K-Theory for the Leaf Space of Foliations formed by the Generic K-Ornits of some indecomposable M D 5 -Groups
aa r X i v : . [ m a t h . K T ] J a n K-THEORY OF THE LEAF SPACE OF FOLIATIONSFORMED BY THE GENERIC K-ORBITS OF SOMEINDECOMPOSABLE
M D -GROUPS Le Anh Vu* and Duong Quang Hoa**Department of Mathematics and InformaticsUniversity of Pedagogy, Ho Chi Minh City, VietnamE-mail: (*) [email protected](**) [email protected]
Abstract
The paper is a continuation of the authors’ work [18]. In [18], we consider foliationsformed by the maximal dimensional K-orbits (
M D -foliations) of connected M D -groups such that their Lie algebras have 4-dimensional commutative derived ideals andgive a topological classification of the considered foliations. In this paper, we study K-theory of the leaf space of some of these M D -foliations and characterize the Connes’C*-algebras of the considered foliations by the method of K-functors. INTRODUCTION
In the decades 1970s − M D n -group) is an n-dimensional solvable real Lie group whose orbits in the co-adjointrepresentation (i.e., the K- representation) are the orbits of zero or maximal dimension. TheLie algebra of an M D n -group is called an M D n -algebra (see [5, Section 4.1]).In 1982, studying foliated manifolds, A. Connes [3] introduced the notion of C*-algebraassociated to a measured foliation. In the case of Reeb foliations (see A. M. Torpe [14]),the method of K-functors has been proved to be very effective in describing the structure ofConnes’ C*-algebras. For every MD-group G, the family of K-orbits of maximal dimensionforms a measured foliation in terms of Connes [3]. This foliation is called MD-foliationassociated to G. Key words : Lie group, Lie algebra,
M D -group, M D -algebra, K-orbit, Foliation, Measured foliation,C*-algebra, Connes’ C*-algebra associated to a measured foliation.2000AMS Mathematics Subject Classification: Primary 22E45, Secondary 46E25, 20C20. M D -foliations associated with all indecomposable connected M D -groups and characterized Connes’ C*-algebras of these foliations in [16]. Recently, Vuand Shum [17] have classified, up to isomorphism, all the M D -algebras having commutativederived ideals.In [18], we have given a topological classification of M D -foliations associated to theindecomposable connected and simply connected M D -groups, such that M D -algebras ofthem have 4-dimensional commutative derived ideals. There are exactly 3 topological typesof the considered M D -foliations, denoted by F , F , F . All M D -foliations of type F arethe trivial fibrations with connected fibre on 3-dimesional sphere S , so Connes’ C*-algebrasof them are isomorphic to the C*-algebra C ( S ) ⊗ K following [3, Section 5], where K denotesthe C*-algebra of compact operators on an (infinite dimensional separable) Hilbert space.The purpose of this paper is to study K-theory of the leaf space and to characterize thestructure of Connes’ C*-algebras C ∗ ( V, F ) of all M D -foliations ( V, F ) of type F by themethod of K-functors. Namely, we