CUBO A Mathematical Journal Vol.10, N o ¯ 03, (103–114). October 2008 The Flip Crossed Products of the C ∗ -Algebras by Almost Commuting Isometries Takahiro Sudo Department of Mathematical Sciences, Faculty of Science, University of the Ryukyus, Nishihara, Okinawa 903-0213, Japan email: sudo@math.u-ryukyu.ac.jp ABSTRACT We study the flip crossed products of the C∗-algebras by almost commuting isometries and obtain some results on their structure, K-theory, and continuity. RESUMEN Estudiamos el produto flip crossed de una C∗-algebra mediante isometrias casi com- mutando y obtenemos algunos resultados sobre su estructura, K-teoria, y continuidad. Key words and phrases: C*-algebra, Continuous field, K-theory, Isometry. Math. Subj. Class.: 46L05, 46L80. 104 Takahiro Sudo CUBO 10, 3 (2008) Introduction Recall that the soft torus Aε of Exel [3] (for any ε ∈ [0, 2] the closed interval) is defined to be the universal C∗-algebra generated by almost commuting two unitaries uε,1 and uε,2 in the sense that ‖uε,2uε,1 − uε,1uε,2‖ ≤ ε. Its K-theory is computed in [3] by showing that it can be represented as a crossed product by Z and applying the Pimsner-Voiculescu six-term exact sequense for the crossed product. It is shown by Exel [4] that there exists a continuous field of C∗-algebras on [0, 2] with fibers the soft tori varying continuously. Furthermore, K-theory and continuity of the crossed products of Aε by the flip (a Z2-action) are considered by Elliott, Exel and Loring [2]. On the other hand, we [8] began to study continuous fields of C∗-algebras by almost commuting isometries and obtained some similar results (but different in some senses) on their structure, K- theory and continuity as those by Exel. In this paper we consider those properties for the flip crossed products of the C∗-algebras generated by almost commuting isometries. Refer to [1], [5], and [9] for some basics in C∗-algebras and K-theory. 1 The flip crossed products by isometries The Toeplitz algebra is defined to be the universal C∗-algebra generated by a (non-unitary) isom- etry, and it is denoted by F, which is also the semigroup C∗-algebra C∗(N) of the semigroup N of natural numbers. The C∗-algebra C(T) of all continuous functions on the 1-torus T is the universal C∗-algebra generated by a unitary, which is also the group C∗-algebra C∗(Z) of the group Z of integers. There is a canonical quotient map from F to C(T) by universality, whose kernel is iso- morphic to the C∗-algebra K of all compact operators on a separable infinite dimensional Hilbert space (cf. [5]). Definition 1.1 For ε ∈ [0, 2], the soft Toeplitz tensor product denoted