Bivariant K-theory of generalized Weyl algebras
aa r X i v : . [ m a t h . K T ] A p r BIVARIANT K-THEORY OF GENERALIZED WEYL ALGEBRAS
JULIO GUTI´ERREZ AND CHRISTIAN VALQUI
Abstract.
We compute the isomorphism class in KK alg of all noncommutative generalizedWeyl algebras A = C [ h ]( σ, P ), where σ ( h ) = qh + h is an automorphism of C [ h ], exceptwhen q = 1 is a root of unity. In particular, we compute the isomorphism class in KK alg ofthe quantum Weyl algebra, the primitive factors B λ of U ( sl ) and the quantum weightedprojective lines O ( WP q ( k, l )). Contents
Introduction 11. Basic results on locally convex algebras 31.1. Locally convex algebras 41.2. Diffotopies 41.3. Extensions of locally convex algebras 51.4. The algebra of smooth compact operators and the smooth Toeplitz algebra 72. Bivariant K-theory of locally convex algebras 102.1. Definition and properties of kk alg KK alg kk alg invariants of generalized Weyl algebras 184.1. The Toeplitz algebra of a smooth generalized crossed product 194.2. The case where P is a non constant polynomial 204.3. The case where P is a constant polynomial 27References 28 Introduction
In [7], Cuntz defined a bivariant K -theory kk alg in the category lca of locally convexalgebras. These are algebras A that are complete locally convex vector spaces over C witha jointly continuous multiplication. To a pair of locally convex algebras A and B , there Mathematics Subject Classification.
Primary 46L87; Secondary 19K35, 58B34.
Key words and phrases.
K-theory, kk-theory, smooth generalized crossed products, generalized Weylalgebras.Julio Guti´errez was supported by Cienciactiva CG 217-2014.Christian Valqui was supported by PUCP-DGI-2017-1-0035. correspond abelian groups kk algn ( A, B ), n ∈ Z and there are bilinear maps kk algn ( A, B ) × kk algm ( B, C ) → kk algn + m ( A, C )for every
A, B and C locally convex algebras and m, n ∈ Z . Using this product, we candefine a category KK alg whose objects are locally convex algebras and whose morphisms aregiven by the graded groups kk alg ∗ ( A, B ). Then the bivariant K -theory kk alg can be seen as afunctor kk alg : lca → KK alg . This functor is universal among split exact, diffotopy invariantand stable functors (see Theorem 7.26 in [8]). In particular, an isomorphism in KK alg inducesan isomorphism in KK L p (see Definition 2.7) and in HP .Joachim Cuntz initiated the construction of different bivariant K-theories in several cate-gories (see [6], [7] and [9]), and in [7] he proved that the Weyl algebra W = C h x, y | xy − yx = 1 i is isomorphic to C in KK alg . By the universal property of KK alg this implies KK L p ( C , W ) = Z and KK L p ( C , W ) = 0.On the other hand, in [10], exact sequences analog to the Pimsner-Voiculescu exact se-quence were constructed for smooth generalized crossed products that satisfy the conditionof being tame smooth. We shall consider generalized Weyl algebras over C [ h ] which aresmooth generalized crossed products (but are not tame smooth in general). Definition . Let D be a ring, σ ∈ Aut ( D ) and a a central element of D . The generalizedWeyl algebra D ( σ, a ) is the algebra generated by x and y over D satisfying xd = σ ( d ) x, yd = σ − ( d ) y, yx = a and xy = σ ( a )for all d ∈ D .In this article, we compute the isomorphism class in KK alg of all non commutative gener-alized Weyl algebras A = C [ h ]( σ, P ), where σ ( h ) = qh + h is an automorphism of C [ h ] and P ∈ C [ h ], except when q = 1 is a root of unity. In the table below we list all posible casesfor A and our results.Conditions Results Observation P is constant P = 0 A ∼ = KK alg C Prop 4.22 A N -graded P = 0 A ∼ = KK alg S C ⊕ C Prop 4.21 A tame smooth P is nonconstantwith r distinctroots q not a root of unity A ∼ = KK alg C r Thm 4.18Prop 4.20 q = 1 and h = 0 A ∼ = KK alg C r Thm 4.18 q = 1, a root of unity No result q = 1 and h = 0 No result A commutativeA generalized Weyl algebra A = C [ h ]( σ, P ) is tame smooth if and only if P ( h ) ∈ C [ h ] is anon zero constant polynomial (see Remark 4.3). Hence, if P is a non constant polynomial, A is a generalized crossed product that is not tame smooth, and so we cannot use the results of[10]. However, in most cases we can construct an explicit faithful representation of A , whichallows us to follow the general strategy of [7] and [10], in order to determine the KK alg classof A .Our main result is Theorem 4.18, which computes the isomorphism class of A in KK alg inthe following two cases: • q = 1 and h = 0. • q is not a root of unity and P has a root different from h − q . IVARIANT K-THEORY OF GENERALIZED WEYL ALGEBRAS 3
In each of these cases we construct an exact triangle SA → A A − → A → A, (0.1)in the triangulated category ( KK alg , S ), where A n is the subspace of degree n of the Z -gradedalgebra A (see Lemma 3.2). In order to construct the exact triangle in (0.1) we follow themethods of [10]: we construct a linearly split extension0 → Λ A → T A → A → T A ∼ = KK alg A and Λ A ∼ = KK alg A A − . The exact triangle in (0.1) yields A ∼ = KK alg A ⊕ S ( A A − ). The main result now followsafter we prove A A − ∼ = KK alg S C r − in Proposition 4.17, since A = C [ h ] ∼ = KK alg C .The main result allows to compute the isomorphism class in KK alg of the quantum Weylalgebra, the primitive factors B λ of U ( sl ) and the quantum weighted projective lines O ( WP q ( k, l )) (see [4]).For the sake of completeness we also discuss the case of N -graded and the case of tamesmooth generalized Weyl algebras.In the case where A = L n ∈ N A n is an N -graded locally convex algebra it can be shownthat A ∼ = KK alg A . This is the case when • P is nonconstant, q is not a root of unity and P has only h − q as a root or • P = 0.In these cases we obtain A ∼ = KK alg C .In the case where P is a nonzero constant polynomial, A is a tame smooth generalizedcrossed product and the results from [10] apply. In this case there is an exact triangle SA → A → A → A, (0.2)in the triangulated category ( KK alg , S ) and we obtain A ∼ = KK alg S C ⊕ C .In the case where q = 1 and h = 0, we have σ = id and so A ∼ = C [ h, x, y ] / ( xy − P ) is acommutative algebra. This case and the case where q = 1 is a root of unity remain open.The article is organized as follows. In section 1, we recall basic results on locally convexalgebras. Lemma 1.18 is a technical result which asserts that the projective tensor productof the Toeplitz algebra T with an algebra with a countable basis is the algebraic tensorproduct. In section 2 we recall the definition and properties of kk alg following [7] and [9]. Insection 3, we define generalized Weyl algebras and construct explicit faithful representationswhen q = 1 and h = 0, and when q is not a root of unity and P has a root different from h − q .In section 4, we compute the isomorphism class in KK alg of all noncommutative generalizedWeyl algebras A = C [ h ]( σ, P ) where σ ( h ) = qh + h except when q = 1 is a root of unity.1. Basic results on locally convex algebras
In this section we recall some constructions in the category of locally convex algebras thatare needed for the definition of the bivariant K -theory kk alg . We follow the discussions in[10] and [7]. In Lemma 1.18 we prove that the projective tensor product of the Toeplitzalgebra T with an algebra with a countable basis is the algebraic tensor product. JULIO GUTI´ERREZ AND CHRISTIAN VALQUI
Locally convex algebras.
Definition . A locally convex algebra A is a complete locally convex vector space over C which is an algebra such that for any continuous seminorm p in A there is a continuousseminorm q in A such that p ( ab ) ≤ q ( a ) q ( b ) for all a, b ∈ A . This is equivalent to requiringthe multiplication to be jointly continuous.A seminorm p of A is called submultiplicative if p ( ab ) ≤ p ( a ) p ( b ), for all a, b ∈ A . If thetopology of A can be defined by a family of submultiplicative seminorms we say that A isan m -algebra.Morphisms in the category of locally convex algebras are continuous homomorphisms. Wedenote by ⊗ π the projective tensor product of locally convex vector spaces (see chapter 43in [15]). This is a completion of the algebraic tensor product. The projective tensor productof two locally convex algebras is again a locally convex algebra. The following are examplesof locally convex algebras.(1) All algebras with a countable basis over C . These are locally convex algebras withthe topology generated by all seminorms (Proposition 2.1 in [7]). Examples includethe Weyl algebra and generalized Weyl algebras over C [ h ] (see Corollary 3.3).(2) C ∞ ([0 , p n ( f ) = || f || + || f ′ || + 12 || f ′′ || + · · · + 1 n ! || f ( n ) || where || f || = sup { f ( t ) | t ∈ [0 , } .(3) We define C [0 ,
1] as the (closed) subalgebra of C ∞ ([0 , A wehave C [0 , ⊗ π A = A [0 , A [0 ,
1] is the algebra of C ∞ functions with valuesin A and all derivatives vanishing at 0 and 1. We define A [0 ,
1) and A (0 ,
1) as thesubalgebras of A [0 ,
1] of functions that vanish at 1, and at 0 and 1 respectively.
Definition . We denote by SA and CA the algebras A (0 ,
1) and A [0 ,
1) and we call themthe suspension and the cone of A respectively.Note that S ( · ) is a functor. Given a morphism of locally convex algebras φ : A → B , thereis a morphism S ( φ ) : SA → SB defined by f φ ◦ f . We can iterate this functor n timesto obtain S n A and S n ( f ).1.2. Diffotopies.
An important feature of the bivariant K -theory kk alg is the invariance with respect todifferentiable homotopies. For more details on diffotopies consult Section 6.1 in [8]. Definition . Let φ , φ : A → B be homomorphisms of locally convex algebras. A diffotopybetween φ and φ is a homomorphism Φ : A → C ∞ ([0 , , B ) such that ev i ◦ Φ = φ i . If thereis a diffotopy between φ and φ we call them diffotopic and write φ ≃ φ .Using a reparameterization of the interval we can assume that all derivatives of Φ at 0 and1 vanish and therefore we can assume that a diffotopy is given by a map Φ : A → B [0 , Definition . Given two locally convex algebras A and B , we denote by h A, B i the set ofdiffotopy classes of continuous homomorphisms from A to B . Given φ : A → B a continuoushomomorphism, we denote by h φ i its diffotopy class. IVARIANT K-THEORY OF GENERALIZED WEYL ALGEBRAS 5
Lemma 1.5.
