CUBO A Mathemati al Journal Vol.15, N o 03, (123�132). O tober 2013 K-theory for the group C∗-algebras of nilpotent dis rete groups Takahiro Sudo Department of Mathemati al S ien es, Fa ulty of S ien e, University of the Ryukyus, Senbaru 1, Nishihara, Okinawa 903-0213, Japan. sudo�math.u-ryukyu.a .jp ABSTRACT We study the K-theory groups for the group C∗-algebras of nilpotent dis rete groups, mainly, without torsion. We determine the K-theory lass generators for the K-theory groups by using generalized Bott proje tions. RESUMEN Estudiamos los grupos de la K-teoría para el grupo de álgebras C∗ de grupos dis retos nilpotentes prin ipalmente sin torsión. Determinamos los generadores de la lase de K-teoría para los grupos de la K-teoría usando proye iones generalizadas de Bott. Keywords and Phrases: group C*-algebra, K-theory, nilpotent dis rete group, Bott proje tion. 2010 AMS Mathemati s Subje t Classi� ation: 46L05, 46L80, 19K14. 124 Takahiro Sudo CUBO 15, 3 (2013) 1 Introdu tion The K-theory groups for the group C∗-algebra of the dis rete Heisenberg nilpotent group are omputed in the paper [1℄ of Anderson and Pas hke by determining the K-thoery lass generators for the K-theory groups by using the Bott proje tion on the two dimensional torus. The K- theory groups for the group C∗-algebras of the generalized dis rete Heisenberg nilpotent groups are omputed in the paper [3℄ of the author by determining the K-theoy lass generators for the K-theory groups by using generalized Bott proje tions on the higher dimensional torus de�ned in [3℄. In this paper, based on those results in the typi al ase of two-step, nilpotent dis rete groups, we study the K-theory groups for the group C∗-algebras of general, nilpotent dis rete groups, mainly, without torsion, and it is found out that we an determine the K-theory lass generators for the K-theory groups by using the generalized Bott proje tions. Moreover, several onsequen es of this main result are also obtained. Notation. We denote by C(X) the C∗-algebra of all ontinuous, omplex-valued fun tions on a ompa t Hausdor� spa e X. Denote by C∗(G) the (full or redu ed) group C∗-algebra of a nilpotent, dis rete group G (that is amenable). Note that C∗(G) is generated by unitaries that orrespond to generators of G. Denote by K0(A) and K1(A) the K0-group and the K1-group of a C ∗ -algebra A respe tively (see [2℄). 2 Finitely generated nilpotent dis rete ase Re all that a k-dimensional non ommutative torus denoted by Tkθ is the universal C ∗ -algebra generated by k unitaries Uj (1 ≤ j ≤ k) with the relations UiUj = e 2πiθij UjUi for i 6= j and θij ∈ R and θ = (θij) ∈ Mk(R) a k × k skew adjoint matrix over the �eld R of real numbers with θii = 0 and θji = −θij (i 6= j). Lemma 2.1. Let G be a �nitely generated, two-step nilpotent dis rete group without torsion and with Z its enter and C∗(G) be the group C∗-algebra of G. Then C∗(G) an be viewed as a ontinuous �eld C∗-algebra over the dual group Z∧ of Z with �bers given by non ommutative tori T n−k θλ with the relations by θλ varing over elements λ ∈ Z ∧ , where Z∧ is an ordinary torus Tk by Pontrjagin duality theorem with 1 ≤ k = rank(Z) the rank of Z, and n = rank(G). Proof. This is ertainly known and may follow from the same way as done by [1℄ in the ase of G the dis rete Heisenberg group of rank 3. Indeed, note that sin e G/Z is ommutative, the ommutator subgroup [G, G] of G is ontained in Z. As a fa t of the unitaty representation theory for G, that is identi�ed with the representation CUBO 15, 3 (2013) K-theory for the