by F ⊗ε F is defined to be the universal C∗-algebra generated by two isometries sε,1, sε,2 such that ‖sε,2sε,1 − sε,1sε,2‖ ≤ ε (ε-commuting). Let π : F⊗ε F → Aε be the canonical onto ∗-homomorphism sending the isometry generators to the unitary generators. Remark. Refer to [8], in which super-softness is further defined and assumed, but it should be unnecessary from the universality argument (as given below). Instead, in fact, another norm estimate of the form ‖sε,2s ∗ ε,1 −s ∗ ε,1sε,2‖≤ ε (ε-∗-commuting) may be required, but we omit such an estimate in what follows. If not assuming the estimate, F⊗ε F should be replaced with C ∗(N2)ε, where C∗(N2) is the semigroup C∗-algebra of N2 (in what follows). Definition 1.2 The flip on F ⊗ε F is the (non-unital) endomorphism σ defined by σ(sε,j ) = s ∗ ε,j for j = 1, 2. Since σ2 is the identity on F ⊗ε F, we denote by (F ⊗ε F) ⋊σ Z2 the crossed product of F ⊗ε F by the action σ of the order 2 cyclic group Z2, i.e., a flip crossed product. CUBO 10, 3 (2008) The Flip Crossed Products of the C∗-Algebras ... 105 Definition 1.3 For ε ∈ [0, 2], we define Eε to be the universal C ∗-algebra generated by an isometry t1 and the elements tn+1 = u nt1(u ∗)n for n ∈ N, where u is an isometry, such that ‖ut1−t1u‖≤ ε. Let αε be the endomorphism of Eε defined by αε(tn) = tn+1 = utnu ∗ for n ∈ N. Let Eε ⋊αε N be the semigroup crossed product of Eε by the action αε of the additive semigroup N of natural numbers. Remark. Note that F⊗2 F (or C ∗(N2)2) is isomorphic to the unital full free product F∗C F, which is also isomorphic to the full semigroup C∗-algebra C∗(N ∗ N) of the free semigroup N ∗ N. As in the above remark, another estimate ‖ut∗ 1 − t∗ 1 u‖≤ ε may be required accordingly. It is shown in [8] that F⊗ε F ∼= Eε ⋊αε N, where the map ϕ from F⊗ε F to Eε ⋊αε N is defined by ϕ(sε,1) = t1 and ϕ(sε,2) = u, and its inverse ψ is given by ψ(tn+1) = s n ε,2sε,1(s ∗ ε,2) n for n ∈ N and n = 0 and ψ(u) = sε,2. Proposition 1.4 For ε ∈ [0, 2], we have the following isomorphism: (F ⊗ε F) ⋊σ Z2 ∼= Eε ⋊αε∗β (N ∗ Z2), where N ∗ Z2 is the free product of N and Z2, and the action β on Eε is given by β(tn) = t ∗ n for n ∈ N. Proof. The crossed product (F ⊗ε F) ⋊σ Z2 is the universal C ∗-algebra generated by isometries sε,1, sε,2 and a unitary ρ such that ‖sε,2sε,1 − sε,1sε,2‖ ≤ ε and ρsε,jρ ∗ = sε,j (j = 1, 2) with ρ2 = 1, while Eε ⋊αε∗β (N ∗ Z2) is the C ∗-algebra generated by isometries t1, u and a unitary v such that ‖ut1 −t1u‖≤ ε and tn+1 = utnu ∗ = unt1(u ∗)n for n ∈ N, and vt1v ∗ = t∗ 1 and vuv∗ = u∗ with v2 = 1. The isomorphism between them is given by sending sε,1,sε,2, and ρ to t1,u, and v respectively (cf. [2]). 