There is a group structure in h A, SB i given by concatenation. The groupstructures in h A, S n B i that we get from concatenation in different variables all agree and areabelian for n ≥ .Proof. See Lemma 6.4 in [8]. (cid:4)
Definition . A locally convex algebra A is called contractible if the identity map is diffo-topic to 0. Examples.
Examples of contractible locally convex algebras are h C [ h ] and CA . The diffo-topies are given by φ s : h C [ h ] → h C [ h ], φ s ( h ) = sh and ψ s : CA → CA , ψ s ( f )( t ) = f ( st ),respectively. Note that the algebras ( h − h ) C [ h ] are isomorphic to h C [ h ] and therefore arealso contractible.We conclude this section with a note on N -graded algebras. Lemma 1.7.
Let A = L n ∈ N A n be an N -graded locally convex algebra, then A is diffotopyequivalent to A . In particular C [ h ] is diffotopy equivalent to C .Proof. The diffotopy is given by the family of morphisms φ t : A → A , t ∈ [0 , a n ∈ A n to t n a n . When t = 1 we recover the identity and when t = 0 the morphismis a retraction of A onto A . (cid:4) Extensions of locally convex algebras.
In this section we define extensions of locally convex algebras and their classifying maps.Extensions play a key role in the definition of kk alg . Definition . An extension of locally convex algebras0 → I → E → B → s : B → E . Similarly we say that anextension of length n → I → E → · · · → E n → B → − ds + sd = id , where d is the differential of the chain complex. Example.
Let A be a locally convex algebra. The extension0 → SA → CA → A → A . It is linearly split with continuous linear section s : A → CA given by a ∈ A f ∈ CA with f ( t ) = (1 − ψ ( t )) a , where ψ : [0 , → [0 ,
1] is a C ∞ bijectionwith f (0) = 0, f (1) = 1 and all derivatives vanishing at 0 and at 1.Now, we define the tensor algebra which has a universal property in the category of locallyconvex algebras. It is a completion of the usual algebraic tensor algebra. Let V be a completelocally convex vector space. The algebraic tensor algebra is defined as T alg V = ∞ M n =1 V ⊗ n . Notice that we are considering a non-unital algebraic tensor algebra. There is a linear map σ : V → T alg V mapping V into the first summand. We topologize T alg V with all seminormsof the form α ◦ φ , where φ is any homomorphism from T alg V into a locally convex algebra B such that φ ◦ σ is continuous on V and α is a continuous seminorm on B . JULIO GUTI´ERREZ AND CHRISTIAN VALQUI
Definition . The tensor algebra
T V is the completion of T alg V with respect to the familyof seminorms { α ◦ φ } defined above.The tensor algebra T V is a locally convex algebra that satisfies the following universalproperty.
Proposition 1.10.
Given a continuous linear map s : V → B from a complete locally convexvector space V to a locally convex algebra B there is a unique morphism of locally convexalgebras τ : T V → B such that τ ◦ σ = s . The morphism τ is defined by τ ( x ⊗ x ⊗· · ·⊗ x n ) = s ( x ) s ( x ) . . . s ( x n ) where x i ∈ V .Proof. See Lemma 6.9 in [8]. (cid:4)
In particular, if A is a locally convex algebra, the identity map id : A → A induces amorphism π : T A → A .We use the universal property of T A to construct a universal extension. There is anextension 0 → J A → T A π → A → J A is defined as the kernel of π : T A → A , which has a canonical continuous linearsection given by σ : A → T A . This extension is universal in the sense that given anyextension of locally convex algebras 0 → I → E → B → s and a morphism α : A → B , there is a morphism of extensions0 / / J A / / γ (cid:15) (cid:15) T A / / τ (cid:15) (cid:15) A / / α (cid:15) (cid:15) / / I / / E / / B / / τ : T A → E is the morphism induced by the continuous linear map s ◦ α : A → E and γ : J A → I is the restriction of τ .Notice that J ( · ) is a functor. Given a morphism α : A → B , consider the extension0 → J B → T B → B → J ( α ) : J A → J B in the natural way. We can iterate this construction n times to obtain J n A and J n ( α ).We observe that the map γ : J A → I is unique up to diffotopy. Given two continuouslinear sections s and s , the smooth family of continuous linear sections s t = ts + (1 − t ) s induces a diffotopy γ t which connects s and s . Hence the corresponding γ ’s are diffotopic. Definition . The morphism γ : J A → I is called the classifying map of the extension0 → I → E → B → α : A → B . It is well defined up to diffotopy.Similarly, we can define the classifying map of an extension0 → I → E → · · · → E n → B → α : A → B to be the map γ : J n A → I in0 / / J n A / / γ (cid:15) (cid:15) T ( J n − A ) / / (cid:15) (cid:15) · · · / / T A / / (cid:15) (cid:15) A / / α (cid:15) (cid:15) / / I / / E / / · · · / / E n / / B / / IVARIANT K-THEORY OF GENERALIZED WEYL ALGEBRAS 7
The algebra of smooth compact operators and the smooth Toeplitz algebra.
We define the algebra K of smooth compact operators which play the role of the C * -algebra of compact operators used in Kasparov’s KK -theory. Then we define the smoothToeplitz algebra T and prove that the projective tensor product of T with an algebra witha countable basis is the algebraic tensor product. Definition . The algebra of smooth compact operators K is defined as the algebra of N × N matrices a = ( a ij ) such that p n ( a ) = P i,j ∈ N (1 + i + j ) n | a i,j | is finite for n ∈ N . Thetopology is defined by the seminorms p n .The algebra K with the seminorms p n is a locally convex algebra, which is isomorphic tothe space s of rapidly decreasing sequences as a locally convex vector space. Lemma 1.13.
The locally convex spaces K , s ⊗ π s and s ⊕ s are isomorphic to s .Proof. The proofs of these facts can be found in [16] Chapter 3 Section 1.1. (cid:4)
We also define the smooth Toeplitz algebra which plays the role of the Toeplitz C * -algebra.The Fourier series gives an isomorphism of locally convex spaces between C ∞ ( S ) and thespace of rapidly decreasing Laurent series (see Theorem 51.3 in [15]) C ∞ ( S ) ∼ = (X i ∈ Z a i z i | X i ∈ Z | i | n | a i | < ∞ , ∀ n ∈ N ) , where z corresponds to the function z : S → C , z ( t ) = t . This space is isomorphic to thespace s of rapidly decreasing sequences. Definition . The smooth Toeplitz algebra T is defined by the direct sum of locally convexvector spaces T = K ⊕ C ∞ ( S ). In order to define the multiplication, we define v k = (0 , z k )and write x for an element ( x,
0) with x ∈ K . We denote the elementary matrices in K by e ij and set e ij = 0 for all i, j <
0. The multiplication is defined by the following relations e ij e kl = δ jk e il , v k e ij = e ( i + k ) ,j , e ij v k = e i, ( j − k ) , for all i, j, k, l ∈ Z and v k v − l = ( v k − l (1 − e − e − . . . e l − ,l − ) , l > v k − l , l ≤ , for all k, l ∈ Z .Denote v and v − by S and S ∗ respectively.There is a linearly split extension0 → K → T → C ∞ ( S ) → C ∞ ( S ) → T is defined by z S .The smooth Toeplitz algebra is generated, as a locally convex algebra, by S and S ∗ . Infact, it satisfies a universal property in the category of m -algebras. Lemma 1.15 (Satz 6.1 in [6]) . T is the universal unital m -algebra generated by two ele-ments S and S ∗ satisfying the relation S ∗ S = 1 whose topology is defined by a family ofsubmultiplicative seminorms { p n } n ∈ N with the condition that there are positive constants C n such that p n ( S k ) ≤ C n (1 + k n ) and p n ( S ∗ n ) ≤ C n (1 + k n ) . (1.1) JULIO GUTI´ERREZ AND CHRISTIAN VALQUI
The following diffotopy is due to [6]. In the context of C * -algebras a homotopy like thisone is used to prove Bott periodicity and the Pimsner-Voiculescu sequence. Lemma 1.16 (Lemma 6.2 in [6]) . There is a unital diffotopy φ t : T → T ⊗ π T such that φ t ( S ) = S S ∗ ⊗ f ( t )( e ⊗ S ) + g ( t )( Se ⊗ φ t ( S ∗ ) = SS ∗ ⊗ f ( t )( e ⊗ S ∗ ) + g ( t )( eS ∗ ⊗ where f, g ∈ C [0 , are such that f (0) = 0 , f (1) = 1 , g (0) = 1 and g (1) = 0 . Note that φ ( S ) = S ⊗ φ ( S ) = S S ∗ ⊗ e ⊗ S . Lemma 1.15 implies that, in orderto define a morphism from T to T ⊗ π T , we only need to check the relations on S and S ∗ and the bounds of (1.1).We finish this section with a result for tensoring algebras with a countable basis over C equipped with the fine topology and the Toeplitz algebra. Although the result is knownto experts, we give all the details since it allows us to prove Lemma 1.18, which is a keyingredient in Proposition 4.5, one of our main technical results. Lemma 1.17.
The locally convex space A ⊗ π s is isomorphic to the space F of sequences { x n } n ∈ N ⊆ A such that || x || ρ,k = X n ∈ N | n | k ρ ( x ( n )) is finite for all k ∈ N and any continuous seminorm ρ on A , where the topology on F isdefined by the seminorms || · || ρ,k .Proof. There is an inclusion φ : A ⊗ s → F defined by a ⊗ α ∈ A ⊗ s
7→ { x n = α n a } ∈ F .Let z = P Nt =1 a ( t ) ⊗ α ( t ) be an element of A ⊗ s . We have || φ ( z ) || ρ,k = X n ∈ N ρ N X t =1 a ( t ) α ( t ) n ! | n | k ≤ X n ∈ N N X t =1 ρ ( a ( t ) ) | α ( t ) n || n | k = N X t =1 ρ ( a ( t ) ) p k ( α ( t ) ) . This implies || φ ( z ) || ρ,k ≤ ( ρ ⊗ p k )( z ). We can write z = P n ∈ N P Nt =1 a ( t ) α ( t ) n ⊗ e n and therefore( ρ ⊗ p k )( z ) ≤ X n ∈ N ( ρ ⊗ p k ) N X t =1 a ( t ) α ( t ) n ⊗ e n ! = X n ∈ N ρ N X t =1 a ( t ) α ( t ) n ! | n | k = || φ ( z ) || ρ,k . This implies that || · || ρ,k = ρ ⊗ p k in the image of A ⊗ s . Since all finite sequences in A arein A ⊗ s , A ⊗ s is dense in F . Since F is a complete space, we conclude A ⊗ π s = F . (cid:4) IVARIANT K-THEORY OF GENERALIZED WEYL ALGEBRAS 9
Lemma 1.18.
