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## The structure of crossed products of irrational rotation algebras by finite subgroups of SL2(Z) (2006)

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### Citations

869 |
Theory of Operator Algebras. I
- Takesaki
- 1979
(Show Context)
Citation Context ...acial states. We will make extensive use of the L 2 -norm (or seminorm) associated with a tracial state τ of a C*-algebra A, given by ‖a‖2,τ = τ(a ∗ a) 1/2 . See the discussion before Lemma V.2.20 of =-=[70]-=- for more on this seminorm in the von Neumann algebra context. All the properties we need are immediate from its identification with the seminorm in which one completes A to obtain the Hilbert space H... |

758 |
C ∗ -algebras and their automorphism Groups
- Pedersen
- 1979
(Show Context)
Citation Context ...ent if there exists a strictly continuous map v: G → UM(A) (the unitary group in the multiplier algebra of A) such that α ′ s = Ad(vs) ◦αs and vst = vsαs(vt) for all s, t ∈ G. (Compare with 8.11.3 of =-=[54]-=-.) It is easily seen that exterior equivalent actions have isomorphic crossed products and isomorphic reduced crossed products, with isomorphisms given on L 1 (G, A) by f ↦→ f · v. Following [21], if ... |

555 | K-theory for operator algebras
- Blackadar
- 1986
(Show Context)
Citation Context ...KKL n (C, C(L/L ′) ⊗ A) ∼ = KK L n (C, C(L/L′ ) ⊗ B). For any G-C*-algebra D, let qD : C(L/L ′) ⊗ D → D be the quotient map given by evaluation at the identity coset L ′ ∈ L/L ′. By Theorem 20.5.5 of =-=[5]-=-, the chain of maps KK L n (C, C(L/L′ ) ⊗ D) resL L ′ −→ KK L′ n (C, C(L/L′ ) ⊗ D) q∗ −→ KK L′ (C, D) is an isomorphism. Moreover, when applied to A and B in place of D, it transforms Kasparov product... |

492 |
The geometries of 3-manifolds
- Scott
- 1983
(Show Context)
Citation Context ...F is a subgroup of SL2(Z), its action on T2 is orientation preserving. Also, there are only finitely many points of T 2 at which the action is not free. Analyzing these, for example as on page 407 of =-=[68]-=-, one can show that F \T 2 is a compact 2-dimensional manifold. The rational cohomology H ∗ (G\EG; Q) agrees with H ∗ (T 2 ; Q) F . Since the action is orientation preserving, F acts trivially on H p ... |

334 |
Equivariant KK-theory and the Novikov conjecture
- Kasparov
- 1988
(Show Context)
Citation Context ...t T ∗ (L/C) denote the cotangent space of the quotient L/C (which is a manifold). Then it follows from the work of Kasparov (see Theorem 2 of [36]; Definitions 4.1 and 5.1 and Theorems 5.2 and 5.7 of =-=[37]-=-) that there exist Dirac and dual-Dirac elements D ∈ KK L( 0 C0(T ∗ (L/C)), C ) and β ∈ KK L( 0 C, C0(T ∗ (L/C)) ) such that D ⊗C β = 1 ∈ KK L( 0 C0(T ∗ (L/C)), C0(T ∗ (L/C)) ) and such that γ = β ⊗C0... |

231 |
Classifying space for proper actions and K-theory of group C∗-algebras
- Baum, Connes, et al.
- 1993
(Show Context)
Citation Context ... K equipped with an action corresponding to ω as above. For an action α: G → Aut(A), we denote by K top ∗ (G; A) the left hand side of the Baum-Connes Conjecture with coefficients A. See Section 9 of =-=[3]-=-, where it is called KG ∗ (EG; A). As there, we let µ: K top ∗ (G; A) → K∗(A⋊α,r G) be the assembly map. Versions of the following definition have appeared in [45, 11]. Definition 1.2. Let [ω] ∈ H2 (G... |

229 | Elements of homotopy theory - Whitehead - 1978 |

190 |
The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized K-functor
- Rosenberg, Schochet
(Show Context)
Citation Context ...for “most” irrational θ [74], and for AΘ ⋊ Z2 for “most” skew symmetric real matrices Θ [6]. We need to show that the crossed products satisfy the Universal Coefficient Theorem, as in Theorem 1.17 of =-=[67]-=-. A much stronger result than the one formulated in the proposition below has been obtained by Meyer and Nest in Proposition 8.5 of [47]. Since the arguments used in the special case we consider here ... |