will express C ∗ ( V, F ) by two repeated extensions of theform 0 / / C ( X ) ⊗ K / / C ∗ ( V, F ) / / B / / , / / C ( X ) ⊗ K / / B / / C ( Y ) ⊗ K / / , then we will compute the invariant system of C ∗ ( V, F ) with respect to these extensions. Ifthe given C*-algebras are isomorphic to the reduced crossed products of the form C ( V ) ⋊ H ,where H is a Lie group, we can use the Thom-Connes isomorphism to compute the connectingmap δ , δ .In another paper, we will study the similar problem for all M D -foliations of type F . M D − FOLIATIONS OF TYPE F M D -groups which associate with M D -foliations of type F (see [18]).In this section, G will be always an connected and simply connected M D -group suchthat its Lie algebras G is an indecomposable M D -algebra generated by { X , X , X , X , X } with G := [ G , G ] = R .X ⊕ R .X ⊕ R .X ⊕ R .X ∼ = R , ad X ∈ End ( G ) ≡ M at ( R ). Namely, G will be one of the following Lie algebras which are studied in [17] and [18].1. G , , λ ,λ ,ϕ ) ad X = cos ϕ − sin ϕ ϕ cos ϕ λ
00 0 0 λ ; λ , λ ∈ R \ { } , λ = λ , ϕ ∈ (0 , π ) . G , , λ,ϕ ) ad X = cos ϕ − sin ϕ ϕ cos ϕ λ
00 0 0 λ ; λ ∈ R \ { } , ϕ ∈ (0 , π ) . G , , λ,ϕ ) ad X = cos ϕ − sin ϕ ϕ cos ϕ λ
10 0 0 λ ; λ ∈ R \ { } , ϕ ∈ (0 , π ) . The connected and simply connected Lie groups corresponding to these algebras aredenoted by G λ , λ , ϕ ) , G λ , ϕ ) , G λ , ϕ ) . All of these Lie groups are M D -groups(see [17]) and G is one of them. We now recall the geometric description of the K-orbits ofG in the dual space G ∗ of G . Let { X ∗ , X ∗ , X ∗ , X ∗ , X ∗ } be the basis in G ∗ dual to the basis { X , X , X , X , X } in G . Denote by Ω F the K-orbit of G including F = ( α, β + iγ, δ, σ ) in G ∗ ∼ = R . • If β + iγ = δ = σ = 0 then Ω F = { F } (the 0-dimensional orbit). • If | β + iγ | + δ + σ = 0 then Ω F is the 2-dimensional orbit as followsΩ F = n(cid:16) x, ( β + iγ ) .e ( a.e − iϕ ) , δ.e aλ , σ.e aλ (cid:17) , x, a ∈ R o when G = G λ ,λ ,ϕ ) , λ , λ ∈ R ∗ , ϕ ∈ (0; π ) . n(cid:16) x, ( β + iγ ) .e ( a.e − iϕ ) , δ.e aλ , σ.e aλ (cid:17) , x, a ∈ R o when G = G λ,ϕ ) , λ ∈ R ∗ , ϕ ∈ (0; π ) . n(cid:16) x, ( β + iγ ) .e ( a.e − iϕ ) , δ.e aλ , δ.ae aλ + σ.e aλ (cid:17) , x, a ∈ R o when G = G λ,ϕ ) , λ ∈ R ∗ , ϕ ∈ (0; π ) . In [18], we have shown that, the family F of maximal-dimensional K-orbits of G formsmeasured foliation in terms of Connes on the open submanifold V = (cid:8) ( x, y, z, t, s ) ∈ G ∗ : y + z + t + s = 0 (cid:9) ∼ = R × (cid:0) R (cid:1) ∗ ( ⊂ G ∗ ≡ R )Furthermore, all foliations (cid:0) V, F , λ ,λ ,ϕ ) (cid:1) , (cid:0) V, F , λ,ϕ ) (cid:1) , (cid:0) V, F , λ,ϕ ) (cid:1) are topologicallyequivalent to each other ( λ , λ , λ ∈ R \ { } , ϕ ∈ (0; π )). Thus, we need only choose a envoyamong them to describe the structure of the C*-algebra. In this case, we choose the foliation (cid:16) V, F , ( , π ) (cid:17) .In [18], we have described the foliation (cid:16) V, F , ( , π ) (cid:17) by a suitable action of R . Namely,we have the following proposition. 3 ROPOSITION 1.