Let s be the locally convex space of rapidly decreasing sequences and A analgebra with a countable basis over C equipped with the fine topology. Then we have A ⊗ π s = A ⊗ s as locally convex spaces. This implies that A ⊗ π T = A ⊗ T and A ⊗ π ( T ⊗ π T ) = A ⊗ ( T ⊗ π T ) as locally convex algebras.Proof. We prove that the space F from Lemma 1.17 is equal to the algebraic tensor product A ⊗ s . Let { v n } n ∈ N be a countable basis of A . Given { x n } n ∈ N a sequence of elements in A with ρ k ( x ) finite for all k ∈ N we have, for n fixed x n = X i ∈ N λ ( i ) n v i where λ ( i ) n = 0 for finitely many i ∈ N .First, we prove that span { x n } n ∈ N is finite dimensional. Suppose this is not the case. Weconstruct subsequences { x n i } and { v m i } such that λ ( m i ) n i = 0. Choose n such that x n = 0and m such that λ ( m ) n = 0. Suppose { x n , . . . , x n k } and { v m , . . . , v m k } have been chosen.span { x i } i>n k is infinite dimensional and therefore it is not contained in span { v i } ≤ i ≤ m k .Choose n k +1 > n k such that x n k +1 / ∈ span { v i } ≤ i ≤ m k . We can choose m k +1 > m k such that λ ( m k +1 ) n k +1 = 0.Now we define a seminorm in Aρ X i ∈ N c i v i ! = X i ∈ N | c i | α i with α i = 0 for i / ∈ { n k } k ∈ N and α n k ≥ | λ ( m k ) n k | − . Thus we have ρ ( x n k ) ≥ ρ ( x ) = X i ∈ N ρ ( x i ) ≥ X i ∈ N ρ ( x n i )diverges. We conclude that span { x n } n ∈ N is finite dimensional.Let N ∈ N be such that span { x n } n ∈ N ⊆ span { v , . . . , v N } . That is x n = N X i =0 λ ( i ) n v i Then x = lim M →∞ M X n =0 x n ⊗ e n = lim M →∞ M X n =0 N X i =0 λ ( i ) n v i ⊗ e n = lim M →∞ N X i =0 v i ⊗ M X n =0 λ ( i ) n e n Consider the seminorm p j ( P c i v i ) = | c j | . Then, since x ∈ A ⊗ π s , X n ∈ N | n | k p i ( x n ) = X n ∈ N | n | k | λ ( i ) n | < ∞ for all k ∈ N . Thus, for a fixed i , the sequences { λ ( i ) n } are rapidly decreasing on n . Therefore P n ∈ N λ ( n ) i e n ∈ s and consequently x = P Ni =1 v i ⊗ s i ∈ A ⊗ s .The equalities A ⊗ π T = A ⊗ T and A ⊗ π ( T ⊗ π T ) = A ⊗ ( T ⊗ π T ) follow because, byLemma 1.13, we have T ∼ = s and T ⊗ π T ∼ = s as locally convex vector spaces. (cid:4) Bivariant K-theory of locally convex algebras
Definition and properties of kk alg . The bivariant K -theory kk alg is constructed by Cuntz in [7]. In this section we give thedefinition of kk alg and state its main properties. A complete treatise of these constructions inthe context of bornological algebras can be found in [8]. The proofs translate to the contextof locally convex algebras in a straightforward manner.There is a canonical map h J k A, K ⊗ π S k B i → h J k +1 A, K ⊗ π S k +1 B i that assigns to each morphism α the classifying map associated with the extension0 → K ⊗ π S k +1 B → K ⊗ π CS k B → K ⊗ π S k B → . Definition . Let A and B be locally convex algebras. We define kk alg ( A, B ) = lim −→ k ∈ N h J k A, K ⊗ π S k B i and for n ∈ Z kk algn ( A, B ) = lim −→ k ∈ N k + n ≥ h J k + n A, K ⊗ π S k B i . The group structure of kk algn ( A, B ) is defined using Lemma 1.5.
Lemma 2.2.
There is an associative product kk algn ( A, B ) × kk algm ( B, C ) → kk algn + m ( A, C ) Proof.
Follows from Lemma 6.32 in [8]. (cid:4)
In view of this associative product we can regard locally convex algebras as objects of acategory KK alg with morphisms between A and B given by elements of kk alg ∗ ( A, B ). Anymorphism φ : A → B of locally convex algebras induces an element kk ( φ ) ∈ kk alg ( A, B )which is associated with the diffotopy class of i ◦ φ : A → B → K ⊗ π B , where i is theinclusion of B into the first corner, i.e. i ( b ) = e ⊗ b . We have kk ( φ ) kk ( ψ ) = kk ( ψ ◦ φ ) (seeTheorem 2.3.1 in [9]) and therefore we have a functor kk alg ∗ : lca → KK alg . In particular the identity of A induces an element kk ( id A ) in kk alg ( A, A ) which is denotedby 1 A . Definition . A functor F from the category of locally convex algebras to an abeliancategory C is called IVARIANT K-THEORY OF GENERALIZED WEYL ALGEBRAS 11 (1) diffotopy invariant if F ( f ) = F ( g ) whenever f and g are diffotopic,(2) half exact for linearly split extensions if F ( A ) → F ( B ) → F ( C )is exact whenever 0 → A → B → C → K -stable if the natural inclusion i : A → K ⊗ π A , sending a to e ⊗ a induces anisomorphism F ( i ) : F ( A ) → F ( K ⊗ π A ).The functor kk alg ∗ : lca → KK alg is diffotopy invariant, half exact for linearly split extensionsand is K -stable.2.2. Bott Periodicity and Triangulated structure of KK alg . The suspension of locally convex algebras determines a functor S : KK alg → KK alg with S ( A ) = SA . Theorem 2.4. [Bott periodicity] There is a natural equivalence between S and the identityfunctor, hence KK alg n ( A, B ) ∼ = KK alg ( A, B ) and KK alg n +1 ( A, B ) ∼ = KK alg ( A, B ) .Proof. See Corollary 7.25 in [8] and the discussion that follows. (cid:4)
By Theorem 2.4, S is an automorphism and S − ∼ = S . We recall the triangulated structureof ( KK , S ).Let f : A → B be a morphism in lca . The mapping cone of f is defined as the locallyconvex algebra C ( f ) = { ( x, g ) ∈ A ⊕ CB | f ( x ) = g (0) } . The triangle SB kk ( ι ) / / C ( f ) kk ( π ) / / A kk ( f ) / / B in ( KK alg , S ), where π : C ( f ) → A is the projection into the first component and ι : SB → C ( f ) is the inclusion into the first component, is called a mapping cone triangle.Let E : 0 → A f → B g → C → lca . This induces an element kk ( E ) ∈ kk alg ( C, A ) that corresponds to the classifying map
J C → A of the extension andhence an element kk ( E ) ∈ kk alg ( SC, A ). The triangle SC kk ( E ) / / A kk ( f ) / / B kk ( g ) / / C in ( KK alg , S ) is called an extension triangle. Proposition 2.5.
The category KK alg with suspension automorphism S : KK alg → KK alg and with triangles isomorphic to mapping cone triangles as exact triangles is a triangulatedcategory. Furthermore, extension triangles are exact.Proof. See Propositions 7.22 and 7.23 in [8]. (cid:4)
Stabilization by Schatten ideals.
In [9], Cuntz and Thom define a related bivariant K -theory in the category lca . We recallthe definition for the case of the Schatten ideals. Let H denote an infinite dimensionalseparable Hilbert Space. Definition . The Schatten ideals L p ⊆ B ( H ), for p ≥
1, are defined by L p = { x ∈ B ( H ) | T r | x | p < ∞} . Equivalently, L p consists of the space of bounded operators such that the sequence of itssingular values { µ n } is in l p ( N ). Definition . Let A and B be locally convex algebras and p ≥
1. We define kk L p n ( A, B ) = kk alg ( A, B ⊗ π L p ) . The groups kk L p ( A, B ), for all p ≥
1, are isomorphic (Corollary 2.3.5 of [9]). Moreover,when p >
1, this bivariant K -theory is related to algebraic K -theory in the following manner: Theorem 2.8 (Theorem 6.2.1 in [9]) . For every locally convex algebra A and p > we have kk L p ( C , A ) = K ( A ⊗ π L p ) . Corollary 2.9 (Corollary 6.2.3 in [9]) . The coefficient ring kk L p ∗ ( C , C ) is isomorphic to Z [ u, u − ] with deg ( u ) = 2 . This implies that kk L p ( C , C ) = Z and kk L p ( C , C ) = 0.Consider the category KK L p whose objects are locally convex algebras and whose mor-phisms are given by the graded groups kk L p ∗ ( A, B ). Since kk L p ∗ : lca → KK L p is diffotopyinvariant, half exact for linearly split extensions and is K -stable (see Lemma 7.20 in [8]), bythe universal property of kk alg ∗ we have a functor KK alg → KK L p .2.4. Weak Morita equivalence.