167 |
C*-algèbres et géométrie differentielle
- Connes
- 1980
(Show Context)
Citation Context ...ruct finitely generated projective C∗( Z2 ⋊ F, ˜ ) Ωa -modules which represent a suitable class Sa. To do this, we exploit below some basic constructions and ideas of Alain Connes and Marc Rieffel in =-=[16, 63]-=- and Sam Walters in [72, 75]. Define a new cocycle Ω1/a ∈ Z2( Z2 , C ( [a, 1], T )) by setting Ω1/a( · , · )(θ) = ω1/θ for all θ ∈ [a, 1]. Set A = C∗ (Z2, Ωa) and B = C∗( Z2 ) , Ω1/a . Then the fiber ... |

163 |
On the classification of inductive limits of sequences of semisimple finitedimensional algebras
- Elliott
- 1976
(Show Context)
Citation Context ...d Z, then θ2 = θ1 or θ2 = 1 − θ1. Thus, it remains to prove that Aθ ⋊α Z2 ∼ = A1−θ ⋊α Z2 for θ ∈ (0, 1) � Q. By Theorem 6.3 and George Elliott’s classification theorem for AF algebras (Theorem 4.3 of =-=[23]-=-), it suffices to exhibit an isomorphism f : K0(Aθ ⋊α Z2) → K0(A1−θ ⋊α Z2) of scaled ordered K0-groups. Since each algebra has a unique tracial state, and since the order on the K0-group of a simple u... |

138 |
Exact sequences for K-groups and Ext-groups of certain cross-products
- Pimsner, Voiculescu
(Show Context)
Citation Context ...([1], [Eθ]) is a basis for K0(Aθ). Choose θ ∈ [a, 1] � Q and let τ : Aθ → R be the canonical tracial state on Aθ. In Theorem 1.4 of [63] it is shown that τ∗([Eθ]) = θ. It follows from the Appendix in =-=[55]-=- that τ∗ is an isomorphism from K0(Aθ) to Z + θZ, so the result follows. □ We are now ready to give an explicit basis for K0(Aθ ⋊ F) for all θ ∈ (0, 1] and for F = Z2, Z3, Z4, Z6. Theorem 4.9. Let θ ∈... |

107 |
E-theory and KK-theory for groups which act properly and isometrically on Hilbert space
- Higson, Kasparov
(Show Context)
Citation Context ...1], T)) such that ωj = Ω( · , · )(j) for j = 0, 1. Suppose further that G satisfies the Baum-Connes conjecture with coefficients. (If G is amenable or a-T-menable, this is automatic by Theorem 1.1 of =-=[33]-=-.) Then K∗(C ∗ r (G, ω0)) ∼ = K∗(C ∗ r (G, ω1)). Indeed, using the homotopy Ω ∈ Z2 (G, C([0, 1], T)) one can construct a C*- (G, Ω) which is a continuous field of C*-algebras over [0, 1] with fibers i... |

104 |
Projective modules over higher-dimensional noncommutative tori
- Rieffel
- 1988
(Show Context)
Citation Context ...ve.) Since ωΘ is a real cocycle, Corollary 1.13 immediately gives the following well known computation of the K-theory of a higher dimensional noncommutative torus, as found, for example, in [24] and =-=[64]-=-. Of course, we do not immediately get the additional information given in [24] and [64]. Corollary ( 1.14. For any d ≥ 1 and any skew symmetric real d × d matrix Θ, we ∗ d C (Z , ωΘ) ) ∼ 2 = Z d−1 fo... |

81 |
Cohomology of groups, Graduate Texts in Mathematics 87
- Brown
- 1982
(Show Context)
Citation Context ..., will be used several times. It actually holds in greater generality, but all we need here is the version below, which follows from the arguments in the proof of Proposition 5.6(b) in Chapter III of =-=[10]-=-. Lemma 3.2. Suppose that G is a discrete group and H and C are subgroups of G such that C has finite index in G. For g ∈ G, let ϕg : C∗ r (H) → C∗ r (gHg−1) denote the isomorphism induced by conjugat... |

81 |
Elementary Differential Topology
- Munkres
- 1963
(Show Context)
Citation Context ...�Uj−1 and their quotients Mj/L are smooth manifolds, and Mj is locally L-homeomorphic to spaces of the form X × L/Lj for suitable compact subgroups Lj ⊆ L. Now Mj/L is triangulable by Theorem 10.6 of =-=[46]-=-, and refining the triangulation allows us to assume that the inverse image of each open simplex S is L-homeomorphic to S ×L/Lj with the trivial action of L on S. By Bott periodicity and the previous ... |

81 |
conjecture de Baum-Connes pour les feuilletages moyennables, K-theory, 17
- Tu, La
- 1999
(Show Context)
Citation Context ...for computing an explicit basis for K0(Aθ ⋊α F). Step (2) (the Universal Coefficient Theorem) will be obtained from the BaumConnes conjecture by slightly extending ideas first used in Section 10.2 of =-=[71]-=- and Lemma 5.4 of [15]. Step (3) (the tracial Rokhlin property) is obtained via Theorem 5.5, according to which an action of a finite group on a simple separable unital nuclear C*-algebra with unique ... |