group C∗-algebras of nilpotent dis rete groups 125 theory of C∗(G), any element λ in Z∧ indu es an irredu ible indu ed representation πλ of G and of C∗(G) and any element of [G, G] is mapped to a omplex number in the one-torus T, so that the image of C∗(G) under πλ is a non ommutive torus T n−k θλ with θλ asso iated to λ. Sin e elements λ ∈ Z∧ = Tk vary ontinuously on Z∧, the norms of πλ([ui, uj]) for ui, uj unitary generators of C∗(G) orresponding to generators of G also vary ontinuously, to make a ontinuous �eld C∗-algebra over Z∧ with �bers non ommutative tori Tn−k θλ . As the main result we obtain Theorem 1. Let G be a �nitely generated, nilpotent dis rete group without torsion and C∗(G) be the group C∗-algebra of G. Then the K-theory lass generators in the K0-group K0(C ∗(G)) are given by the lass of the identity of C∗(G) and the lasses of generalized Bott proje tions ombinatori ly orresponding to abelian subalgebras of C∗(G) that orrespond to even subsets of mutually ommuting generators, even numbered, in the set of generators of G. Moreover, the K-theory lass generators in the K1-group K1(C ∗(G)) are given by the lass of unitary generators of C∗(G) that orrespond to ea h of generators of G, or orrespond both to generators of G and to the generalized Bott proje tions, ea h of whi h is obtained ombinatori ly from both the generalized Bott proje tion and ea h of generators of G whi h is not involved in the generalized Bott proje tion. The statement above an be understood pre isely by helpful examples and remark below the following proof. Proof. Re all that under the assumption on G, the group G is isomorphi to a su essive semi-dire t produ t of Z the group of integers: G ∼= Z ⋊ Z ⋊ · · · ⋊ Z rossed by Z rank(G) − 1 times, with rank(G) the rank of G. Then C∗(G) ∼= C ∗(Z) ⋊ Z ⋊ · · · ⋊ Z a su essive rossed produ t C∗-algebra by Z, and C∗(Z) ∼= C(T) by the Fourier transform. Set rank(G) = n. Let U = {g1, · · · , gn} be the set of generators of G. Note that sin e G is dis rete, the generators of G an be identi�ed with orresponding unitary generators of C∗(G) (via the left regular, or universal representation on the orresponding Hilbert spa e sin e G is amenable). Suppose that V is an even subset of U with some mutually ommuting generators of G. Denote by C∗(V) the C∗-algebra generated by elements of V. Then C∗(V) is an abelian subalgebra of C∗(G) and is isomorphi to C(T|V|) the C∗-algebra of all ontinuous, omplex-valued fun tions on the |V|-dimensional torus T|V|, where |V| is the ardinality of V. We assign su h even 126 Takahiro Sudo CUBO 15, 3 (2013) subset V ea h to the generalized Bott proje tion PV in M2(C(T |V|)) the 2 × 2 matrix algebra over C(T|V|), involving all elements of V. See the remark below for the de�nition of PV. It follows that K0(C(T |V|)) an be embedded in K0(C ∗(G)) anoni ally. Therefore, the K0- group lass [PV] an be viewed in K0(C ∗(G)). It also follows that if V 6= V ′ even subsets in U, then [PV] 6= [PV ′], i.e., PV is not equivalent to PV ′. Indeed, if PV is equivalent to PV ′, then we an dedu e a ontradition, by observing that the oordinates of T |V| orresponding to V are di�erent from those of T |V ′ | of V ′. If G is ommutative, then G ∼= Zn, and C∗(G) ∼= C(Tn) by the Fourier transform, and it is shown by [3℄ that the K0-group lasses of generalized Bott proje tions on the even dimensional tori T 2k (2 ≤ 2k ≤ n) ombinatori ly in Tn and the lass of the identity generate all lasses in K0(C(T n)). By the lemma above, if G is a �nitely generated, two-step nilpotent dis rete