In the context of separable C * -algebras, two algebras A and B are strong Morita equivalentif and only if K ⊗ A ∼ = K ⊗ B (they are stably isomorphic). Therefore a strong Moritaequivalence of separable C * -algebras induces an equivalence in KK . In the case of locallyconvex algebras we recall the definition of weak Morita equivalence from [7], which still giveus an isomorphism between two objects in KK alg .A Morita context yields the data required in order to define maps A → K ⊗ π B . Definition . Let A and B be locally convex algebras. A Morita context from A to B consists of a locally convex algebra E that contains A and B as subalgebras and twosequences ( ξ i ) i ∈ N and ( η j ) j ∈ N of elements of E that satisfy(1) η j Aξ i ⊂ B for all i, j .(2) The sequence ( η j aξ i ) is rapidly decreasing for each a ∈ A . That is, for each continuousseminorm α in B , α ( η j aξ i ) is rapidly decreasing in i, j .(3) For all a ∈ A , ( P ξ i η i ) a = a .A Morita context (( ξ i ) , ( η j )) from A to B determines a homomorphism A → K⊗ π B definedby a P i,j ∈ N e ij ⊗ η j aξ i . Thus it determines an element kk (( ξ i ) , ( η j )) of kk alg ( A, B ).In the next proposition, we give conditions for a Morita context to determine an equiva-lence in KK alg . IVARIANT K-THEORY OF GENERALIZED WEYL ALGEBRAS 13
Proposition 2.11.
Let (( ξ i ) , ( η j )) be a Morita context from A to B in E . If (( ξ ′ l ) , ( η ′ k )) is a Morita context from B to A in the same locally convex algebra and if Aξ i ξ ′ l ⊂ A and η ′ k η j A ⊂ A for all i, j, k, l ; then kk (( ξ i ) , ( η j )) · kk (( ξ ′ l ) , ( η ′ k )) = 1 A . Therefore, if we also have Bξ ′ l ξ i ⊂ B and η k η ′ j B ⊂ B for all i, j, k, l , then kk (( ξ i ) , ( η j )) isinvertible in kk alg .Proof. See Lemma 7.2 in [7]. (cid:4)
Quasihomomorphisms.
The definition of a quasihomomorphism goes back to [5]. We give the definition of a quasi-homomorphisms in the context of locally convex algebras from [9]. A quasihomomorphismfrom A to B determines an element in kk alg ( A, B ). As a matter of fact it determines amorphism from E ( A ) to E ( B ) for any split-exact functor E : lca → C where C is an additivecategory. The reader can also see Section 4 in [10] and Section 3.3.1 in [8]. Definition . Let A , B and D be locally convex algebras with B a closed subalgebra of D . A quasihomomorphism from A to B in D is a pair of homomorphisms ( α, ¯ α ) from A to D such that α ( x ) − ¯ α ( x ) ∈ B , α ( x ) B ⊂ B and Bα ( x ) ⊂ B for all x ∈ A . We denote suchquasihomomorphism by ( α, ¯ α ) : A ⇒ D ⊲ B .The original definition of quasihomomorphisms required B to be an ideal in D (Definition2.1 in [5]). Note that if B is an ideal then the conditions α ( x ) B ⊂ B and Bα ( x ) ⊂ B aresatisfied automatically. On the other hand we only need to check these conditions in a setof algebraic generators of A . Remark . Let G ⊂ A be a subset that generates A as a locally convex algebra. If α ( x ) − ¯ α ( x ) ∈ B , α ( x ) B ⊂ B and Bα ( x ) ⊂ B for all x ∈ G , then the conditions are alsosatisfied for all x ∈ A .Next we see how a quasihomomorphism ( α, α ′ ) : A ⇒ D ⊲ B determines an element kk ( α, ¯ α ) ∈ kk alg ( A, B ). As a matter of fact, we work with split exact functors from lca toan additive category C . An extension 0 → A → B π → C → lca is split if there is amorphism of locally convex algebras s : C → B such that πs = id C . Definition . Let C be an additive category. A sequence A → B → C in C is split exact ifit is isomorphic to the sequence A → A ⊕ C → C with the natural inclusion and projection.A functor E : lca → C is called split exact if it sends split extensions in lca to split exactsequences in C . Lemma 2.15. [Section 3.2 in [9] ] Let E be a split exact functor from lca to an additivecategory C . Then a quasihomomorphism ( α, α ′ ) : A ⇒ D ⊲ B determines a morphism E ( α, ¯ α ) : E ( A ) → E ( B ) in C .Proof. Let D ′ be the closed subalgebra of A ⊕ D generated by all elements ( a, α ( a )) and(0 , b ) with a ∈ A and b ∈ B . Then we have an exact sequence0 → B → D ′ → A → B ⊆ D ′ given by b (0 , b ) and the projection π : D ′ → A defined by π ( a, x ) = a . This extension has two splits α ′ , ¯ α ′ : A → D ′ defined by α ′ ( a ) = ( a, α ( a ))and ¯ α ′ ( a ) = ( a, ¯ α ( a )). Because of the split-exactness of E , E ( B ) → E ( D ′ ) is a kernel of E ( D ′ ) → E ( A ). Therefore, the morphism E ( α ′ ) − E ( ¯ α ′ ) : E ( A ) → E ( D ′ ) defines a morphism E ( α, ¯ α ) : E ( A ) → E ( B ). (cid:4) The following proposition summarizes some properties of quasihomomorphisms.
Proposition 2.16.
Let E be a split exact functor from lca to an additive category C and ( α, α ′ ) : A ⇒ D ⊲ B be a quasihomomorphism from A to B in D . We have(1) E ( α, ¯ α ) = − E ( ¯ α, α ) .(2) Let φ = α − ¯ α . If φ ( x ) ¯ α ( y ) = ¯ α ( y ) φ ( x ) = 0 for all x, y ∈ A , then φ is a homomor-phism and E ( α, ¯ α ) = E ( φ ) .(3) For any morphism φ : A ′ → A , ( α ◦ φ, ¯ α ◦ φ ) : A ′ → B is a quasihomomorphism from A ′ to B in D and E ( α ◦ φ, ¯ α ◦ φ ) = E ( α, ¯ α ) ◦ E ( φ ) . (4) If ψ : D → F is a morphism such that ψ | B : B → C ⊂ F and the morphisms ψ ◦ α, ψ ◦ ¯ α : A → F define a quasihomomorphism from A to C in F , then E ( ψ ◦ α, ψ ◦ ¯ α ) = E ( ψ | B ) ◦ E ( α, ¯ α ) . (5) Let α and ¯ α be homomorphisms from A to D [0 , such that α ( x ) − ¯ α ( x ) ∈ B [0 , , α ( x ) B [0 , ⊂ B [0 , and B [0 , α ( x ) ⊂ B [0 , for all x ∈ A . If E is diffotopyinvariant, then E ( α , ¯ α ) = E ( α , ¯ α ) (where α t = ev t ◦ α ).Proof. For (1) − (4) see Proposition 21 in [11].To prove (5), we consider the evaluation maps ev t : D [0 , → D . They restrict to theevaluation maps ev t : B [0 , → B . To apply (3) we need to check that the morphisms ev t ◦ α, ev t ◦ ¯ α : A → D define a quasihomomorphism from A to B in D . First noticethat ( ev t ◦ α )( a ) − ( ev t ◦ ¯ α )( a ) = ( ev t ◦ ( α − ¯ α ))( a ) is in B because ( α − ¯ α )( a ) ∈ B [0 , b ∈ B . We want to prove that the product ( ev t ◦ α )( a ) b is in B .Consider a function φ ∈ B [0 ,
1] such that ev t ◦ f = b . Then ( ev t ◦ α )( a ) b = ev t ◦ ( α ( a ) f ) and α ( a ) f ∈ B [0 , B ( ev t ◦ α )( a ) ⊆ B .We can now apply (3) and we obtain E ( ev t ◦ α, ev t ◦ ¯ α ) = E ( ev t ) ◦ E ( α, ¯ α ). Since E isdiffotopy invariant, we have E ( ev ) = E ( ev ), which concludes the result. (cid:4) Generalized Weyl algebras
Generalized Weyl algebras were introduced by Bavula (see [2]) and have been amplystudied. Examples of generalized Weyl algebras include the Weyl algebra, the quantumWeyl algebra, the quantum plane, the enveloping algebra of sl U ( sl ), its primitive factors B λ = U ( sl ) / h C − λ i where C is the Casimir element (see Example 4.7 in [12]) and thequantum weighted projective lines O ( WP q ( k, l )) (see [4]).In our context, generalized Weyl algebras provide a family of examples of Z -graded algebrasthat are smooth generalized crossed products and do not satisfy the condition of being tamesmooth (and therefore they are outside the framework of [10]). Definition . Let D be a ring, σ ∈ Aut ( D ) and a a central element of D . The generalizedWeyl algebra D ( σ, a ) is the algebra generated by x and y over D satisfying xd = σ ( d ) x, yd = σ − ( d ) y, yx = a and xy = σ ( a ) (3.1)for all d ∈ D . Examples.
The following are examples of generalized Weyl algebras:
IVARIANT K-THEORY OF GENERALIZED WEYL ALGEBRAS 15 (1) The Weyl algebra A ( C ) = C h x, y | xy − yx = 1 i is isomorphic to C [ h ]( σ, h ), with σ ( h ) = h − A q ( C ) = C h x, y | xy − qyx = 1 i is isomorphic to C [ h ]( σ, h − σ ( h ) = qh .(3) The quantum plane C h x, y | xy = qyx i is isomorphic to C [ h ]( σ, h ), with σ ( h ) = qh .(4) The primitive quotients of U ( sl ) (see Example 3.2 in [2]), B λ = U ( sl ) / h c − λ i , λ ∈ C , are isomorphic to C [ h ]( σ, P ), with σ ( h ) = h − P ( h ) = − h ( h + 1) − λ/ O ( WP q ( k, l ))is isomorphic to C [ h ]( σ, P ) with P ( h ) = h k Q l − i =0 (1 − q − i h ) and σ ( h ) = q l h (seeTheorem 2.1 in [4] and Example 3.8 in [3]). Lemma 3.2.
A generalized Weyl algebra has a Z -grading A = L n ∈ Z A n where A = D and A n = ( Dy n n > Dx n n < . (3.2) Proof.
Consider the grading in A = D ( σ, a ) defined by setting the degree of y equal to 1,the degree of x equal to −
1, and the degree of all elements of D equal to 0. That is, thedegree of the monomial Q ni =1 d i x α i y β i , with d i ∈ D , is equal to P ni =1 β i − P ni =1 α i . Since therelations defining A are compatible with the grading, the algebra A is Z -graded.Now consider the following relations in A . We have x n y n = σ n ( a ) σ n − ( a ) . . . σ ( a ) y n x n = σ − ( n − ( a ) σ − ( n − ( a ) . . . a Using induction on the length of the monomial Q ni =1 d i x α i y β i we prove (3.2). Note that Dy n = y n D and Dx n = x n D . (cid:4) In the case of generalized Weyl algebras over C [ h ], we have the following result. Corollary 3.3.