The foliation (cid:16) V, F , ( , π ) (cid:17) can be given by an action of the commu-tative Lie group R on the manifold V .Proof. One needs only to verify that the following action λ of R on V gives the foliation (cid:16) V, F , ( , π ) (cid:17) λ : R × V → V (( r, a ) , ( x, y + iz, t, s )) ( x + r, ( y + iz ) .e − ia , t.e a , s.e a )where ( r, a ) ∈ R , ( x, y + iz, t, s ) ∈ V ∼ = R × ( C × R ) ∗ ∼ = R × ( R ) ∗ . Hereafter, forsimplicity of notation, we write ( V, F ) instead of (cid:16) V, F , ( , π ) (cid:17) .It is easy to see that the graph of ( V, F ) is indentical with V × R , so by [3, Section 5],it follows from Proposition 1 that: COROLLARY 1 (analytical description of C ∗ ( V, F )) . The Connes C ∗ -algebra C ∗ ( V, F ) can be analytically described the reduced crossed of C ( V ) by R as follows C ∗ ( V, F ) ∼ = C ( V ) ⋊ λ R . (cid:3) C ∗ ( V, F ) AS TWO REPEATED EXTENSIONS V , W , V , W be the following submanifolds of V V = { ( x, y, z, t, s ) ∈ V : s = 0 } ∼ = R × R × R × R ∗ ,W = V \ V = { ( x, y, z, t, s ) ∈ V : s = 0 } ∼ = R × ( R ) ∗ × { } ∼ = R × ( R ) ∗ ,V = { ( x, y, z, t, ∈ W : t = 0 } ∼ = R × R × R ∗ ,W = W \ V = { ( x, y, z, t, ∈ W : t = 0 } ∼ = R × ( R ) ∗ . It is easy to see that the action λ in Proposition 1 preserves the subsets V , W , V , W .Let i , i , µ , µ be the inclusions and the restrictions i : C ( V ) → C ( V ) , i : C ( V ) → C ( W ) ,µ : C ( V ) → C ( W ) , µ : C ( W ) → C ( W )where each function of C ( V ) (resp. C ( V )) is extented to the one of C ( V ) (resp. C ( W ))by taking the value of zero outside V (resp. V ).It is known a fact that i , i , µ , µ are λ -equivariant and the following sequences areequivariantly exact:(2.1.1) 0 / / C ( V ) i / / C ( V ) µ / / C ( W ) / / / / C ( V ) i / / C ( W ) µ / / C ( W ) / / . V , F ) , ( W , F ) , ( V , F ) , ( W , F ) the foliations-restrictions of( V, F ) on V , W , V , W respectively. THEOREM 1. C ∗ ( V, F ) admits the following canonical repeated extensions ( γ ) 0 / / J b i / / C ∗ ( V, F ) c µ / / B / / , ( γ ) 0 / / J b i / / B c µ / / B / / , where J = C ∗ ( V , F ) ∼ = C ( V ) ⋊ λ R ∼ = C ( R ∪ R ) ⊗ K,J = C ∗ ( V , F ) ∼ = C ( V ) ⋊ λ R ∼ = C ( R ∪ R ) ⊗ K,B = C ∗ ( W , F ) ∼ = C ( W ) ⋊ λ R ∼ = C ( R + ) ⊗ K,B = C ∗ ( W , F ) ∼ = C ( W ) ⋊ λ R , and the homomorphismes b i , b i , b µ , b µ are defined by (cid:16) b i k f (cid:17) ( r, s ) = i k f ( r, s ) , k = 1 , c µ k f ) ( r, s ) = µ k f ( r, s ) , k = 1 , Proof.
We note that the graph of ( V , F ) is indentical with V × R , so by [3, section 5], J = C ∗ ( V , F ) ∼ = C ( V ) ⋊ λ R . Similarly, we have B ∼ = C ( W ) ⋊ λ R ,J ∼ = C ( V ) ⋊ λ R ,B ∼ = C ( W ) ⋊ λ R , From the equivariantly exact sequences in 2.1 and by [2, Lemma 1.1] we obtain therepeated extensions ( γ ) and ( γ ).Furthermore, the foliation ( V , F ) can be derived from the submersion p : V ≈ R × R × R × R ∗ → R ∪ R p ( x, y, z, t, s ) = ( y, z, t, sign s ) . Hence, by a result of [3, p.562], we get J ∼ = C ( R ∪ R ) ⊗ K . The same argument showsthat J ∼ = C (cid:0) R ∪ R (cid:1) ⊗ K, B ∼ = C ( R + ) ⊗ K. COMPUTING THE INVARIANT SYSTEM OF C ∗ ( V, F ) DEFINITION.
The set of elements { γ , γ } corresponding to the repeated extensions ( γ ),( γ ) in the Kasparov groups Ext ( B i , J i ) , i = 1 , C ∗ ( V, F ) and denoted by Index C ∗ ( V, F ). REMARK.
Index C ∗ ( V, F ) determines the so-called stable type of C ∗ ( V, F ) in the set of allrepeated extensions 0 / / J / / E / / B / / , / / J / / B / / B / / . The main result of the paper is the following.