2 Theorem 1.5 For 0 ≤ ε < 2, we obtain the K-theory isomorphisms: K0((F ⊗ε F) ⋊σ Z2) ∼= Z 9 , K1((F ⊗ε F) ⋊σ Z2) ∼= 0. Moreover, Kj((F ⊗ε F) ⋊σ Z2) ∼= Kj((F ⊗ F) ⋊σ Z2) for j = 0, 1. Proof. Since F ⊗ε F ∼= Eε ⋊αε N and αε is a corner endomorphism on Eε, note that Eε ⋊αε N is isomorphic to a corner of (Eε ⊗ K) ⋊ρ∧ ε ⊗id Z, i.e., p((Eε ⊗ K) ⋊ρ∧ ε ⊗id Z)p for a certain projection p, where ρ∧ε is the dual action of the circle action on Eε ⋊αε N and id is the identity action on K (this is a variation of [6], and see also [7]). Hence, (Eε ⋊αε N) ⋊σ Z2 is isomorphic to 106 Takahiro Sudo CUBO 10, 3 (2008) p((Eε ⊗ K) ⋊ρ∧ ε ⊗id Z)p ⋊σ Z2. Therefore, Kj((Eε ⋊αε N) ⋊σ Z2) ∼= Kj(p((Eε ⊗ K) ⋊ρ∧ ε ⊗id Z)p ⋊σ Z2) ∼= K Z2 j (p((Eε ⊗ K) ⋊ρ∧ ε ⊗id Z)p) ∼= K Z2 j (p((Eε ⊗ K) ⋊ρ∧ ε ⊗id Z)p⊗ K) ∼= K Z2 j (((Eε ⊗ K) ⋊ρ∧ ε ⊗id Z) ⊗ K) ∼= Kj((Eε ⊗ K) ⋊ρ∧ ε ⊗id Z ⋊ Z2), where KZ2 j (·) is the equivariant K-theory, and note that p((Eε⊗K) ⋊ρ∧ ε ⊗id Z)p is stably isomorphic to (Eε ⊗ K) ⋊ρ∧ ε ⊗id Z, and (Eε ⊗ K) ⋊ρ∧ ε ⊗id Z ⋊ Z2 ∼= (Eε ⊗ K) ⋊σ′ ε ∗σ⊗id (Z2 ∗ Z2) ∼= (Eε ⋊σ′ ε ∗σ (Z2 ∗ Z2)) ⊗ K since Z ⋊ Z2 ∼= Z2 ∗ Z2, where σ ′ ε(1) = ρ ∧ ε (1)σ(1) (cf. [2]). Set Fε = Eε ⋊σ′ε∗σ (Z2 ∗ Z2). There exists the following six-term exact sequence (A) (cf. [2]): K0(Eε) −−−−→ K0(Eε ⋊σ′ ε Z2) ⊕K0(Eε ⋊σ Z2) −−−−→ K0(Fε) x     y K1(Fε) ←−−−− K1(Eε ⋊σ′ ε Z2) ⊕K1(Eε ⋊σ Z2) ←−−−− K1(Eε). Consider the following exact sequence: 0 → Iε → Eε → π(Eε) = B ′ ε → 0, where π is the canonical quotient map from Eε to the quotient π(Eε) = B ′ ε, where B ′ ε is the universal C ∗-algebra generated by unitaries un+1 = w nv(w∗)n for n ∈ N and n = 0, where π(tn+1) = π(u) nπ(t1)π(u ∗)n = un+1 with v = π(t1) and w = π(u). As shown in [8], K-theory groups of Iε are the same as those of K. Since this quotient is invariant under the action β = σ′ε or σ, we have the following exact sequence: (B) : 0 → Iε ⋊β Z2 → Eε ⋊β Z2 → π(Eε) ⋊β Z2 → 0 and Iε ⋊β Z2 ∼= Iε ⊗C ∗(Z2) and the group C ∗-algebra C∗(Z2) is isomorphic to C 2 via the Fourier transform. As shown in [2], it is deduced that π(Eε) ⋊β Z2 is homotopy equivalent to the crossed product C(T) ⋊β′ Z2, where β ′(z) = z−1 for z ∈ T. It follows that Kj (π(Eε) ⋊β Z2) is isomorphic to Kj (C(T) ⋊β′ Z2). Since the points {±1} in T is fixed under the action β ′, we have 0 → C0(T \{±1}) ⋊β′ Z2 → C(T) ⋊β′ Z2 →⊕ 2C∗(Z2) → 0, where C0(T\{±1}) is the C ∗-algebra of all continuous functions on T\{±1} vanishing at infinity, and C0(T \{±1}) ⋊β′ Z2 ∼= C0(R) ⊗ (C 2 ⋊β′ Z2) ∼= C0(R) ⊗M2(C) and C ∗(Z2) ∼= C 2. Hence the following six-term exact sequence is obtained: 0 −−−−→ K0(C(T) ⋊β′ Z2) −−−−→ Z 4 x     y 0 ←−−−− K1(C(T) ⋊β′ Z2) ←−−−− Z, CUBO 10, 3 (2008) The Flip Crossed Products of the C∗-Algebras ... 