The generalized Weyl algebra A = C [ h ]( σ, P ) , with P ∈ C [ h ] , has a countablebasis over C .Proof. A basis is given by the elements h n , h n y m and h n x m for n ∈ N , m ≥ (cid:4) There are several ways of writing the same generalized Weyl algebra. The conjugation of σ by an automorphism τ of D gives rise to an isomorphism of generalized Weyl algebras. Lemma 3.4.
Let σ , τ be automorphisms of D and let a be a central element of D . Then τ ( a ) is central in D and D ( σ, a ) ∼ = D ( τ στ − , τ ( a )) . Proof.
Let x ′ and y ′ be the generators of D ( τ στ − , τ ( a )) over D . There is a morphism φ : D ( σ, a ) → D ( τ στ − , τ ( a )) defined by x x ′ , y y ′ , d τ ( d ), for all d ∈ D . Weneed to check that φ is compatible with the relations (3.1). Using the relations defining D ( τ στ − , τ ( a )) we have x ′ τ ( d ) = ( τ στ − )( τ ( d )) x ′ = τ ( σ ( d )) x ′ y ′ τ ( d ) = ( τ σ − τ )( τ ( d )) y ′ = τ ( σ − ( d )) y ′ x ′ y ′ = τ ( a ) y ′ x ′ = ( τ στ − )( τ ( a )) = τ ( σ ( a )) .φ − is defined by x ′ x , y ′ y , d τ − ( d ) for all d ∈ D . (cid:4) In the case D = C [ h ], we use Lemma 3.4 to write a given generalized Weyl algebra in acanonical form. Any automorphism of C [ h ], is of the form σ ( h ) = qh + h with q, h ∈ C and q = 0. We have three cases(1) σ is conjugate to id if and only if σ = id ,(2) if q = 1 and h = 0, then σ is conjugate to h h − q = 1, then σ is conjugate to h qh .Combining this with Lemma 3.4, we obtain the following result. Proposition 3.5 (Compare with Proposition 2.1.1 in [14].) . Let A = C [ h ]( σ, P ) , with P ∈ C [ h ] and σ ( h ) = qh + h with q, h ∈ C and q = 0 . The following facts hold:(1) If σ = id , then A ∼ = C [ h, x, y ] / ( yx − P ) .(2) If q = 1 and h = 0 then A ∼ = C [ h ]( σ , P ) with σ ( h ) = h − and P ( h ) = P ( − h h ) .(3) If q = 1 then A ∼ = C [ h ]( σ , P ) with σ ( h ) = qh and P ( h ) = P ( h + h − q ) . (cid:4) Proof. (1) is straightforward, for (2) and (3) use Lemma 3.4 with τ ( h ) = − h h and τ ( h ) = h + h − q , respectively. (cid:4) By Proposition 3.5, we can assume that σ = id , σ ( h ) = h − σ ( h ) = qh for some q = 0. Proposition 3.6.
Let A = C [ h ]( σ, P ) , with P ∈ C [ h ] . The following facts hold:(1) If σ ( h ) = h − and P is a non constant polynomial, then A ∼ = C [ h ]( σ, P ) with P (0) = 0 .(2) If σ ( h ) = qh and P has a nonzero root, then A ∼ = C [ h ]( σ, P ) with P (1) = 0 .Proof. Follows from Lemma 3.4. In (1) we set τ ( h ) = h − λ for any root λ of P , and in (2)we set τ ( h ) = λh , where λ is a non zero root of P . (cid:4) Note that P in Proposition 3.5 has a non zero root if and only if P has a root differentfrom h − q .It is worth mentioning that generalized Weyl algebras over C [ h ] have been classified up toisomorphism (see [2]and [14]).To finish this section, we construct faithful representations for the generalized Weyl alge-bras covered in cases (1) and (2) of Proposition 3.6. We define V N as the vector space ofsequences of complex numbers indexed by N . Let U , U − ∈ End ( V N ) be the shift to the IVARIANT K-THEORY OF GENERALIZED WEYL ALGEBRAS 17 right and the shift to the left respectively. Note that U − U = 1, U U − = 1 − e . U = · · · U − = · · · Additionally, we use the following elements N = P i ∈ N ( − i ) e i,i and G = P i ∈ N q i e i,i for q = 0 not a root of unity. N = · · · − − − G = · · · q q
00 0 0 q ... . . . Lemma 3.7.
The following relations are satisfied in
End ( V N ) .(1) U N = ( N + 1) U ,(2) U − N = ( N − U − ,(3) U G = ( q − G ) U ,(4) U − G = ( qG ) U − . (cid:4) As a consequence of Lemma 3.7, we obtain that the subalgebras E and E of End ( V N )generated by {U , U − , N } and {U , U − , G } , respectively, have countable basis over C andtherefore they are locally convex algebras with the fine topology. Lemma 3.8.
We have the following representations for generalized Weyl algebras A = C [ h ]( σ, P ( h )) in the following cases.(1) If σ ( h ) = h − and P is a nonzero polynomial with P (0) = 0 , then there is a faithfulrepresentation ρ : A → E such that ρ ( h ) = N, ρ ( x ) = U − and ρ ( y ) = P ( N ) U = U P ( N − (2) If σ ( h ) = qh with q = 0 not a root of unity and P (1) = 0 , then there is a faithfulrepresentation ρ : A → E such that ρ ( h ) = G, ρ ( x ) = U − and ρ ( y ) = P ( G ) U = U P ( qG ) Proof.
For (1), first we notice that we have an injective homomorphism C [ h ] ֒ → End ( V N )defined by h N .This homomorphism is injective because all the entries in the diagonal of matrix N aredifferent. With P (0) = 0 we will see that the relations of C [ h ]( σ, P ( h )) hold. To prove this,we use the relations of Lemma 3.7. For a polynomial α ( h ) ∈ C [ h ] we have ρ ( xα ( h )) = U − α ( N ) = α ( N − U − = ρ ( α ( h − x ) ρ ( yα ( h )) = P ( N ) U α ( N ) = α ( N + 1) P ( N ) U = ρ ( α ( h + 1) y ) ρ ( yx ) = U P ( N − U − = U U − P ( N ) = (1 − e ) P ( N ) = P ( N ) = ρ ( P ( h )) ρ ( xy ) = U − U P ( N −
1) = P ( N −
1) = ρ ( P ( h − P (0) = 0 in the third row to guarantee (1 − e ) P ( N ) = P ( N ). Now we prove that ρ is injective. Let α = X n ≥ p n ( h ) y n + X m< q m ( h ) x m be an element of A . Then we have ρ ( α ) = X n ≥ p n ( P ( N ))( P ( N ) U ) n + X m< q m ( P ( N )) U m − . Note that ( P ( N ) U ) n = Q n ( N ) U n where Q n ( N ) = P ( N ) P ( N + 1) . . . P ( N + ( n − ρ ( α ) = 0 then q m = 0 and because Q n = 0, we have p n = 0. Therefore α = 0and so ρ is injective.(2) is proved in a similar way: we have an injective homomorphism C [ h ] ֒ → End ( V N )defined by h G . This homomorphism is injective because q = 0 not a root of unity implythat all the entries in the diagonal of matrix G are different. Using P (1) = 0, it is easy tosee that the relations of D ( σ, a ) hold. We also need to use the relations of Lemma 3.7. Weprove that ρ is injective in a similar way. In this case we note that ( P ( G ) U ) n = Q n ( G ) U n where Q n ( G ) = P ( G ) P ( q − G ) . . . P ( q − ( n − G ). (cid:4) kk alg invariants of generalized Weyl algebras In this section, we compute the isomorphism class in KK alg of generalized Weyl algebras A = C [ h ]( σ, P ) where σ ( h ) = qh + h is an automorphism of C [ h ] and P ∈ C [ h ]. Wesummarize our results in the table below.Conditions Results Observation P is constant P = 0 A ∼ = KK alg C Prop 4.22 A N -graded P = 0 A ∼ = KK alg S C ⊕ C Prop 4.21 A tame smooth P is nonconstantwith r distinctroots q not a root of unity A ∼ = KK alg C r Thm 4.18Prop 4.20 q = 1 and h = 0 A ∼ = KK alg C r Thm 4.18In Section 4.1, we recall the definition of T B from [10]. Generalized Weyl algebras A = C [ h ]( σ, P ) are smooth generalized crossed products and in Proposition 4.5 we construct alinearly split extension 0 → Λ A → T A → A → . In the case where P is a non constant polynomial, A is a generalized crossed product that isnot tame smooth so we cannot apply the results of [10] directly. In this case we follow themethods of [7] and [10] to obtainΛ A ∼ = KK alg A A − (Theorem 4.8 ) and T A ∼ = KK alg A (Theorem 4.14 )in the cases where P is non constant and • q = 1 and h = 0 or • q is not a root of unity and P has a root different from h − q .With these isomorphisms we construct in Theorem 4.15 an exact triangle SA → A A − → A → A IVARIANT K-THEORY OF GENERALIZED WEYL ALGEBRAS 19 in the triangulated category ( KK alg , S ) (see Proposition 2.5). This implies A = A ⊕ S ( A A − ) . In Proposition 4.17, we prove that A A − ∼ = KK alg S C r − , and since by Lemma 1.7 we knowthat A ∼ = KK alg C , we obtain our main result Theorem 4.18: in these cases A ∼ = KK alg C r .We also determine the KK alg -class of A when A is N -graded. In this case Lemma 1.7 givesus A ∼ = KK alg A . This is the case when • P is nonconstant, q is not a root of unity and P has only h − q as a root or • P = 0.In both cases we obtain A ∼ = KK alg C in Propositions 4.22 and 4.20.If P is a nonzero constant polynomial, A is a tame smooth generalized crossed productand the results from [10] apply. In this case we obtain A ∼ = KK alg S C ⊕ C in Proposition 4.21.4.1. The Toeplitz algebra of a smooth generalized crossed product.