THEOREM 2.
Index C ∗ ( V, F ) = { γ , γ } , where γ = (cid:18) (cid:19) in the group Ext ( B , J ) = Hom ( Z , Z ) ; γ = (1 , in the group Ext ( B , J ) = Hom ( Z , Z ) . To prove this theorem, we need some lemmas as follows.
LEMMA 1.
Set I = C ( R × R ∗ ) and A = C (cid:0) ( R ) ∗ (cid:1) The following diagram is commutative . . . / / K j ( I ) / / β (cid:15) (cid:15) K j (cid:0) C ( R ) ∗ (cid:1) / / β (cid:15) (cid:15) K j ( A ) / / β (cid:15) (cid:15) K j +1 ( I ) / / β (cid:15) (cid:15) . . .. . . / / K j +1 (cid:0) C ( V ) (cid:1) / / K j +1 (cid:0) C ( W ) (cid:1) / / K j +1 (cid:0) C ( W ) (cid:1) / / K j (cid:0) C ( V ) (cid:1) / / . . . where β is the isomorphism defined in [13, Theorem 9.7] or in [2, corollary VI.3], j ∈ Z / Z .Proof. Let k : I = C (cid:0) R × R ∗ (cid:1) −→ C (cid:0)(cid:0) R (cid:1) ∗ (cid:1) v : C (cid:0)(cid:0) R (cid:1) ∗ (cid:1) −→ A = C (cid:0)(cid:0) R (cid:1) ∗ (cid:1) be the inclusion and restriction defined similarly as in 2.1.One gets the exact sequence0 / / I k / / C (cid:0) ( R ) ∗ (cid:1) v / / A / / C ( V ) ∼ = C (cid:0) R × R × R ∗ (cid:1) ∼ = C ( R ) ⊗ I ,C ( W ) ∼ = C (cid:0) R × (cid:0) R (cid:1) ∗ (cid:1) ∼ = C ( R ) ⊗ A ,C ( W ) ∼ = C (cid:0) R × (cid:0) R (cid:1) ∗ (cid:1) ∼ = C ( R ) ⊗ C (cid:0) R (cid:1) ∗ . / / C ( R ) ⊗ I id ⊗ k / / C ( R ) ⊗ C ( R ) ∗ id ⊗ v / / C ( R ) ⊗ A / / . Now, using [13, Theorem 9.7; Corollary 9.8] we obtain the assertion of Lemma 1.
LEMMA 2.
Set I = C ( R × R ∗ ) and A = C ( S ) The following diagram is commutative . . . / / K j ( I ) / / β (cid:15) (cid:15) K j (cid:0) C ( S ) (cid:1) / / β (cid:15) (cid:15) K j ( A ) / / β (cid:15) (cid:15) K j +1 ( I ) / / β (cid:15) (cid:15) . . .. . . / / K j ( C ( V )) / / K j ( C ( V )) / / K j ( C ( W )) / / K j +1 ( C ( V )) / / . . . where β is the Bott isomorphism, j ∈ Z / Z .Proof. The proof is similar to that of lemma 1, by using the exact sequence (2.1.1) anddiffeomorphisms: V ∼ = R × ( R ) ∗ ∼ = R × R + × S , W ∼ = R × ( R ) ∗ ∼ = R × R + × S .Before computing the K-groups, we need the following notations. Let u : R → S be themap u ( z ) = e πi ( z/ √ z ) , z ∈ R Denote by u + (resp. u − ) the restriction of u on R + (resp. R − ). Note that the class [ u + ](resp. [ u − ]) is the canonical generator of K ( C ( R + )) ∼ = Z (resp. K ( C ( R − )) ∼ = Z ). Let usconsider the matrix valued function p : ( R ) ∗ ∼ = S × R + → M ( C ) (resp. p : S ∼ = D/S → M ( C )) defined by: p ( x ; y ) ( resp. p ( x, y )) = 12 − cos π p x + y x + iy √ x + y sin π p x + y x − iy √ x + y sin π p x + y π p x + y . Then p (resp. p ) is an idempotent of rank 1 for each ( x ; y ) ∈ ( R ) ∗ (resp. ( x ; y ) ∈ D/S ).Let [ b ] ∈ K ( C ( R )) be the Bott element, [1] be the generator of K ( C ( S )) ∼ = Z . LEMMA 3 (See [15, p.234]) . (i) K ( B ) ∼ = Z , K ( B ) = 0 ,(ii) K ( J ) ∼ = Z is generated by ϕ β (cid:0) [ b ] ⊠ [ u + ] (cid:1) and ϕ β (cid:0) [ b ] ⊠ [ u − ] (cid:1) ; K ( J ) = 0 ,(iii) K ( B ) ∼ = Z is generated by ϕ β (cid:0) [1] ⊠ [ u + ] (cid:1) ; K ( B ) ∼ = Z is generated by ϕ β (cid:0) [ p ] − [ ε ] (cid:1) ,where ϕ j , j ∈ Z / Z , is the Thom-Connes isomorphism (see[2]), β is the isomorphism inLemma 1, ε is the constant matrix (cid:18) (cid:19) and ⊠ is the external tensor product (see, forexample, [2,VI.2]). LEMMA 4. (i) K (cid:0) C ∗ ( V, F ) (cid:1) ∼ = Z , K (cid:0) C ∗ ( V, F ) (cid:1) ∼ = Z , ii) K ( J ) = 0; K ( J ) ∼ = Z is generated by ϕ β (cid:0) [ b ] ⊠ [ u + ] (cid:1) and ϕ β (cid:0) [ b ] ⊠ [ u − ] (cid:1) ,(iii) K ( B ) = 0; K ( B ) ∼ = Z is generated by ϕ β [¯1] and ϕ β (cid:0) [¯ p ] − [ ε ] (cid:1) ,where ¯1 is unit element in C ( S ) , ϕ is the Thom-Connes isomorphism, β is the Bottisomorphism.Proof. (i) K i (cid:0) C ∗ ( V, F ) (cid:1) ∼ = K i (cid:0) C ( S ) (cid:1) ∼ = Z , i = 0 , K (cid:0) C ( S ) (cid:1) = Z [¯1] + Z [ q ] , where q ∈ P (cid:0) C ( S ) (cid:1) . Otherwise, in [9, p.48,53,56]; [13, p.162], one has shown that the map dim : K (cid:0) C ( S ) (cid:1) → Z is a surjective group homomorphism which satisfied dim[¯1] = 1 , ker(dim) = Z and non-zero element q ∈ P (cid:0) C ( S ) (cid:1) in the kernel of the map dim has the form [ q ] = [¯ p ] − [ ε ].Hence, the result is derived straight away because β and ϕ are isomorphisms. Proof of theorem 2
1. Computation of ( γ ). Recall that the extension ( γ ) in theorem 1 gives rise to a six-termexact sequence 0 = K ( J ) / / K (cid:0) C ∗ ( V, F ) (cid:1) / / K ( B ) δ (cid:15) (cid:15) K ( B ) δ O O K (cid:0) C ∗ ( V, F ) (cid:1) o o K ( J ) o o By [11, Theorem 4.14], the isomorphismsExt( B , J ) ∼ = Hom (cid:0) ( K ( B ) , K ( J ) (cid:1) ∼ = Hom( Z , Z )associates the invariant γ ∈ Ext( B , J ) to the connecting map δ : 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 ∈ Z / Z ): . . . / / K j ( J ) / / K j (cid:0) C ∗ ( V, F ) (cid:1) / / K j ( B ) / / K j +1 ( J ) / / . . .. . . / / K j (cid:0) C ( V ) (cid:1) / / ϕ j O O K j (cid:0) C ( V ) (cid:1) / / ϕ j O O K j (cid:0) C ( W ) (cid:1) / / ϕ j O O K j +1 (cid:0) C ( V ) (cid:1) / / ϕ j +1 O O . . . In view of Lemma 2, the following diagram is commutative . . . / / K j (cid:0) C ( V ) (cid:1) / / K (cid:0) C ( V ) (cid:1) / / K j (cid:0) C ( W ) (cid:1) / / K j +1 (cid:0) C ( V ) (cid:1) / / . . .. . . / / K j ( I ) / / β O O K j (cid:0) C ( S ) (cid:1) / / β