is a Lie group, then we can use the ThomConnes isomorphism to compute the connecting map δ0 , δ1 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 MD5groups G which are studied in [17] and [18] Then, the Lie algebra G of G 5,4,14(λ ,μ ,ϕ ) will be the one of the Lie algebras ) G5,4,14 (λ ,μ ,ϕ (see [17] or [18]) Namely, G is the Lie algebra generated X5} with G1 := [ G,G ] = □.X ⊕□.X ⊕□.X ⊕□.X ≅ □ and as follows  cos ϕ  sin ϕ  adX := ;  0  ad ∈ End − sin ϕ 0 cosϕ { X1, X2, X3, X4, by ( G 1) ≡ Mat X1 (□)  λ, μ ∈□ , μ > 0,ϕ ∈( 0,π ) λ − μ μ λ  0 We now recall the geometric description of the K-orbits of G in the dual space G* of G Let {X } * , X2*, X3*, X 4*, X 5* in G Denote by ΩF be the basis in G* dual to the basis { X1 , X2 , X3 , X4 , X5 F = ( α , β + iγ ,δ + iσ ) the K-orbit of G including } in G ≅ □ ×□ × □ ≅ □ * - If β + iγ = δ + iσ = then ΩF = { F} (the 0-dimension orbit), - If β + iγ + δ + iσ ≠ then ΩF is the 2-dimension orbit as follows Ω = {( ( ) x,F( β + iγ ) e a.eiϕ , ( δ + iσ ) ea( λ−iμ) ) ,□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 V = { ( x, y, z,t, s ) ∈ G * : y + z + t + s ≠ 0} ≅ □ × ( □ ) * Furthermore, all the foliations } ∈( 0; π ) , { ( V , F 5,4,14 (λ ,μ ,ϕ ) ), λ , μ ∈□ , μ > 0,ϕ 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 Kfunctors In this case, we   π  choose the foliation V   ,F    5,4,14 0,1,     In [18], we also describe the foliation V, F  V ,F π   5,4,14 0,1,   commutative Lie group □ 2 by suitable action of □  can be given by an action of the  Proposition 2.1 The foliation π    5,4,14 0,1,   Namely, we have the following assertion     2  on the manifold V    is given by the action Proof One needs only to verify that the foliation V ,F5,4,14 0,1, π       λ : □ × V →V of □ on V as follows  λ ( (r,a),( x , y + iz, t + is ) is) e−ia ) : = ( x + r,( ) , y + iz) e−ia , ( t + x, y + iz,t + is ) ∈V ≅ □ × ( □ × □ )  ≅ □ × ( □ )  Hereafter, for   π  simply, we write F3 instead of V   ,F where ( r, a ) ∈□ and (   5,4,14 0,1,    It is easy to see that the graph of F3 is identified with V × □ , 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 C0 ( V ) by □2 as follows C*(F3) ≅ C 0( V ) ⋊ □ C*(F3) as a single extension 3.1 Let V1, W1 be the following sub-manifolds of V V1= { ( x, y + iz, t + is) ∈V : t + is ≠ 0} ≅ □ ×□ × □ * , W1 = V \V1 = { ( x, y + iz,t + is) ∈V : t + is = 0} ≅ □ ×□ * It is easy to see that the action λ in Proposition 2.1 preserves the subsets V1, W1 Let i, μ be the inclusion and the restriction i : C0 ( V1 ) →C0 ( V ) , where each function of zero outside V1 μ : C0 ( V ) →C0 ( W1 ) C0 ( V1 ) is extended to the one of C0 ( V ) by taking the value of It is known a fact that i, μ are λ- equivariant and the following sequence is equivariantly exact: (3.1) →C ( V ) i→C ( V ) μ→C ( W ) →0 0 3.2 Now we denote by ( V1,F1 ) , respectively ( W1, F1 ) restrictions of the foliations F3 on V1 , W1 , Theorem 3.1 C*( F3) admits the following canonical extension ( γ) →J ‸ →C *( F ) i μ□→B →0 , where J = C* ( V ,F C B = C* ( W ,F 1 ≅C ) 1 ) (□× ) ⋊ λ ( W1 ) ) □2≅C □ λ( ⋊ ⊗K , □ + ⊗K , + □ and the homomorphism ‸i, μ□ ( ‸i f ) ( r, s ) □2≅ C 0 C ( V) 3) ≅ *( F C0 ⋊ (V) ≅ λ is defined by ( μ□ f ) ( r, s ) = if ( r, s ) , = μ f ( r, s ) Proof Note that the graph of F3 is identified with V × □ , so by [3, section 5], we have: J = C* ( 1V , 1F ) ≅0 C (1 V )λ ⋊ □ , B = C* ( W ,F 1 ≅C ) ( W1 ) ⋊ □2 λ From λ-equivariantly exact sequence in 3.1 and by [2, Lemma 1.1] we obtain the single extension ( γ1 ) Furthermore, the foliations ( V1,F1 ) and ( W1, F1 ) can be come from the submersions p :V ≅ □ × □ × □ * →□ × □ q ( x, re iϕ , r 'eiϕ' )  and + ( re iϕ :W ≅ □ × □ * →□ , r ') ( + x, reiϕ )  r Hence, by a result of [3, p.562], we get J = C* ( V1 ,F1 C ) B = C* ( W ,F 1 ≅C ) ≅0 ( V ⋊ ) ( W1 ) ⋊ □2≅ C λ (□× ) □2 ≅C λ( □ Computing the invariant system of □ ⊗K , + ) ⊗K + C * ( F3 ) Definition 4.1 The set of element { γ1} corresponding to the single extension in ( γ1 ) ( ) the Kasparov group Ext ( B, J ) is called the system of invariant of C * F ( ) C * ( F3 determines the so-called table type of ) and denoted by Index C * F3 Remark 4.2 Index ) C * ( F3 in the set of all single extension →J →E →B →0 The main result of the paper is the following ( ) Theorem 4.3 Index C * F3 = { γ 1} , where γ1 = ( 0,1) in the group Ext ( B, J ) = Hom ( □ , □ ) ⊕ Hom ( □ , □ ) To prove this theorem, we need some lemmas as follows Lemma 4.4 Set I =C and (□ × S1 ) A=C (S ) The following diagram is commutative → K ( ) I ( C( S3 )) → K j +1 A) j β2 k : C0 (□ j ( ) β2 ) I → β2 →K j ( C0 ( W1 ) ) →K j+1 ( C0 ( V1 ) ) j ∈□ / 2□ is the Bott isomorphism, Proof Let → K j β2 →K j ( C0 ( V1 ) ) →K j ( C0 ( V ) → where β2 ( → K × S ) →C ( S ) , v : C ( S ) →C ( S ) be the inclusion and restriction defined similarly as in 3.1 One gets the exact sequence ( ) →I k→C S v→A →0 Note that C ( V1 C ) ≅0 ( □ × □ + ) ⊗ C C0 ( V ) ≅ C ( □ ×□ C (W C (□× ) ) + ≅ 0 (□ × S1 ) (□× ) ≅C ⊗C ≅C + ⊗I □ + × S ) 0≅ C + □ ( □ ×□ ) ( S ) (□× ) □ ⊗C ( S ) ⊗A + So, the extension (3.1) can be identified to the following one →C (□ + + × □ ) ⊗ I Id ⊗k→0C ( +□ × □ ) ⊗ C ( S ) Id0 ⊗v→C (□ ×□ A →0 So, the assertion of lemma is derived from the naturalness of Bott isomorphism Remark 4.5 i) K ii) j K (C (□ × S1 ) ( C( S ) ) ≅ □, ) ≅ K0 j (C (S )) j ∈□ / 2□ ≅ □ , j ∈□ / 2□ ) ⊗ j iii) ( K C ( S1 ) ) ≅□ is generated by ϕ β by ϕ1β2 [ Id ] [ 1] , K ( C ( S ) ) ≅□ is generated (where is a unit element in C ) (S ϕ j , j ∈□ / 2□ , is the Thom-Connes ; isomorphism; Id is the identity of S1 ) Proof of Theorem 4.3 Recall that the extension to a six-term exact sequence ( γ1 ) in theorem 3.1 gives the rise K ( J ) →K ( C * ( F3 ) (4.1) ) →K ( B ) K1 ( C * ( F δ1 ) ) ← K1 ( B) ← K1 ( δ0 J) By [11, Theorem 4.14], the isomorphism Ext ( B, J ) ≅ Hom□ ( K ( B) , K ( J ) ) ⊕ Hom□ ( K ( B) ,K ( J)) associates the invariant γ1 ∈ Ext ( B, J ) to the pair ( δ , δ1 ) ∈ 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 ( j ∈□ / 2□ ) : → K j ( J ) → K j ( C * ( F3 ) ( B) ϕj ) → K j+1 ( J ) → →K j ϕj ϕ ϕj j +1 →K j ( C0 ( V1 ) ) →K j ( C0 ( V ) ) →K j ( C0 ( W1 ) ) →K j +1 ( C0 ( V1 ) → In view of Lemma 4.4, the following diagram is commutative →K j ( C0 ( V1 ) )) ) →K j ( C0 ( V ) ) →K j ( C0 ( W1 ) ) →K j +1 ( C0 ( V1 → β2 → K j ( I) β2 → K )) (C( S j → β2 β2 → Kj( K j+1 ( A) Consequently, instead of computing the pair sum → I) ( δ0 ,δ1 ) Hom□ ( K0 ( B) , K1 ( J ) ) ⊕Hom□ ( K1 ( B) , K0 ( J ) ) , from the direct it is sufficient to compute the pair ( δ , δ1 ) ∈ Hom□ ( K0 ( A) , K1 ( I ) ) ⊕ Hom□ ( K1 ( A) , K0 ( I ) words, the sixterm exact sequence (4.1) can be identified with the following one ) In other ) ( (□ K C0 (4.2) × S1 ) ) (C( S ))  →K δ1 (C( S )) δ0 ( C ( S ) ) ← K ( C ( S ) ) ← ( □ (C K K → K 1 × S1 ) ) By remark 3.5, this sequence becomes □ → □ → □ (4.3)δ δ0 □ ← □ ← □ By the exactness, the sequence (4.3) will be the one of the following ones □ 0→ □ 1→ □ δ1 = δ0 = □ ←0 □ ←1 □ or 1→ □ □ 0→ □ δ0 = δ1 = □ ←1 □ ←0 □ ( C ( S1 ) ) , b = a 0  ∈ GL ( C ( S ) ) Now we choose a = eiϕ ∈ GL e a ⊕b =  iϕ −1 Then −iϕ  e  Let u = u ( x, y, z, t ) = u ( cosθ1 cosθ2 cosϕ, cosθ1 cosθ2 sinϕ, cosθ1 sinθ2 ,sinθ1 )  eiϕ e iθ cosθ =  sin  θ  − sin θ2 e − iϕ e − iθ cosθ2 ∈ GL0  − is a pre-image of a ⊕ b So, u =   e−iϕ e −iθ cosθ  Let q = I ⊕ = 1  0    02 2 ( C( S ) ) sinθ2   eiϕ eiθ cosθ2  −sinθ We get −  cos θ p =uqu = −iϕ −iθ1  e e cos2θ sinθ eiϕeiθ1 cosθ sinθ  2 sin θ2 2 ( (  ∈P C □ × S  )) + Then rank (p) = So δ ( [ a]1 ) = [ p] − [ I1 K- theoretical exact sequence associate to □ 0→ □ 1→ □ δ1 = □ δ0 = ←0 □ ← □ The proof is completed ( γ1 ) ] is ≠ ∈K ( C( □ × S1 ) ) Therefore, REFERENCES Brown L G.; Douglas R G., Fillmore P A (1977), “Extension of C*-algebra and Khomology”, Ann of Math, 105, pp 265 – 324 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 Connes A (1982), “A Survey of Foliations and Operator Algebras”, Proc Sympos Pure Mathematics, 38, pp 521 – 628 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) Diep D N (1999), Method of Non-commutative Geometry for Group C*-algebras Research Notes in Mathematics Series, vol 416 Cambridge: Chapman and HallCRC Press 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) Kasparov G G (1981), “The operator K-functor and extensions of C*-algebras”, Math USSR Izvestija, 16(3), pp 513 – 572 Kirillov A A (1976), Elements of the Theory of Representations, Springer – Verlag Pub., Berlin – Heidenberg – New York Rordam, M., Larsen F., Laustsen N (2000), An Introduction to K –Theory for C*– Algebras, Cambridge University Press, United Kingdom Rosenberg J (1976), “The C*-algebras of some real p-adic solvable groups”, Pacific J Math., 65(1), pp 175 – 192 Rosenberg J (1982), “Homological invariants of extension of C*-algebras”, Proc Sympos Pure Math., 38, AMS Providence R.I., pp 35 – 75 10 11 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) 17 18 19 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 Vu L A., Hoa D Q (2009), “The topology of foliations formed by the generic Korbits of a subclass of the indecomposable MD5-groups”, Science in China, series A: Mathematics, 52 (2), pp 351-360 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  ... have given a topological classification of MD 5-foliations associated to the indecomposable connected and simply connected MD5- groups, such that MD 5algebras of them have 4-dimensional commutative... 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... for Crossed Products of a C* –algebra by an Action of □ ”, Adv In Math., 39, pp 31 – 55 Connes A (1982), ? ?A Survey of Foliations and Operator Algebras? ??, Proc Sympos Pure Mathematics, 38, pp 521 –