group without torsion, then C∗(G) an be viewed as a ontinuous �eld C∗-algebra over the dual group Z∧ of the enter Z of G with �bers given by non ommutative tori, that are su essive rossed produ t C∗-algebras by Z, generated by unitaries orresponding to generators of G not in Z, where their relations vary over Z∧. It is also shown by [3℄ that even in this ase, the same holds as in the ommutative ase. Indeed, the lass of the identity and the lasses of generalized Bott proje tions in M2(C ∗(G)) generate all lasses in K0(C ∗(G)), be ause it is noti ed in [3℄ that the lasses of the genearalized Rie�el proje tions de�ned in [3℄ and the lass of the identity generate all lasses in the K0-group of a �ber, a non ommutative torus, and the lasses of the genearalized Rie�el proje tions an not ontribute to a lass of K0(C ∗(G)) sin e those proje tions are not ontinuous over Z∧. Therefore, a proje tion for a lass of K0(C ∗(G)) an not involve the generalized Rie�el proje tions in �bers. We now onsider the general ase by indu tion. Suppose that the theorem on K0 is true when rank(G) ≤ n. Let rank(G) = n + 1. Let [p] ∈ K0(C ∗(G)) for a proje tion p in a matrix algebra over C∗(G). If p is generated by k unitaries orresponding to k generators of G with k ≤ n, then p is ontained in the group C∗-algebra C∗(H) of a nilpotent subgroup H of G generated by V the set of the k generators of G, that is C∗(H) = C∗(V) ⊂ C∗(G). By indu tion hypothesis, the lass [p] is spanned by the lass of the identity and the lasses of generalized Bott proje tions in M2(C ∗(H)). We now assume that the proje tion p involves all elements of U. We also may assume that G is not two-step nilpotent. Therefore, the quotient group G/Z is not ommutative and nilpotent. There is a quotient map q from C∗(G) to C∗(G/Z) and is extended to their matrix algebras. Then q(p) is a proje tion that involves all generators of G/Z. But by indu tion, and sin e G/Z is non ommutative, the K0-group lasses of K0(C ∗(G/Z)) an not involve all generators of G/Z. This is a ontradi tion. Hen e, there is no su h proje tion p. In fa t, this redu tion an be ontinued until that p is ontained in an abelian subalgebra of C∗(G) that is generated by unitaries orresponding to a set of mutually ommuting generators of G CUBO 15, 3 (2013) K-theory for the group C∗-algebras of nilpotent dis rete groups 127 The K1-group ase for C ∗(G) is treated similarly as in the K0-group ase above. Indeed, when G is ommutative, it is shown by [3℄ that the K1-group K1(C ∗(G)) an be generated by the lasses represented by either unitary generators of C∗(G) orresponding to generators of G or the unitaires that ombinatori ly orrespond to both generalized Bott proje tions and ea h of unitary generators of C∗(G) orresponding to generators of G. See the remark below for the de�nition of the unitaries. Moreover, even in the ase of G two-step nilpotent, the same holds for K1(C ∗(G)). And the general ase an be proved by the same way as in the proof for that ase of K0(C ∗(G)). In fa t, the onstru tion of generators of K1(C ∗(G)) an be made by bije tively orreponding to the generators of K0(C ∗(G)) onstru ted, in a suitable and ombinatori way (see the examples below). Remark. Re all from [3℄ (or [1℄ originally) that the Bott proje tion P in M2(C(T 2)) is de�ned as a proje tion-valued fun tion from T 2 to M2(C): P(w, z) = Ad(U(w, z)) ( 1 0 0 0 ) ∈ M2(C), (w, z) ∈ T 2, where U(w, z) = Y(t, z)∗ with w = e2πit ∈ T for t ∈ [0, 1] and Y(t, z) = exp ( iπt 2 K(z) ) exp ( iπt 2 S ) K(z) = ( 0 z z 0 ) , S = K(1). Moreover, the