In [10], Gabriel and Grensing define smooth generalized crossed products. These areinvolutive locally convex algebras analog to C * -algebra generalized crossed products in [1].In the same article [10], sequences analog to the Pimsner-Voiculescu exact sequence areconstructed for smooth generalized crossed products that are tame smooth (see definition 18in [10]). Definition . A gauge action γ on a locally convex algebra B is a pointwise continuousaction of S on B . An element b ∈ B is called gauge smooth if the map t γ t ( b ) is smooth.If we have a gauge action on B , then B n = { b ∈ B | γ t ( b ) = t n b, ∀ t ∈ S } defines a natural Z -grading of B . Definition . A smooth generalized crossed product is a locally convex algebra B with aninvolution and a gauge action such that • B and B generate B as a locally convex involutive algebra, • all b are gauge smooth and the induced map B → C ∞ ( S , B ) is continuous.Generalized Weyl algebras A = C [ h ]( σ, P ) are locally convex algebras when given the finetopology. They have an involution defined by y ∗ = x , x ∗ = y and d ∗ = d for all d ∈ D .There is an action of S defined by γ t ( ω n ) = t n ω n for ω n ∈ A n . With this action, generalizedWeyl algebras over C [ h ] are smooth generalized crossed products. Remark . Generalized Weyl algebras A = C [ h ]( σ, P ) are only tame smooth when P isconstant (see definition 18 in [10]). If P is non constant, we have A A − = ( P ) ( A = C [ h ].This implies that A is not tame smooth because tame smooth generalized crossed products B have a frame in degree 1 which implies that B B − = B . Definition . Let B be a smooth generalized crossed product. We define T B to be theclosed subalgebra of T ⊗ π B generated by 1 ⊗ B , S ⊗ B and S ∗ ⊗ B − .We tensor the linearly split extension 0 → K → T → C ∞ ( S ) → B to obtain0 → K ⊗ π B → T ⊗ π B p → C ∞ ( S ) ⊗ π B → . (4.1)which is still a linearly split extension. Proposition 4.5.
Let A be a generalized Weyl algebra C [ h ]( σ, P ) . Then there is a linearlysplit extension → Λ A ι → T A ¯ p → A → where Λ A is the ideal L i,j ≥ e i,j ⊗ A i +1 A − ( j +1) of T A , ι is the inclusion of Λ A in T A and ¯ p isthe restriction of p to T A .Proof. By Corollary 3.3, A has a countable basis over C . Using Lemma 1.18, we concludethat the projective tensor products of (4 .
1) are all algebraic. The image of T A is generatedby 1 ⊗ A , z ⊗ A and z − ⊗ A − and it is isomorphic to A via z n ⊗ a n a n . The kernelof π is the intersection of K ⊗ A and T A . The elements of T A are of the form 1 ⊗ a + P k,l ≥ S k +1 S ∗ l +1 ⊗ a k +1 a − ( l +1) . Now we note that S k +1 S l +1 = ( S k − l (1 − e , − · · · − e l,l ) , k ≥ l (1 − e , − · · · − e k,k ) S ∗ ( l − k ) , k < l Using the vector space decomposition
T ⊗ A = K ⊗ A ⊕ C ∞ ( S ) ⊗ A we note that an elementof the kernel is of the form X k ≥ l S k − l ( − e , − · · · − e l,l ) ⊗ a k +1 a − ( l +1) + X k Lemma 4.7. The elements of Λ A can be written uniquely as sums P e i,j ⊗ y i +1 P i,j ( h ) x j +1 Proof. Follows from Lemma 3.2. (cid:4) Define j : A A − → Λ A by j ( a ) = e ⊗ a . We embed Λ A in a suitable algebra so thatwe can construct a Morita equivalence to its subalgebra j ( A A − ) = e ⊗ A A − . Considerthe faithful representation ρ : A → E from Lemma 3.8 (where E = E if q = 1 and E = E if q = 1). Tensoring with 1 T we obtain an injective morphism 1 T ⊗ ρ : T ⊗ A → T ⊗ E whichrestricts to an injective morphism ¯ ρ : T A ֒ → T ⊗ E .Now, we show Λ A ∼ = KK alg A A − . Theorem 4.8. There is a Morita equivalence between Λ A and j ( A A − ) , therefore there isan invertible element θ ∈ kk alg (Λ A , A A − ) which is an inverse of kk ( j ) .Proof. We write the proof in the case σ ( h ) = h − P (0) = 0. The case σ ( h ) = qh and P (1) = 0 can be proven in a similar way since the matrices involved in the representationsatisfy corresponding algebraic relations (Lemma 3.7).Using the representation from Lemma 3.8, we obtain a faithful representation¯ ρ : Λ A → T ⊗ E . The Morita equivalence is given by ξ i = ξ ′ i = e i, ⊗ U i and η j = η ′ j = e ,j ⊗ U i − . We checkthat these sequences satisfy the conditions in Definition 2.10 and Proposition 2.11.First, we establish that ξ i , η j defines a Morita context between Λ A and e ⊗ A A − according with Definition 2.10. Let w = P e i,j ⊗ y i +1 P i,j ( h ) x j +1 be an element of Λ A .(1) η j ¯ ρ ( w ) ξ i ∈ e ⊗ A A − . We have η j ¯ ρ ( w ) ξ i = e ⊗ U j − [ P ( N ) U ] j +1 P j,i ( N ) U i +1 − U i Note that ( P ( N ) U ) j +1 = U j +11 R j +1 ( N ) where R j +1 ( N ) = P ( σ ( N )) . . . P ( σ j +1 ( N )) = P ( σ ( N )) R ′ j +1 ( N )and therefore we have η j ¯ ρ ( w ) ξ i = e ⊗ U R j +1 ( N ) P j,i ( N ) U − = e ⊗ U P ( σ ( N )) R ′ j +1 ( N ) P j,i ( N ) U − = e ⊗ P ( N ) U R ′ j +1 ( N ) P j,i ( N ) U − = ¯ ρ ( e ⊗ yR ′ j +1 ( h ) P j,i ( h ) x ) ∈ ¯ ρ ( e ⊗ A A − )(2) The terms η j ¯ ρ ( w ) ξ i are rapidly decreasing. This is because the elements of Λ A arefinite sums.(3) ( P ξ i η i ) ¯ ρ ( w ) = ¯ ρ ( w ). We have( X ξ i η i ) ¯ ρ ( w ) = (cid:16)X e i,i ⊗ U i U i − (cid:17) (cid:16)X e k,l ⊗ ( U P ( N )) k +1 P k,l ( N ) U l +1 − (cid:17) = X e k,l ⊗ U k U k − U k +11 R k +1 ( N ) P k,l ( N ) U l +1 − = X e k,l ⊗ U k +11 R k +1 ( N ) P k,l ( N ) U l +1 − = ¯ ρ ( w ) Now we check the conditions of Proposition 2.11. We show that ¯ ρ ( w ) ξ k ξ ′ l and η ′ k η ′ l ¯ ρ ( w )are still elements of ¯ ρ (Λ A ).¯ ρ ( w ) ξ k ξ ′ l = (cid:16)X e i,j ⊗ ( U P ( N )) i +1 P i,j ( N ) U j +1 − (cid:17) ( e k, ⊗ U k )( e l, ⊗ U l )which is 0 unless l = 0 and in that case we obtain¯ ρ ( w ) ξ k ξ ′ l = X e i, ⊗ ( U P ( N )) i +1 P i,k ( N ) U − = ¯ ρ (cid:16)X e i, ⊗ y i +1 P i,k ( h ) x (cid:17) ∈ ¯ ρ (Λ A )and similarly we compute η ′ k η l ¯ ρ ( w ) = ( e ,k ⊗ U k − )( e ,l ⊗ U l − ) (cid:16)X e i,j ⊗ ( U P ( N )) i +1 P i,j ( N ) U j +1 − (cid:17) which is 0 unless k = 0 and in that case we obtain¯ ρ ( w ) ξ k ξ ′ l = X e ,j ⊗ U l − ( U P ( N )) l +1 P l,j ( N ) U j +1 − = ¯ ρ (cid:16)X e ,j ⊗ yR ′ l +1 ( h ) P l,j ( h ) x j +1 (cid:17) ∈ ¯ ρ (Λ A ) . The Morita context from e ⊗ A A − to Λ A is defined by ( ξ ′ i , η ′ j ). So far we have proved kk (( ξ i ) , ( η j )) · kk (( ξ ′ i ) , ( η ′ j )) = 1 Λ A . Let z = e ⊗ yP , x ∈ e ⊗ A A − . Then ¯ ρ ( z ) ξ ′ l ξ k =¯ η l η ′ k ρ ( z ) = 0 unless l = k = 0 and in this case ¯ ρ ( z ) ξ ′ ξ = ¯ ρ ( z ) η η ′ = ¯ ρ ( z ). Thus we have kk (( ξ ′ i ) , ( η ′ j )) · kk (( ξ i ) , ( η j )) = 1 e ⊗ A A − . (cid:4) Next, we show T A ∼ = KK alg A . Define j : A → T A by j ( a ) = 1 ⊗ a . We show that thisinclusion induces an invertible element kk ( j ) ∈ kk alg ( T A , A ). Lemma 4.9. There is a quasihomomorphism ( id, Ad ( S ⊗ T A ⇒ T ⊗ A ⊲ C , where C = M i,j ∈ N e i,j ⊗ A i A − j . Here Ad ( S ⊗ is the restriction of Ad ( S ⊗ 1) : T ⊗ A → T ⊗ A defined by x ( S ⊗ x ( S ∗ ⊗ .Proof. We have A i A − j A j A − k ⊆ A i A − k because A − j A j ⊆ A , therefore C is a subalgebra. Toprove that the pair ( id, Ad ( S ⊗ ⊗ A ) C , ( S ⊗ A ) C and ( S ∗ ⊗ A − ) C are subsets of C .Now we let a i ∈ A i and we check( id − Ad ( S ⊗ ⊗ a ) = e ⊗ a ∈ C ( id − Ad ( S ⊗ S ⊗ a ) = Se ⊗ a ∈ C ( id − Ad ( S ⊗ S ∗ ⊗ a − ) = eS ∗ ⊗ a − ∈ C . (cid:4) Define i : A → C by i ( a ) = e ⊗ a . Proposition 4.10. There is a Morita equivalence between C and i ( A ) . Therefore there isan invertible element κ ∈ kk alg ( C , A ) . IVARIANT K-THEORY OF GENERALIZED WEYL ALGEBRAS 23 Proof. Using Lemma 