107 where Kj (C0(R) ⊗ M2(C)) ∼= Kj+1(C) (mod 2) and Kj(⊕ 2 C 2) ∼= ⊕4Kj (C). It follows that K0(C(T) ⋊β′ Z2) ∼= Z 3 and K1(C(T) ⋊β′ Z2) ∼= 0 (cf. [2]). Therefore, for the above exact sequence (B), we obtain the diagram: Z 2 −−−−→ K0(Eε ⋊β Z2) −−−−→ Z 3 x     y 0 ←−−−− K1(Eε ⋊β Z2) ←−−−− 0, where Kj(K ⊗C ∗(Z2)) ∼= Kj (C 2). Hence we obtain K0(Eε ⋊β Z2) ∼= Z 5 and K1(Eε ⋊β Z2) ∼= 0. This implies that the diagram (A) is Z −−−−→ Z5 ⊕ Z5 −−−−→ K0(Fε) x     y K1(F0) ←−−−− 0 ⊕ 0 ←−−−− 0 where it is shown in [8] that K0(Eε) ∼= Z and K1(Eε) ∼= 0. It follows that K0(Fε) ∼= Z 9 and K1(Fε) ∼= 0. It follows from this and the first part shown above that K0((F⊗ε F) ⋊σ Z2) ∼= Z 9 and K1((F ⊗ε F) ⋊σ Z2) ∼= 0. The second claim follows from the case ε = 0 and the same argument as above. Note that F ⊗ F ∼= F ⋊id N, where id is the trivial action. 2 Corollary 1.6 For 0 ≤ ε < 2, the natural onto ∗-homomorphism ϕε,0 from (F ⊗ε F) ⋊σ Z2 to (F ⊗ F) ⋊σ Z2 sending sε,j to s0,j (j = 1, 2) induces the isomorphism between their K-groups. Proposition 1.7 There exists a continuous field of C∗-algebras on the closed interval [0, 2] such that its fibers are (F ⊗ε F) ⋊σ Z2 for ε ∈ [0, 2], and for any a ∈ (F ⊗2 F) ⋊σ Z2, the sections [0, 2] ∋ ε 7→ ϕε(a) ∈ (F⊗ε F) ⋊σ Z2 are continuous, where ϕε : (F⊗2 F) ⋊σ Z2 → (F⊗ε F) ⋊σ Z2 is the natural onto ∗-homomorphism sending s2,j to sε,j (j = 0, 1). Proof. As shown before, (F ⊗ε F) ⋊σ Z2 ∼= (Eε ⋊αε N) ⋊σ Z2. Furthermore, this is isomorphic to p((Eε ⊗ K) ⋊ρ∧ ε ⊗id Z)p ⋊σ Z2. Hence it follows that ((Eε ⋊αε N) ⋊σ Z2) ⊗ K ∼= (p((Eε ⊗ K) ⋊ρ∧ ε ⊗id Z)p ⋊σ Z2) ⊗ K ∼= (p((Eε ⊗ K) ⋊ρ∧ ε ⊗id Z)p⊗ K) ⋊σ⊗id Z2 ∼= (((Eε ⊗ K) ⋊ρ∧ ε ⊗id Z) ⊗ K) ⋊σ⊗id Z2 ∼= ((Eε ⊗ K ⊗ K) ⋊ρ∧ ε ⊗id⊗id Z) ⋊σ⊗id Z2 ∼= ((Eε ⊗ K) ⋊ρ∧ ε ⊗id Z) ⋊σ Z2 ∼= (Eε ⊗ K) ⋊σ′ ε ∗σ⊗id (Z2 ∗ Z2). It is deduced from [2] that there exists a continuous field of C∗-algebras on [0, 2] such that its fibers are (Eε ⊗ K) ⋊σ′ ε ∗σ⊗id (Z2 ∗ Z2) for ε ∈ [0, 2], and for any b ∈ (E2 ⊗ K) ⋊σ′ 2 ∗σ⊗id (Z2 ∗ Z2), the 108 Takahiro Sudo CUBO 10, 3 (2008) sections [0, 2] ∋ ε 7→ ψε(b) ∈ (Eε⊗K)⋊σ′ ε ∗σ⊗id (Z2∗Z2) are continuous, where ψε is the unique onto ∗-homomorphism from (E2 ⊗ K) ⋊σ′ 2 ∗σ⊗id (Z2 ∗ Z2) to (Eε ⊗ K) ⋊σ′ ε ∗σ⊗id (Z2 ∗ Z2). Cutting down this continuous field by cutting down the fibers from ((Eε ⋊αε N) ⋊σ Z2)⊗ K to (Eε ⋊αε N) ⋊σ Z2 by minimal projections, we obtain the desired continuous field. 