generalized Bott proje tion Qk in M2(C(T 2k)) is de�ned in [3℄ by a proje tion-valued fun tion from T 2k to M2(C): Qk(z1, · · · , z2k) = Ad(U1(z1, z2))Ad(U2(z3, z4)) · · · Ad(Uk(z2k−1, z2k)) ( 1 0 0 0 ) where Uj(·, ·) = U(·, ·) for 1 ≤ j ≤ k. Furthermore, the unitary Vk in M2(C(T 2k+1)) obtained from the generalized Bott proje tion Qk and a unitary generator u of C ∗(G) orresponding to a generator of G is de�ned in [3℄ by Vk = ( 1 0 0 1 ) + (u − 1) ⊗ Qk ∈ M2(C(T 2k+1 )). Example 2.2. If G = Zn, then C∗(G) ∼= C(Tn) by the Fourier transform, and K∗(C(T n)) ∼= Z2 n−1 for ∗ = 0, 1 ( [4℄). Note that the generators of K0(C(T n)) are given by the lass of the identity and the lasses of generalized Bott proje tions de�ned as above and the generators of K1(C(T n)) are given by the lasses of unitary generators of C∗(Zn) orresponding to generators of Zn and the 128 Takahiro Sudo CUBO 15, 3 (2013) lasses of the unitaries asso iated to both generalized Bott proje tions and the unitary generators of C∗(Zn) de�ned as above (see [3℄). More pre isely, when n = 4, the generators of K0(C ∗(Z4)) ∼= Z8 is given by the following lasses: [1], [P12], [P13], [P14], [P23], [P24], [P34], [Q1234], where 1 is the identity of C∗(Z4) and ea h Pij over T 4 is identi�ed with the Bott proje tion over T 2 that orresponds to i, j oordinates in T4, and Q1234 is the generalized Bott proje tion over T 4 . Also, the generators of K1(C ∗(Z4)) ∼= Z8 is given by the following lasses: [u1], [u2], [u3], [u4], [V123], [V124], [V134], [V234], where ea h uj is the unitary generator of C ∗(Z4) orresponding to generators of Z4 and ea h unitary Vijk in M2(C ∗(Z4)) is obtained by Pij and uk. Note that Vijk may be obtained from either Pjk and ui, or Pik and uj. Example 2.3. Let G be the dis rete Heisenberg group of rank 3: G =        1 a c 0 1 b 0 0 1     | a, b, c ∈ Z    . Then Z = Z and G/Z ∼= Z2. Also, C∗(G) is viewed as a ontinuous �led C∗-algebra over T = Z∧ with �bers non ommutative 2-tori T2θλ. It is omputed by [1℄ (and also [3℄) that K0(C ∗ (G)) ∼= Z 3, K1(C ∗ (G)) ∼= Z 3, and the generators of K0(C ∗(G)) is given by the lass of the identity of C∗(G) and two lasses of the Bott proje tions over T 2 , where their domains are di�erent in the sense as one T 2 = T×Z∧ with the �rst fa tor T orresponding to one of two generators of the �bers and the other T 2 = T × Z∧ with the �rst fa tor T orresponding to the other of two generators of the �bers, and the generators of K1(C ∗(G)) is given by two lasses of unitary generators of C∗(G) orresponding both to generators of G and to one of two Bott proje tions and the lass of the unitary of M2(C ∗(G)) obtained from both the hosen Bott proje tion and the rest of unitary generators of C∗(G) orresponding to generators of G. Namely, K0(C ∗(G)) ∼= 〈[1], [P13], [P23]〉, K1(C ∗(G)) ∼= 〈[u1], [u3], [V123]〉, where the equations mean that the left hand sides are generated by the lasses in the bra kets in the right hand sides, and the third oordinate in T 3 orresponds to Z∧ and the unitary V123 is CUBO 15, 3 (2013) K-theory for the group C∗-algebras of nilpotent dis rete groups 129 obtained from the Bott proje tion P13 and u2. Note that the above set of generators of K1(C ∗(G)) may be repla ed with {[u2], [u3], [V ′ 123]}, where V ′ 123 is obtained from the Bott proje tion P23 and u1. Example 2.4. Let G × G be the dire t produ t of G the dis rete Heisenberg nilpotent group of rank 3. Then C∗(G×G) ∼= C∗(G)⊗C∗(G) the tensor produ t of C∗(G). Sin e Kj(C ∗(G)) (j = 0, 1) are torsion free, the Künneth theorem in K-theory