72 |
Group Extensions and Cohomology for Locally Compact Groups
- Moore
- 1976
(Show Context)
Citation Context ...T1 denote the projective unitary group. If V : G → U is an ω-representation, then V determines an action α: G → PU ∼ = Aut(K) by αs = Ad(V (s)) for s ∈ G. (Continuity follows from Proposition 5(a) of =-=[50]-=-.) With ω(s, t) = ω(s, t), the full and reduced crossed products K ⋊α G and K ⋊α,r G are then isomorphic to C ∗ (G, ω) ⊗ K and C ∗ r(G, ω) ⊗ K. On the level of L 1 -algebras the isomorphisms are given... |

58 |
The Baum-Connes conjecture via localisation of categories
- Meyer, Nest
(Show Context)
Citation Context ...atisfy the Universal Coefficient Theorem, as in Theorem 1.17 of [67]. A much stronger result than the one formulated in the proposition below has been obtained by Meyer and Nest in Proposition 8.5 of =-=[47]-=-. Since the arguments used in the special case we consider here are much easier, we include the short proof. Proposition 6.1. Let G be an amenable (or, more generally, an a-T-menable) group which can ... |

53 |
The cancellation theorem for projective modules over irrational rotation C∗-algebras
- Rieffel
- 1983
(Show Context)
Citation Context ...ruct finitely generated projective C∗( Z2 ⋊ F, ˜ ) Ωa -modules which represent a suitable class Sa. To do this, we exploit below some basic constructions and ideas of Alain Connes and Marc Rieffel in =-=[16, 63]-=- and Sam Walters in [72, 75]. Define a new cocycle Ω1/a ∈ Z2( Z2 , C ( [a, 1], T )) by setting Ω1/a( · , · )(θ) = ω1/θ for all θ ∈ [a, 1]. Set A = C∗ (Z2, Ωa) and B = C∗( Z2 ) , Ω1/a . Then the fiber ... |

48 |
Actions of finite groups on the hyperfinite type II1
- Jones
- 1980
(Show Context)
Citation Context ... and with ε0 in place of ε. We regard A as a subalgebra of N = πτ(A) ′′ , and we let τ also denote the extension of τ to N. The algebra N is hyperfinite by Lemma 2.16 of [51]. We apply Lemma 5.2.1 of =-=[35]-=-. The term “equivariant s.m.u.” [system of matrix units] is defined in Section 1.5 of [35], and N(ψ) is defined in Section 1.2 of [35]. (Also note that pages 44 and 45 of [35] are switched.) We take t... |

48 |
Non-commutative tori—a case study of non-commutative differentiable manifolds
- Rieffel
- 1990
(Show Context)
Citation Context ... = K top ∗ (G). Let Θ be any skew symmetric real d × d matrix. Then Θ determines a 2-cocycle on Z d via ωΘ(m, n) = exp(πi〈Θm, n〉) for m, n ∈ Z d . The corresponding d-dimensional noncommutative torus =-=[65]-=- is the twisted group C*-algebra AΘ = C ∗ (Z d , ωΘ). Equivalently, AΘ is the universal C*algebra generated by unitaries u1, u2, . . .,ud subject to the relations ukuj = exp(2πiθj,k)ujukCROSSED PRODU... |

46 |
Tracially AF
- Lin
(Show Context)
Citation Context ...3) Proof that the action of the group has the tracial Rokhlin property of [59]. Given these steps, one uses Theorem 2.6 of [59] to prove that the crossed product has tracial rank zero in the sense of =-=[39, 40]-=-, and then Huaxin Lin’s classification theorem (Theorem 5.2 of [41]) to conclude that the crossed product is AF. Our results have been motivated by previous studies of the algebras Aθ ⋊α F and AF θ by... |

45 |
Extensions and low dimensional cohomology theory of locally compact groups
- Moore
- 1964
(Show Context)
Citation Context ...NITE GROUPS 5 Recall that two cocycles ω, ω ′ ∈ Z 2 (G, T) are cohomologous if there exists a Borel function u: G → T such that, with ∂u(s, t) = u(s)u(t)u(st), we have ω ′ = ∂u · ω. (See Chapter 1 of =-=[48]-=-; the action of G on T is trivial.) It is then easy to verify that the map L 1 (G, ω ′ ) → L 1 (G, ω), given by f ↦→ u · f, extends to isomorphisms of the full and reduced twisted group C*-algebras. I... |