O O K j ( A ) / / β O O K j +1 ( I ) / / β O O . . . δ : K ( B ) → K ( J ), it is sufficient to compute δ : K ( A ) → K ( I ). Thus, by the proof of Lemma 4, we have to define δ (cid:0) [¯ p ] − [ ε ] (cid:1) = δ (cid:0) [¯ p ] (cid:1) (because δ (cid:0) [ ε ] (cid:1) = (0; 0) and δ (cid:0) [¯1] (cid:1) = (0; 0)). By the usual definition (see[13,p.170]), for [¯ p ] ∈ K ( A ) , δ (cid:0) [¯ p ] (cid:1) = (cid:2) e πi ˜ p (cid:3) ∈ K ( I ) where ˜ p is a preimage of ¯ p in (amatrix algebra over) C ( S ), i.e. v ˜ p = ¯ p .We can choose ˜ p ( x, y, z ) = z √ z ¯ p ( x, y ) , ( x, y, z ) ∈ S .Let ˜ p + (resp. ˜ p − ) be the restriction of ˜ p on R × R + (resp. R × R − ). Then we have δ (cid:0) [¯ p ] (cid:1) = (cid:2) e πi ˜ p (cid:3) = (cid:2) e πi ˜ p + (cid:3) + (cid:2) e πi ˜ p − (cid:3) ∈ K (cid:16) C (cid:0) R (cid:1) ⊗ C (cid:0) R + (cid:1)(cid:17) ⊕ K (cid:16) C (cid:0) R (cid:1) ⊗ C (cid:0) R − (cid:1)(cid:17) = K ( I )By [13, Section 4], for each function f : R ± → Q n ^ C (cid:0) R (cid:1) such that lim x →± f ( t ) =lim x →±∞ f ( t ), where Q n ^ C (cid:0) R (cid:1) = n a ∈ M n ^ C (cid:0) R (cid:1) , e πia = Id o , the class [ f ] ∈ K (cid:0) C ( R ) ⊗ C ( R ± ) (cid:1) can be determined by [ f ] = W f . [ b ] ⊠ [ u ± ], where W f = 12 πi Z R ± T r (cid:0) f ′ ( z ) f − ( z ) (cid:1) dz is the winding number of f .By simple computation, we get δ (cid:0) [ p ] (cid:1) = [ b ] ⊠ [ u + ] + [ b ] ⊠ [ u − ]. Thus γ = (cid:16) (cid:17) ∈ Hom Z ( Z , Z ).2. Computation of ( γ ). The extension ( γ ) gives rise to a six-term exact sequence K ( J ) / / K ( B ) / / K ( B ) δ (cid:15) (cid:15) K ( B ) δ O O K ( B ) o o K ( J ) = 0 o o By [11, Theorem 4.14], γ = δ ∈ Hom (cid:0) K ( B ) , K ( J ) (cid:1) = Hom Z ( Z , Z ). Similarly topart 1, taking account of Lemma 1 and 3, we have the following commutative diagram( j ∈ Z / Z ) . . . / / K j ( J ) / / K j ( B ) / / K j ( B ) / / K j +1 ( J ) / / . . .. . . / / K j (cid:0) C ( V ) (cid:1) / / ϕ j O O K j (cid:0) C ( W ) (cid:1) / / ϕ j O O K j (cid:0) C ( W ) (cid:1) / / ϕ j O O K j +1 (cid:0) C ( V ) (cid:1) / / ϕ j +1 O O . . .. . . / / K j − ( I ) / / β O O K j − (cid:0) C ( R ) ∗ (cid:1) / / β O O K j − ( A ) / / β O O K j ( I ) / / β O O . . . Thus we can compute δ : K ( A ) → K ( I ) instead of δ : K ( B ) → K ( J ). By theproof of Lemma 3, we have to define δ (cid:0) [ p ] − [ ǫ ] (cid:1) = δ (cid:0) [ p ] (cid:1) (because δ (cid:0) [ ǫ ] (cid:1) = (0 , δ (cid:0) [ p ] (cid:1) = [ b ] ⊠ [ u + ] + [ b ] ⊠ [ u − ]. Thus γ = (1 , ∈ Hom Z ( Z , Z ) ∼ = Z . The proof is completed. (cid:3) EFERENCES
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