for C∗-algebras (see [2℄) implies that K0(C ∗(G × G)) ∼= [K0(C ∗(G)) ⊗ K0(C ∗(G))] ⊕ [K1(C ∗(G)) ⊗ K1(C ∗(G))] ∼= [Z 3 ⊗ Z3] ⊕ [Z3 ⊗ Z3] ∼= Z 18, K1(C ∗ (G × G)) ∼= [K0(C ∗ (G)) ⊗ K1(C ∗ (G))] ⊕ [K1(C ∗ (G)) ⊗ K0(C ∗ (G))] ∼= [Z 3 ⊗ Z3] ⊕ [Z3 ⊗ Z3] ∼= Z 18. Our theorem tells us that K0(C ∗ (G × G)) ∼= 〈[1], [P13], [P23], [P46], [P56], [P14], [P15], [P16], [P24], [P25], [P26], [P34], [P35], [P36], [Q1346], [Q1356], [Q2346], [Q2356]〉, where the subindi es 1, 2, 3 orrespond to the unitary generators uj of C ∗(G)⊗C and the subindi es 4, 5, 6 orrespond to the unitary generators uj of C⊗C ∗(G) and both subindi es 3 and 6 orresponds to the enter Z of G. Also, K1(C ∗(G × G)) ∼= 〈[u1], [u3], [V123], [u4], [u6], [V456], [V(P14, u2)], [V(P15, u2)], [V(P16, u2)], [V(P24, u3)], [V(P25, u3)], [V(P26, u3)], [V(P34, u5)], [V(P35, u6)], [V(P36, u4)], [V(Q1346, u2)], [V(Q1356, u2)], [V(Q2346, u1)]〉, where ea h V(Pij, uk) means the unitary obtained from Pij and uk and ea h V(Qijkl, um) means the unitary obtained from Qijkl and um. Note that the unions of subindei es su h as (1, 2, 4) of (14, 2) and (1, 2, 5) of (15, 2) are taken only on e among ombinations of (i, j, k) with i < j < k. Also, the hoi e of adding um to either Pij or Qijkl may be di�erent to make the same set of unions of subindi es, and the set of generators of K0(C ∗(G × G)) orresponds to the set of generators of K1(C ∗(G × G)) bije tively. Corollary 2.5. If G is a �nitely generated, dis rete nilpotent group without torsion, then K0(C ∗(G)) ∼= K1(C ∗(G)). Example 2.6. The isomorphism in the orollary above does not hold if G has torsion. Indeed, if G = Zn = Z/nZ (n ≥ 2) a y li group, then C ∗(G) ∼= Cn, so that K0(C ∗(G)) ∼= Zn but K1(C ∗(G)) ∼= 0. 130 Takahiro Sudo CUBO 15, 3 (2013) Corollary 2.7. If G is a �nitely generated, dis rete nilpotent group without torsion, then the K-theory groups K0(C ∗(G)) and K1(C ∗(G)) are torsion free. Proof. This follows from the onstru tion of the generators of K0(C ∗(G)) and K1(C ∗(G)) obtained in the theorem above. Remark. Possibly, in the last orollary, the group G may have torsion. Example 2.8. We onsider a version of the dis rete Heisenberg nilpotent group with torsion (see [1℄ or [3℄ for the dis rete Heisenberg nilpotent group). Let G = Z22 ⋊α Z2 be a semi-dire t produ t of the produ t group Z 2 2 of the y i group Z2 = Z/2Z by an a tion of Z2 de�ned by αt(b + tc, c) for b, c, t ∈ Z2. Then the group C ∗ -algebra C∗(G) is isomorphi to the rossed produ t C(Z∧2 × Z ∧ 2 ) ⋊α∧ Z2 via the Fourier transform, where the dual a tion α ∧ of Z2 on the produ t spa e of the dual group Z ∧ 2 ∼= Z2 is de�ned by α ∧ t (z, w) = (z, z tw) for z, w ∈ Z∧2 via the duality α∧t (ϕz,w)(b, c) = ϕz,w(b + tc, c) = z b+tcwc = ϕz,ztw(b, c) where ϕz,w ∈ Z ∧ 2 ×Z ∧ 2 indenti�ed with (z, w) ( f. [5℄). We then obtain the following de omposition: C(Z∧2 × Z ∧ 2 ) ⋊α∧ Z2 ∼= [C ⊗ (C 2 ⋊α∧ Z2)] ⊕ [C ⊗ (C 2 ⋊α∧ Z2)] ∼= [C 2 ⊗ C∗(Z2)] ⊕ [M2(C)] ∼= [C 2 ⊗ C2] ⊕ M2(C) ∼= C 4 ⊕ M2(C) where the a tion α∧ on C2 in the �rst dire t summand is trivial and that in the se ond is the shift. Therefore, K0(C ∗ (G)) ∼= Z 5, but K1(C ∗ (G)) ∼= 0. Hen e the K-theory groups are torsion free. In this ase, the dire t sum fa tor Z 4 in Z 5 = K0(C ∗(G)) omes from C4 in C∗(G) whi h is a maximal abelian subalgebra of C∗(G) but the other dire t sum fa tor Z in Z5 = K0(C ∗(G)) omes from M2(C) in C ∗(G) whi h is a non ommutative subalgebra of C∗(G). Therefore, the ase with torsion is ertainly di�erent from the torsion free ase onsidered above, but the nilpotent ase with torsion is