2 2 The flip crossed products by n isometries The n-fold tensor product ⊗nF of F is the universal C∗-algebra generated by mutually commuting and ∗-commuting n isometries, while the universal C∗-algebra generated by mutually commuting n isometries is just the semigroup C∗-algebra C∗(Nn) of the semigroup Nn. The C∗-algebra C(Tn) of all continuous functions on the n-torus Tn is the universal C∗-algebra generated by mutually commuting n unitaries, which is also the group C∗-algebra C∗(Zn) of the group Zn. There is a canonical quotient map from ⊗nF to C(Tn) ∼= ⊗nC(T) by universality, Definition 2.1 For ε ∈ [0, 2], the soft Toeplitz n-tensor product denoted by ⊗nε F is defined to be the universal C∗-algebra generated by n isometries sε,j (1 ≤ j ≤ n) such that ‖sε,ksε,j−sε,jsε,k‖≤ ε (1 ≤ j,k ≤ n). Remark. Note that, in fact, the norm estimates of the form ‖sε,ks ∗ ε,j −s ∗ ε,jsε,k‖≤ ε may be further required (and in what follows). If not assuming these estimates, ⊗nε F should be replaced with C∗(Nn)ε in the same sense (and in what follows). Definition 2.2 The flip on ⊗nε F is the (non-unital) endomorphism σ defined by σ(sε,j ) = s ∗ ε,j for 1 ≤ j ≤ n. Since σ2 is the identity on ⊗nε F, we denote by (⊗ n ε F) ⋊σ Z2 the crossed product of ⊗ n ε F by the action σ of Z2. Definition 2.3 For ε ∈ [0, 2], we define Emε to be the universal C ∗-algebra generated by n isome- tries t (j) 1 (1 ≤ j ≤ m) and the partial isometries t (j) n+1 = u nt (j) 1 (u∗)n for n ∈ N, where u is an isometry such that ‖ut (j) 1 − t (j) 1 u‖ ≤ ε and ‖t (k) 1 t (j) 1 − t (j) 1 t (k) 1 ‖ ≤ ε (1 ≤ j,k ≤ m). Let αε be the endomorphism of Emε defined by αε(t (j) n ) = t (j) n+1 = ut (j) n u ∗ for n ∈ N. Let Emε ⋊αε N be the semigroup crossed product of Emε by the action αε of N. Remark. Note that ⊗n 2 F (or C∗(Nn)2) is isomorphic to the unital full free product ∗ n C F, which is also isomorphic to the full semigroup C∗-algebra C∗(∗nN) of the free semigroup ∗nN. As in the above remark, the additional estimates ‖u(t (j) 1 )∗ − (t (j) 1 )∗u‖≤ ε and ‖t (k) 1 (t (j) 1 )∗ − (t (j) 1 )∗t (k) 1 ‖ ≤ ε may be required accordingly. It is shown as in [8] that ⊗m+1ε F ∼= Emε ⋊αε N as in the case in Section 1. Proposition 2.4 For ε ∈ [0, 2], we have (⊗m+1ε F) ⋊σ Z2 ∼= E m ε ⋊αε∗β (N ∗ Z2), CUBO 10, 3 (2008) The Flip Crossed Products of the C∗-Algebras ... 109 where the action β on Emε is given by β(t (j) n ) = (t (j) n ) ∗ for n ∈ N and 1 ≤ j ≤ m. Proof. This is shown as in the proof of Proposition 1.4 similarly. 