3.8, we think of C represented in T ⊗ E (where E = E if q = 1and E = E if q = 1). The Morita equivalence is given by ξ i = ξ ′ i = e i, ⊗ U i and η j = η ′ j = e ,j ⊗ U i − . The proof that these elements determine a Morita equivalence is similarto the proof of Theorem 4.8. The Morita context (( ξ i ) , ( η j )) determines a morphism C →K ⊗ π i ( A ) that in turn determines the element kk (( ξ i ) , ( η j )) ∈ kk alg ( C , A ). We define κ = kk (( ξ i ) , ( η j )) kk ( i ) − where i : A → e ⊗ A . (cid:4) Proposition 4.11. Let κ ∈ kk ( C , A ) as in Proposition 4.10, then kk ( j ) kk ( id, Ad ( S ⊗ κ = 1 A . This implies that kk ( j ) has a right inverse and that kk ( id, Ad ( S ⊗ has a left inverse.Proof. We have ( id − Ad ( S ⊗ j ( a )) = e ⊗ a , thus kk ( j ) kk ( id, Ad ( S ⊗ kk ( i ). By Proposition 4.10, kk ( i ) κ = 1 A . (cid:4) To show that kk ( j ) is invertible, we construct a right inverse for kk ( id, Ad ( S ⊗ T A and asubalgebra ¯ C of ( T ⊗ π T ) ⊗ A and prove that ¯ C is Morita equivalent to T A . To constructthis diffotopic family we use the diffotopy φ t : T → T ⊗ π T of Lemma 1.16.Consider the map Φ t = φ t ⊗ id A : T ⊗ A → ( T ⊗ π T ) ⊗ A where φ t is the diffotopy oflemma 1.16. Since φ ( S ) = S ⊗ 1, then Φ ( x ⊗ a ) = x ⊗ ⊗ a . Lemma 4.12. There is a diffotopic family of quasihomomorphisms (Φ t , Φ ◦ Ad ( S ⊗ T A ⇒ ( T ⊗ π T ) ⊗ A ⊲ ¯ C . Where ¯ C is the subalgebra M i,j,p,q ∈ N e i,j ⊗ S p S ∗ q ⊗ A i + p A − ( j + q ) . Proof. We check that (Φ t , Φ ◦ Ad ( S ⊗ T A . First we note that Φ t (1 ⊗ A ) ¯ C , Φ t ( S ⊗ A ) ¯ C and Φ t ( S ∗ ⊗ A − ) ¯ C are subsets of ¯ C .Finally, we compute(Φ t − Φ ◦ Ad ( S ⊗ ⊗ a ) = e ⊗ ⊗ a ∈ ¯ C (Φ t − Φ ◦ Ad ( S ⊗ S ⊗ a ) = f ( t )( e ⊗ S ⊗ a ) + g ( t )( Se ⊗ ⊗ a ) ∈ ¯ C (Φ t − Φ ◦ Ad ( S ⊗ S ∗ ⊗ a − ) = ¯ f ( t )( e ⊗ S ∗ ⊗ a − ) + ¯ g ( t )( eS ∗ ⊗ ⊗ a − ) ∈ ¯ C . (cid:4) Define η : T A → ¯ C as the restriction of the injective morphism T ⊗ A → ( T ⊗ π T ) ⊗ A given by η ( x ⊗ a ) = e ⊗ x ⊗ a . Proposition 4.13. There is a Morita equivalence between ¯ C and η ( T A ) . Therefore kk ( η ) ∈ kk alg ( T A , ¯ C ) is invertible.Proof. Using Lemma 3.8, we have an injective morphism ¯ C → ( T ⊗ π T ) ⊗ E (where E = E if q = 1 and E = E if q = 1)The Morita equivalence is given by ξ i = ξ ′ i = e i, ⊗ ⊗ U i and η j = η ′ j = e ,j ⊗ ⊗ U j − . The proof is similar to the proof of Theorem 4.8. (cid:4) Theorem 4.14. kk ( j ) ∈ kk alg ( A , T A ) is invertible. Proof. By 4.11, we know that kk ( j ) has a right inverse and kk ( id, Ad ( S ⊗ k ( id, Ad ( S ⊗ φ ( S ) = S ⊗ 1, then if a i ∈ A i and a − j ∈ A − j , we haveΦ ( e i,j ⊗ a i a − j ) = e i,j ⊗ ⊗ a i a − j ∈ ¯ C and therefore Φ ( C ) ⊆ ¯ C , thus by item (4) of Proposition 2.16 we have kk ( id, Ad ( S ⊗ kk (Φ | C ) = kk (Φ , Φ ◦ Ad ( S ⊗ . By item (5) of Proposition 2.16, we obtain kk (Φ , Φ ◦ Ad ( S ⊗ kk (Φ , Φ ◦ Ad ( S ⊗ . We have φ ( S ) = S S ∗ ⊗ e ⊗ S and therefore Φ − Φ ◦ Ad ( S ⊗ 1) = η . By item (2) ofProposition 2.16, kk (Φ , Φ ◦ Ad ( S ⊗ kk ( η ) and by Lemma 4.13, kk ( η ) is invertible. (cid:4) With the isomorphisms in KK alg fromTheorems 4.8 and 4.14, we construct the desiredexact triangle. Theorem 4.15. For a generalized Weyl algebra A = C [ h ]( σ, P ( h )) with P a non constantpolynomial and • q = 1 and h = 0 or • q is not a root of unity and P has a root different from h − q there is an exact triangle SA → A A − → A → A. Proof. The linearly split extension0 → Λ A ι → T A ¯ p → A → SA kk ( E ) → Λ A kk ( ι ) → T A kk (¯ p ) → A, where kk ( E ) ∈ kk alg ( A, Λ A ) = kk alg ( SA, Λ A ) is the element defined by the extension (4.2).By Theorem 4.8, the inclusion j : A A − → Λ A defined by j ( x ) = e ⊗ x induces aninvertible element kk ( j ) ∈ kk alg ( A A − , Λ A ). By Theorem 4.14, the inclusion j : A → T A defined by j ( a ) = 1 ⊗ a induces an invertible element kk ( j ) ∈ kk alg ( A , T A ). We define φ by the commutative diagram in KK alg Λ A kk ( ι ) / / T Akk ( j ) − (cid:15) (cid:15) A A − kk ( j ) O O φ / / A and claim that φ = kk ( i ) − kk ( σ ) . (4.3)For this we use Proposition 4.11 to obtain kk ( j ) − = kk ( id, Ad ( S ⊗ κ and therefore kk ( j ) kk ( ι ) kk ( j ) − = kk ( j ) kk ( ι ) kk ( id, Ad ( S ⊗ κ. IVARIANT K-THEORY OF GENERALIZED WEYL ALGEBRAS 25 Let x = R ( h ) ∈ A A − ⊆ C [ h ]. The product kk ( j ) kk ( ι ) kk ( id, Ad ( S ⊗ φ, ψ ) : A A − ⇒ T ⊗ A⊲ C , where φ ( x ) = e ⊗ x and ψ ( x ) = e ⊗ x .Since φ and ψ are orthogonal, we obtain kk ( φ, ψ ) = kk ( φ ) − kk ( ψ ). Now we multiply thisdifference by κ which is given by the Morita equivalence of Proposition 4.10. Thus we havethat kk ( φ ) κ and kk ( ψ ) κ are determined by maps A A − → C → K⊗ A that send x e ⊗ x and x e ⊗ ρ − ( U − R ( G ) U ) = e ⊗ R ( σ ( h )) (here we use the representation ρ of Lemma3.8). Thus we conclude φ = kk ( i ) − kk ( σ ), proving (4.3).Now we prove that φ = 0. Both i and σ factor through a contractible subalgebra of C [ h ]. This is because we have i ( A A − ) = P ( h ) C [ h ] and σ ( A A − ) = P ( σ ( h )) C [ h ] andthe polynomials P ( h ) and P ( σ ( h )) have some linear factors L ( h ) and L ( σ ( h )). Thus themorphisms i and σ factor through the subalgebras L ( h ) C [ h ] and L ( σ ( h )) C [ h ] which arecontractible. Therefore we have kk ( i ) = kk ( σ ) = 0. (cid:4) Lemma 4.16. Let ( T , Σ) be a triangulated category. If there is an exact triangle Σ X → Y → Z → X then X ∼ = Z ⊕ Σ − Y .Proof. See Corollary 1.2.7 in [13]. (cid:4) Now we compute the isomorphism class of A A − in KK alg . Proposition 4.17. Let A = C [ h ]( σ, P ) where P is a nonconstant polynomial with r differentroots, then A A − ∼ = KK alg S C r − . Proof. Let P ( h ) = c ( h − h ) n . . . ( h − h r ) n r . Without loss of generality we can assume c = 1.Since A A − = ( P ( h )) we have a linearly split extension0 → A A − → C [ h ] π → C [ h ] / ( P ( h )) → . (4.4)By the Chinese Reminder Theorem, there is an isomorphism φ : C [ h ] / ( P ( h )) → r Y i =1 C [ h ] / ( h − h i ) n i . We have the following commutative diagram0 / / ( h − h i ) n i C [ h ] / / (cid:15) (cid:15) C [ h ] q i / / = (cid:15) (cid:15) C [ h ] / ( h − h i ) n i / / µ i (cid:15) (cid:15) / / ( h − h i ) C [ h ] / / C [ h ] ev hi / / C / / h − h i ) n i C [ h ] and ( h − h i ) C [ h ] are contractible, kk ( q i ) and kk ( ev h i ) are invertible,therefore kk ( µ i ) ∈ kk alg ( C [ h ] / ( h − h i ) n i , C ) is invertible. By the additivity of KK alg , thehomomorphism µ : Q ri =1 C [ h ] / ( h − h i ) n i → C r given by µ i in the i -th component induces aninvertible element kk ( µ ). Note that µ ◦ π : C [ h ] → C r is given by ev h i in the i -th component. Since all evaluation maps ev h i induce the same kk alg -isomorphism kk ( ev ) in kk alg ( C [ h ] , C ),we have the commutative diagram in KK alg C [ h ] kk ( π ) / / kk ( ev ) (cid:15) (cid:15) C [ h ] /P ( h ) kk ( µ ) (cid:15) (cid:15) C kk ( △ ) / / C r where △ : C → C r is the diagonal morphism △ (1) = (1 , . . . , C [ h ] by C and C [ h ] / ( P ( h )) by C r in the exact triangle corresponding to (4.4), we obtain an exact trianglein KK alg S C r → A A − → C kk ( △ ) → C r . (4.5)The linearly split extension 0 → C △ → C r → C r − → S C r − → C kk ( △ ) → C r → C r − . Permuting this triangle we obtain the exact triangle S C r → S C r − → C kk ( △ ) → C r . (4.6)Since both triangles (4.5) and (4.6) complete the morphism kk ( △ ) : C → C r , by the axiomTR3 of triangulated categories we have A A − ∼ = KK alg S C r − . (cid:4) Theorem 4.18. Let A = C [ h ]( σ, P ( h )) be generalized Weyl with σ ( h ) = qh + h and P anon constant polynomial such that • q = 1 and h = 0 or • q is not a root of unity and P has a root different from h − q ,then A ∼ = KK alg C r .Proof. The result follows from Theorem 4.15, Lemma 4.16 and Proposition 4.17. (cid:4) Corollary 4.19. Let A be as in Theorem 4.18. Then A ∼ = C r in KK L p and so kk L p ( C , A ) = Z r and kk L p ( C , A ) = 0 . (cid:4) Corollary 4.19 implies K ( A ⊗ π L p ) = Z r . This is compatible with Theorem 4.5 of [12],which computes K ( A ) = Z r for A = C [ h ]( σ, P ) when σ ( h ) = h − P has r simpleroots. Examples. We apply Theorem 4.18 in the following cases.