just the same as the abelian ase with torsion as in the example above, in the K-theory level. Corollary 2.9. If G is a �nitely generated, dis rete nilpotent group without torsion, then both K0(C ∗(G)) and K1(C ∗(G)) are isomorphi to a �nitely generated, free abelian group, i.e., Zm for some positive integer m. CUBO 15, 3 (2013) K-theory for the group C∗-algebras of nilpotent dis rete groups 131 3 In�nitely generated ase We assume that G is a ountable dis rete group. Theorem 2. Let G be an in�nitely generated, nilpotent dis rete group without torsion. Then both K0(C ∗(G)) and K1(C ∗(G)) of the group C∗-algebra C∗(G) are isomorphi to an indu tive limit of �nitely generated free abelian groups: K0(C ∗ (G)) ∼= K1(C ∗ (G)) ∼= lim −→ Z mn, for some positive integers mn with mn < mn+1, where the onne ting maps Z mn → Zmn+1 are inje tive. Therefore, K0(C ∗(G)) ∼= K1(C ∗(G)) ∼= ⊕ ∞ Z, whi h is the in�nite dire t sum of Z, as a group. Proof. Let U = {g1, g2, · · · } be an in�nite set of generators of G and set Un = {g1, g2, . . . , gn}, where gn+1 is not generated by g1, · · · , gn. Let C ∗(Un) denote the C ∗ -algebra generated by the elements of C∗(G) that orrespond to the elements of Un. Then C ∗(Un) is a C ∗ -subalgebra of C∗(G). There is the anoni al in lusion in from C ∗(Un) → C ∗(Un+1). It follows that C ∗(G) is an indu tive limit of the C∗-subalgebras C∗(Un) under the in lusions in. By ontinuity of K-theory, we have Kj(C ∗(G)) ∼= lim −→ Kj(C ∗(Un)) for j = 0, 1. By the theorem in the previous se tion, we see that both Kj(C ∗(Un)) for j = 0, 1 are isomorphi to Z mn for some positive integer mn and mn ≤ mn+1 and also that there is a anoni al in lusion from Kj(C ∗(Un)) ∼= Z mn to Kj(C ∗(Un+1)) ∼= Z mn+1 . We need to he k that mn 6= mn+1 for ea h n. Note that the group Hn generated by elements of Un an be written as a su essive semi-dire t produ t by Z: Hn ∼= Z ⋊ Z ⋊ · · · ⋊ Z rossed by Z n − 1 times. Then Hn+1 ∼= Hn ⋊ Z. It follows that the a tion of Z on Hn an not be non-trivial on every generator of Hn. Be ause, if non-trivial, Hn+1 is not nilpotent (but solvable). Indeed, then there is no enter in Hn+1, a ontradi tion to the nilpotentness of Hn+1. Therefore, there is a generator of Hn su h that the a tion of Z is trivial on it. Therefore, we an onstru t a new Bott proje tion from these ommuting elements of Hn+1, and not from Hn. It follows that mn < mn+1. Example 3.1. If G is an in�nitely generated abelian dis rete group, then C∗(G) is isomorphi to an indu tive limit of C(Tn) with the anoni al in lusion from C(Tn) to C(Tn+1). Then Kj(C ∗ (G)) ∼= lim −→ Kj(C(T n )) ∼= lim −→ Z 2 n−1 ∼= ⊕ ∞ Z 132 Takahiro Sudo CUBO 15, 3 (2013) for j = 0, 1. If G is an indu tive limit of the produ t groups ΠnZ2 with the anoni al in lusion from Π n Z2 to Πn+1Z2, then G is ommutative and in�nitely generated and has torsion. Then C∗(G) ∼= lim −→ C∗(ΠnZ2) ∼= lim−→ ⊗nC∗(Z2) ∼= lim −→ ⊗nC2 ∼= lim −→ C 2 n ∼= ⊕ ∞ C where the last side means the in�nite dire t sum of C, so that Kj(C ∗ (G)) ∼= Kj(⊕ ∞ C) ∼= ⊕ ∞Kj(C) ∼= { ⊕∞Z if j = 0, 0 if j = 1. Re eived: Mar h 2012. A epted: September 2013. Referen es [1℄ J. Anderson and W. Pas hke, The rotation algebra, Houston J. Math. 15 (1989), 1-26. [2℄ B. Bla kadar, K-theory for Operator Algebras, Se ond Edition, Cambridge, (1998). [3℄ T. Sudo, K-theory of ontinuous �elds of quantum tori, Nihonkai Math. J. 15 (2004), 141- 152. [4℄ N. E. Wegge-Olsen, K-theory and C∗-algebras, Oxford Univ. Press (1993). [5℄ D. P. Williams, Crossed produ ts of C∗-algebras, SURV 134, Amer. Math. So . (2007).