2 Theorem 2.5 For 0 ≤ ε < 2, we obtain (inductively) K0((⊗ m+1 ε F) ⋊σ Z2) ∼= Z 2 m+2 +3, K1((⊗ m+1 ε F) ⋊σ Z2) ∼= 0. Moreover, Kj((⊗ m+1 ε F) ⋊σ Z2) ∼= Kj((⊗ m+1F) ⋊σ Z2) for j = 0, 1. Proof. Since ⊗m+1ε F ∼= Emε ⋊αε N, note that E m ε ⋊αε N is isomorphic to a corner of (E m ε ⊗K)⋊ρ∧ε ⊗idZ, i.e., p((Emε ⊗ K) ⋊ρ∧ε ⊗id Z)p for a certain projection p, where ρ ∧ ε is the dual action of the circle action on Emε ⋊αε N and id is the identity action on K (this is a variation of [6], and see also [7]). Hence, (Emε ⋊αε N) ⋊σ Z2 is isomorphic to p((E m ε ⊗ K) ⋊ρ∧ε ⊗id Z)p ⋊σ Z2. Therefore, Kj((E m ε ⋊αε N) ⋊σ Z2) ∼= Kj(p((E m ε ⊗ K) ⋊ρ∧ε ⊗id Z)p ⋊σ Z2) ∼= K Z2 j (p((Emε ⊗ K) ⋊ρ∧ε ⊗id Z)p) ∼= K Z2 j (p((Emε ⊗ K) ⋊ρ∧ε ⊗id Z)p⊗ K) ∼= K Z2 j (((Emε ⊗ K) ⋊ρ∧ε ⊗id Z) ⊗ K) ∼= Kj((E m ε ⊗ K) ⋊ρ∧ε ⊗id Z ⋊ Z2), where p((Emε ⊗ K) ⋊ρ∧ε ⊗id Z)p is stably isomorphic to (E m ε ⊗ K) ⋊ρ∧ε ⊗id Z, and (Emε ⊗ K) ⋊ρ∧ε ⊗id Z ⋊ Z2 ∼= (E m ε ⊗ K) ⋊ρ∧ε ∗σ⊗id (Z2 ∗ Z2) ∼= (E m ε ⋊ρ∧ε ∗σ (Z2 ∗ Z2)) ⊗ K since Z ⋊ Z2 ∼= Z2 ∗Z2 (cf. [2]). Set F m ε = E m ε ⋊ρ∧ε ∗σ (Z2 ∗Z2). There exists the following six-term exact sequence (A)m (cf. [2]): K0(E m ε ) −−−−→ K0(E m ε ⋊ρ∧ε Z2) ⊕K0(E m ε ⋊σ Z2) −−−−→ K0(F m ε ) x     y K1(F m ε ) ←−−−− K1(E m ε ⋊ρ∧ε Z2) ⊕K1(E m ε ⋊σ Z2) ←−−−− K1(E m ε ). We now have the following exact sequence: 0 → Imε ⋊ Z2 → E m ε ⋊ Z2 → π(E m ε ) ⋊ Z2 → 0, where the map π is sending isometries of Emε to unitaries with the same norm estimates by universality, and Imε is the kernel of π, and the action of Z2 is given by ρ ∧ ε or σ. Furthermore, it follows that Imε ⋊ Z2 ∼= Imε ⊗C ∗(Z2) and the K-theory of I m ε is the same as that of K. It is deduced that π(Emε ) ⋊ Z2 is homotopy equivalent to C(T m) ⋊σ Z2, where β(zj ) = (z −1 j ) for (zj ) ∈ T m. Since the points (±1, · · · ,±1) ∈ Tm are fixed under α, we have 0 → C0(T m \ (±1, · · · ,±1)) ⋊ Z2 → C(T m) ⋊ Z2 →⊕ 2 m C ∗(Z2) → 0, 110 Takahiro Sudo CUBO 10, 3 (2008) where C0(X) is the C ∗-algebra of all continuous functions on a locally compact Hausdorff space X vanishing at infinity (in what follows). Set Xm+1 = T m \ (±1, · · · ,±1). By considering invariant subspaces in Xm+1 under β, we obtain a finite composition series {Lj} m j=1 of C0(Xm+1) ⋊ Z2 such that L0 = {0}, Lj = C0(Xj ) × Z2, and Lj/Lj−1 ∼= ⊕ mCm−j+1C0((T \{±1}) m−j+1) ⋊ Z2, where mCm−j+1 mean the combinations. Furthermore, C0((T \{±1}) m−j+1) ⋊ Z2 ∼= C0(R m−j+1) ⊗ (C(Πm−j+1{±i}) ⋊ Z2) and C(Πm−j+1{±i}) ⋊ Z2 ∼= ⊕ m−j+1(C2 ⋊ Z2) ∼= ⊕ m−j+1M2(C), where T\{±1} is homeomorphic to iR∪(−i)R so that the above isomorphisms are deduced from considering orbits under β in this identification. Set C(m,j) = mCm−j+1(m− j + 1). Thus, the following six-term exact sequences are obtained: K0(Lj−1) −−−−→ K0(Lj ) −−−−→ Km−j+1(⊕ C(m,j) C) x     y Km−j+2(⊕ C(m,j) C) ←−−−− K1(Lj ) ←−−−− K1(Lj−1). Now consider the case m = 2. Then 0 → C0(T 2 \ (±1,±1)) ⋊ Z2 → C(T 2) ⋊ Z2 →⊕ 2 2 C∗(Z2) → 0. Furthermore, 0 → C0(X1)⋊Z2 → C0(X2)⋊Z2 → C0(X2\X1)⋊Z2 → 0, where X2 = T 2\(±1,±1), X1 = (T \ {±1}) 2, and C0(X2 \ X1) ⋊ Z2 is isomorphic to ⊕ 2C0(T \ {±1}) ⋊ Z2. We have the following six-term exact sequence: Z 2 −−−−→ K0(C0(X2) ⋊ Z2) −−−−→ 0 x     y Z 2 ←−−−− K1(C0(X2) ⋊ Z2) ←−−−− 0, which implies K0(C0(X2) ⋊ Z2) ∼= 0 and K1(C0(X2) ⋊ Z2) ∼= 0. Thus, 0 −−−−→ K0(C(T 2) ⋊ Z2) −−−−→ Z 2 3 x     y 0 ←−−−− K1(C(T 2) ⋊ Z2) ←−−−− 0, which implies K0(C(T 2) ⋊ Z2) ∼= Z 2 3 and K1(C(T 2) ⋊ Z2) ∼= 0. Therefore, Z 2 −−−−→ K0(E 2 ε ⋊ Z2) −−−−→ Z 2 3 x     y 0 ←−−−− K1(E 2 ε ⋊ Z2) ←−−−− 0. CUBO 10, 3 (2008) The Flip Crossed Products of the C∗-Algebras ... 111 It follows that K0(E 2 ε ⋊ Z2) ∼= Z2 3 +2 and K1(E 2 ε ⋊ Z2) ∼= 0. Therefore, Z −−−−→ Z2 3 +2 ⊕ Z2 3 +2 −−−−→ K0(F 2 ε ) x     y K1(F 2 ε ) ←−−−− 0 ⊕ 0 ←−−−− 0. Hence, it follows that K0(F 2 ε ) ∼= Z2 4 +3 and K1(F 2 ε ) ∼= 0. Next consider the case m = 3. Then 0 → C0(T 3 \ (±1,±1,±1)) ⋊ Z2 → C(T 3) ⋊ Z2 →⊕ 2 3 C∗(Z2) → 0. Furthermore, 0 → C0(X2) ⋊ Z2 → C0(X3) ⋊ Z2 → C0(X3 \ X2) ⋊ Z2 → 0, where X3 = T 3 \ (±1,±1,±1), and 0 → C0(X1) ⋊ Z2 → C0(X2) ⋊ Z2 → C0(X2 \X1) ⋊ Z2 → 0, where X1 = (T \{±1}) 3. We have the following six-term exact sequence: 0 −−−−→ K0(C0(X2) ⋊ Z2) −−−−→ Z 6 x     y 0 ←−−−− K1(C0(X2) ⋊ Z2) ←−−−− Z 3, which implies K0(C0(X2) ⋊ Z2) ∼= Z 3 and K1(C0(X2) ⋊ Z2) ∼= 0. Furthermore, Z 3 −−−−→ K0(C0(X3) ⋊ Z2) −−−−→ 0 x     y Z 3 ←−−−− K1(C0(X3) ⋊ Z2) ←−−−− 0, which implies K0(C0(X3) ⋊ Z2) ∼= 0 and K1(C0(X3) ⋊ Z2) ∼= 0. Thus, 0 −−−−→ K0(C(T 3) ⋊ Z2) −−−−→ Z 2 4 x     y 0 ←−−−− K1(C(T 3) ⋊ Z2) ←−−−− 0, which implies K0(C(T 3) ⋊ Z2) ∼= Z 2 4 and K1(C(T 2) ⋊ Z2) ∼= 0. Therefore, Z 2 −−−−→ K0(E 3 ε ⋊ Z2) −−−−→ Z 2 4 x     y 0 ←−−−− K1(E 3 ε ⋊ Z2) ←−−−− 0. It follows that K0(E 3 ε ⋊ Z2) ∼= Z2 4 +2 and K1(E 3 ε ⋊ Z2) ∼= 0. Therefore, Z −−−−→ Z2 4 +2 ⊕ Z2 4 +2 −−−−→ K0(F 3 ε ) x     y K1(F 3 ε ) ←−−−− 0 ⊕ 0 ←−−−− 0. 112 Takahiro Sudo CUBO 10, 3 (2008) Hence, it follows that K0(F 3 ε ) ∼= Z2 5 +3 and K1(F 3 ε ) ∼= 0. Next consider the case m = 4. Then 0 → C0(T 4 \ (±1,±1,±1,±1)) ⋊ Z2 → C(T 4) ⋊ Z2 →⊕ 2 4 C ∗(Z2) → 0. Furthermore, 0 → C0(X3) ⋊ Z2 → C0(X4) ⋊ Z2 → C0(X4 \ X3) ⋊ Z2 → 0, where X4 = T 4 \ (±1,±1,±1,±1), and 0 → C0(X1) ⋊ Z2 → C0(X2) ⋊ Z2 → C0(X2 \X1) ⋊ Z2 → 0, where X1 = (T \{±1}) 4. We have the following six-term exact sequence: Z 4 −−−−→ K0(C0(X2) ⋊ Z2) −−−−→ 0 x     y Z 12 ←−−−− K1(C0(X2) ⋊ Z2) ←−−−− 0, which implies K0(C0(X2) ⋊ Z2) ∼= 0 and K1(C0(X2) ⋊ Z2) ∼= Z 8. Furthermore, 0 −−−−→ K0(C0(X3) ⋊ Z2) −−−−→ Z 12 x     y 0 ←−−−− K1(C0(X3) ⋊ Z2) ←−−−− Z 8, which implies K0(C0(X3) ⋊ Z2) ∼= Z 4 and K1(C0(X3) ⋊ Z2) ∼= 0. Furthermore, Z 4 −−−−→ K0(C0(X4) ⋊ Z2) −−−−→ 0 