(1) The quantum Weyl algebra A q with q = 1 not a root of unity, is isomorphic to C in KK alg .(2) In the case of the primitive factors B λ of U ( sl ), we have P ( h ) = − h ( h + 1) − λ/ λ = 1, then B λ ∼ = C in KK alg . If λ = 1, then B λ ∼ = C in KK alg . This implies kk L p ( C , B λ ) = Z ⊕ Z and kk L p ( C , B λ ) = 0.(3) The quantum weighted projective line O ( WP q ( k, l )) is isomorphic to C [ h ]( σ, P ) with σ ( h ) = q l h and P ( h ) = h k l − Y i =0 (1 − q − i h ) . IVARIANT K-THEORY OF GENERALIZED WEYL ALGEBRAS 27 In the case q = 1 is not a root of unity, we have O ( WP q ( k, l )) ∼ = C l +1 in KK alg . Thisimplies kk L p ( C , O ( WP q ( k, l ))) = Z l +1 and kk L p ( C , O ( WP q ( k, l ))) = 0. (Comparewith Corollary 5.3 of [4].)In the case where q is not a root of unity and P has only h − q as a root we have thefollowing result. Proposition 4.20. The generalized Weyl algebra A = C [ h ]( σ, P ( h )) , with σ ( h ) = qh + h such that q = 1 and P has only h − q as a root, is isomorphic to C in KK alg .Proof. By Proposition 3.5, A is isomorphic to C [ h ]( σ , P ) with σ ( h ) = qh and P ( h ) = ch n with c ∈ C ∗ and n ≥ 1. The algebra C [ h ]( σ , P ) is N graded with deg h = 2, deg x = n anddeg y = n . To prove this we check that the defining relations xh = qhx, yh = q − hy, yx = ch n and xy = cq n h n are compatible with the grading.The result follows from Lemma 1.7, since the degree 0 subalgebra of A is equal to C . (cid:4) Example. The quantum plane C [ h ]( σ, h ) with σ ( h ) = qh is isomorphic to C in KK alg .4.3. The case where P is a constant polynomial. If P is a nonzero constant polynomial, then A = C [ h ]( σ, P ) is a tame smooth generalizedcrossed product and we can apply the results from [10]. Proposition 4.21. Let A = C [ h ]( σ, P ) where P = 0 is a constant polynomial, then A ∼ = S C × C in KK alg . This implies A ∼ = S C × C in KK L p and therefore we have kk L p ( C , A ) = Z and kk L p ( C , A ) = Z .Proof. In this case A is a tame smooth generalized crossed product with frame ξ i = y i and¯ ξ i = x i for i ∈ N . This frame satisfies the conditions of Definition 18 in [10], therefore wehave a linearly split extension 0 → Λ A ι → T A ¯ p → A → , that yields an exact triangle SA kk ( E ) → Λ A kk ( ι ) → T A kk (¯ p ) → A By Theorem 27 of [10], j : C [ h ] → Λ A , defined by j ( x ) = e ⊗ x induces an invertibleelement kk ( j ) and by Theorem 33 of [10], j : C [ h ] → T A defined by j ( x ) = 1 ⊗ x inducesan invertible element kk ( j ). We have a commutative diagram in KK alg Λ A kk ( ι ) / / T A C [ h ] kk ( j ) O O α / / C [ h ] . kk ( j ) O O We prove that α = 1 C [ h ] − kk ( σ ) and that 1 C [ h ] = kk ( σ ), thus concluding that α =0. By Theorem 33 of [10], we have kk ( j ) − = kk ( id, Ad ( S ⊗ kk ( ι ) − . The product kk ( j ) kk ( ι ) kk (1 , Ad ( S ⊗ φ, ψ ) : C [ h ] ⇒ T ⊗ A ⊲ C , where φ ( Q ) = e ⊗ Q and ψ ( Q ) = e ⊗ Q for all Q ∈ C [ h ]. Since φ and ψ are orthogonal kk ( φ, ψ ) = kk ( φ ) − kk ( ψ ). We now compose kk ( φ ) and kk ( ψ ) with kk ( j ) − . Theorem 27of [10] characterizes kk ( j ) − as given by a Morita equivalence defined byΞ i = S i ⊗ y i and Ξ i = S ∗ i ⊗ x i . therefore kk ( φ ) kk ( j ) − is defined by the morphism Q Q and kk ( ψ ) kk ( j ) − is defined by Q xQy = σ ( Q ). This implies that α = 1 C [ h ] − kk ( σ ).The commutative diagram C [ h ] σ / / ev (cid:15) (cid:15) C [ h ] ev (cid:15) (cid:15) C id / / C , implies that kk ( σ ) = 1 C [ h ] and thus α = 0.This implies the existence of an exact triangle in KK alg SA → C → C → A. Using Lemma 4.16, we obtain A ∼ = S C ⊕ C in KK alg . (cid:4) In the case where P = 0 we have the following result. Proposition 4.22. The generalized Weyl algebra A = C [ h ]( σ, P ( h )) with P = 0 is isomor-phic to C in KK alg .Proof. The relations xh = σ ( h ) x, yh = σ − ( h ) y, yx = 0 and xy = 0are compatible with the grading determined by deg h = 0, deg x = 1 and deg y = 1, thereforethe algebra A is N -graded. The result follows from Lemma 1.7 and the fact that the degree0 subalgebra of A is equal to C [ h ]. (cid:4) References [1] Beatriz Abadie, Søren Eilers, and Ruy Exel, Morita equivalence for crossed products by Hilbert C ∗ -bimodules , Trans. Amer. Math. Soc. (1998), no. 8, 3043–3054.[2] V. V. Bavula and D. A. Jordan, Isomorphism problems and groups of automorphisms for generalizedWeyl algebras , Trans. Amer. Math. Soc. (2001), no. 2, 769–794.[3] Tomasz Brzezi´nski, Circle and line bundles over generalized Weyl algebras , Algebr. Represent. Theory (2016), no. 1, 57–69.[4] Tomasz Brzezi´nski and Simon A. Fairfax, Quantum teardrops , Comm. Math. Phys. (2012), no. 1,151–170.[5] Joachim Cuntz, Generalized homomorphisms between C ∗ -algebras and KK -theory , Dynamics and pro-cesses (Bielefeld, 1981), Lecture Notes in Math., vol. 1031, Springer, Berlin, 1983, pp. 31–45.[6] , Bivariante K -Theorie f¨ur lokalkonvexe Algebren und der Chern-Connes-Charakter , Doc. Math. (1997), 139–182 (German, with English summary).[7] , Bivariant K -theory and the Weyl algebra , K -Theory (2005), no. 1-2, 93–137.[8] Joachim Cuntz, Ralf Meyer, and Jonathan M. Rosenberg, Topological and bivariant K -theory , Ober-wolfach Seminars, vol. 36, Birkh¨auser Verlag, Basel, 2007.[9] Joachim Cuntz and Andreas Thom, Algebraic K -theory and locally convex algebras , Math. Ann. (2006), no. 2, 339–371.[10] Olivier Gabriel and Martin Grensing, Six-term exact sequences for smooth generalized crossed products ,J. Noncommut. Geom. (2013), no. 2, 499–524. IVARIANT K-THEORY OF GENERALIZED WEYL ALGEBRAS 29 [11] Martin Grensing, Universal cycles and homological invariants of locally convex algebras , J. Funct. Anal. (2012), no. 8, 2170–2204.[12] Timothy J. Hodges, Noncommutative deformations of type- A Kleinian singularities , J. Algebra (1993), no. 2, 271–290.[13] Amnon Neeman, Triangulated categories , Annals of Mathematics Studies, vol. 148, Princeton UniversityPress, Princeton, NJ, 2001.[14] Lionel Richard and Andrea Solotar, Isomorphisms between quantum generalized Weyl algebras , J. Alge-bra Appl. (2006), no. 3, 271–285.[15] Fran¸cois Tr`eves, Topological vector spaces, distributions and kernels , Academic Press, New York-London,1967.[16] Manuel Valdivia, Topics in locally convex spaces , North-Holland Mathematics Studies, vol. 67, North-Holland Publishing Co., Amsterdam-New York, 1982. Notas de Matem´atica [Mathematical Notes], 85. Instituto de Matem´atica y Ciencias Afines (IMCA) Calle Los Bi´ologos 245. Urb SanC´esar. La Molina, Lima 12, Per´u. E-mail address : [email protected] Pontificia Universidad Cat´olica del Per´u, Secci´on Matem´aticas, PUCP, Av. Universi-taria 1801, San Miguel, Lima 32, Per´u. E-mail address ::