x     y Z 4 ←−−−− K1(C0(X4) ⋊ Z2) ←−−−− 0, which implies K0(C0(X4) ⋊ Z2) ∼= 0 and K1(C0(X4) ⋊ Z2) ∼= 0. Thus, 0 −−−−→ K0(C(T 4) ⋊ Z2) −−−−→ Z 2 5 x     y 0 ←−−−− K1(C(T 4) ⋊ Z2) ←−−−− 0, which implies K0(C(T 4) ⋊ Z2) ∼= Z 2 5 and K1(C(T 4) ⋊ Z2) ∼= 0. Therefore, Z 2 −−−−→ K0(E 4 ε ⋊ Z2) −−−−→ Z 2 5 x     y 0 ←−−−− K1(E 4 ε ⋊ Z2) ←−−−− 0. It follows that K0(E 4 ε ⋊ Z2) ∼= Z2 5 +2 and K1(E 4 ε ⋊ Z2) ∼= 0. Therefore, Z −−−−→ Z2 5 +2 ⊕ Z2 5 +2 −−−−→ K0(F 4 ε ) x     y K1(F 4 ε ) ←−−−− 0 ⊕ 0 ←−−−− 0. CUBO 10, 3 (2008) The Flip Crossed Products of the C∗-Algebras ... 113 Hence, it follows that K0(F 4 ε ) ∼= Z2 6 +3 and K1(F 4 ε ) ∼= 0. The case for m general can be treated by the step by step argument as shown above. The argument for K-theory is inductive in a sense that it involves essentially suspensions and direct sums inductively. The second claim follows from considering the case ε = 0 and the same argument as above. 2 Corollary 2.6 For 0 ≤ ε < 2, the natural onto ∗-homomorphism ϕε,0 from (⊗ m+1 ε F) ⋊σ Z2 to (⊗m+1F) ⋊σ Z2 sending sε,j to s0,j (1 ≤ j ≤ m + 1) induces the isomorphism between their K-groups. Proposition 2.7 There exists a continuous field of C∗-algebras on the closed interval [0, 2] such that fibers are (⊗m+1ε F) ⋊σ Z2 for ε ∈ [0, 2], and for any a ∈ (⊗ m+1 2 F) ⋊σ Z2, the sections [0, 2] ∋ ε 7→ ϕε(a) ∈ (⊗ m+1 ε F) ⋊σ Z2 are continuous, where ϕε : (⊗ m+1 2 F) ⋊σ Z2 → (⊗ m+1 ε F) ⋊σ Z2 is the natural onto ∗-homomorphism sending s2,j to sε,j (1 ≤ j ≤ m + 1). Proof. As shown before, (⊗m+1ε F) ⋊σ Z2 ∼= (Emε ⋊αε N) ⋊σ Z2. Furthermore, this is isomorphic to p((Emε ⊗ K) ⋊ρ∧ε ⊗id Z)p ⋊σ Z2. Hence it follows that ((Emε ⋊αε N) ⋊σ Z2) ⊗ K ∼= (p((E m ε ⊗ K) ⋊ρ∧ε ⊗id Z)p ⋊σ Z2) ⊗ K ∼= (p((E m ε ⊗ K) ⋊ρ∧ε ⊗id Z)p⊗ K) ⋊σ⊗id Z2 ∼= (((E m ε ⊗ K) ⋊ρ∧ε ⊗id Z) ⊗ K) ⋊σ⊗id Z2 ∼= ((E m ε ⊗ K ⊗ K) ⋊ρ∧ε ⊗id⊗id Z) ⋊σ⊗id Z2 ∼= ((E m ε ⊗ K) ⋊ρ∧ε ⊗id Z) ⋊σ Z2 ∼= (E m ε ⊗ K) ⋊ρ∧ε ∗σ⊗id (Z2 ∗ Z2). It is deduced from [2] that there exists a continuous field of C∗-algebras on [0, 2] such that fibers are (Emε ⊗ K) ⋊ρ∧ε ∗σ⊗id (Z2 ∗ Z2) for ε ∈ [0, 2], and for any b ∈ (E m 2 ⊗ K) ⋊ρ∧ 2 ∗σ⊗id (Z2 ∗ Z2), the sections [0, 2] ∋ ε 7→ ψε(b) ∈ (E m ε ⊗ K) ⋊ρ∧ε ∗σ⊗id (Z2 ∗ Z2) are continuous, where ψε is the unique onto ∗-homomorphism from (Em 2 ⊗ K) ⋊ρ∧ 2 ∗σ⊗id (Z2 ∗ Z2) to (E m ε ⊗ K) ⋊ρ∧ε ∗σ⊗id (Z2 ∗ Z2). Cutting down this continuous field by cutting down the fibers from ((Emε ⋊αε N) ⋊σ Z2) ⊗ K to (Emε ⋊αε N) ⋊σ Z2 by minimal projections, we obtain the desired continuous field. 2 Received: May 2008. Revised: June 2008. 114 Takahiro Sudo CUBO 10, 3 (2008) References [1] B. 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