43 |
The tracial topological rank of C*-algebras
- Lin
(Show Context)
Citation Context ...3) Proof that the action of the group has the tracial Rokhlin property of [59]. Given these steps, one uses Theorem 2.6 of [59] to prove that the crossed product has tracial rank zero in the sense of =-=[39, 40]-=-, and then Huaxin Lin’s classification theorem (Theorem 5.2 of [41]) to conclude that the crossed product is AF. Our results have been motivated by previous studies of the algebras Aθ ⋊α F and AF θ by... |

38 |
Equivariant K-Theory and Freeness of Group Actions on C*-Algebras
- Phillips
- 1987
(Show Context)
Citation Context ...Mj of any finite complex in the triangulation of Mj/L, or the inverse image of the topological interior of one. Using continuity of equivariant K-theory for compact group actions (Theorem 2.8.3(6) of =-=[56]-=- or Proposition 11.9.2 of [5]), we find that (1.3) is an isomorphism when C0(V ) is replaced by C0(Mj). Again using six term exact sequences and the Five Lemma, we conclude that (1.3) is an isomorphis... |

38 |
Produits croisés d’une C∗-algèbre par un groupe d’automorphismes
- Zeller-Meier
- 1968
(Show Context)
Citation Context ...ions on K We begin by recalling some basic facts about group C*-algebras twisted by cocycles. This is a special case of the theory of crossed product C*-algebras twisted by cocycles of [52]. Also see =-=[77]-=- for the case in which the group is discrete. Let G be a second countable locally compact group, with modular function ∆G: G → (0, ∞). Let ω: G×G → T be a Borel 2-cocycle on G. (Recall the algebraic c... |

35 | Twisted Crossed Products of C∗-Algebras - Packer, Raeburn - 1990 |

33 | The tracial Rokhlin property for actions of finite groups
- Phillips
(Show Context)
Citation Context ...tion of the K-theory of the crossed product. (2) Proof that the crossed product satisfies the Universal Coefficient Theorem. (3) Proof that the action of the group has the tracial Rokhlin property of =-=[59]-=-. Given these steps, one uses Theorem 2.6 of [59] to prove that the crossed product has tracial rank zero in the sense of [39, 40], and then Huaxin Lin’s classification theorem (Theorem 5.2 of [41]) t... |

32 | The Connes–Kasparov conjecture for almost connected groups and for linear p-adic groups
- Chabert, Echterhoff, et al.
- 2003
(Show Context)
Citation Context ...metimes possible to show that the Baum-Connes conjecture holds for K (with respect to any action of G on K) without knowing that the conjecture holds for all coefficients. For example, it is shown in =-=[14]-=- that every almost connected group satisfies the conjecture for K but the Baum-Connes conjecture with arbitrary coefficients is not known in general for those groups. We now consider homotopies betwee... |

32 | An equivariant Brauer group and actions of groups on
- Crocker, Kumjian, et al.
- 1997
(Show Context)
Citation Context ... α x : G → Aut(K) for x ∈ X, defined by α x s(g(x)) = ( αs(g) ) (x) for g ∈ C0(X, K) and x ∈ X. Thus, we should understand a C0(X)-linear action as a continuous family of actions on K. It is shown in =-=[17]-=- (see Lemma 3.1 and Theorem 3.6) and [21] that EG(X) is a group with multiplication [α] · [β] = [α ⊗X β], where (α ⊗X β) x = α x ⊗ β x for x ∈ X. The following result, basically due to Mackey, is well... |

29 |
group C∗-algebras, and higher signatures (conspectus), in Novikov conjectures, index theorems and rigidity
- Kasparov, K-theory
- 1993
(Show Context)
Citation Context ...oof. Let C denote the maximal compact subgroup of L and let T ∗ (L/C) denote the cotangent space of the quotient L/C (which is a manifold). Then it follows from the work of Kasparov (see Theorem 2 of =-=[36]-=-; Definitions 4.1 and 5.1 and Theorems 5.2 and 5.7 of [37]) that there exist Dirac and dual-Dirac elements D ∈ KK L( 0 C0(T ∗ (L/C)), C ) and β ∈ KK L( 0 C, C0(T ∗ (L/C)) ) such that D ⊗C β = 1 ∈ KK L... |

28 |
Noncommutative spheres
- Bratteli, Kishimoto
- 1992
(Show Context)
Citation Context ...on theorem (Theorem 5.2 of [41]) to conclude that the crossed product is AF. Our results have been motivated by previous studies of the algebras Aθ ⋊α F and AF θ by several authors. See, for example, =-=[7, 9, 26, 27, 28, 29, 30, 31, 32, 38, 62, 72, 73, 74]-=-. The most was known about the crossed product by the flip: its (unordered) K-theory has been computed in [38], and the crossed product has been proved to be an AF algebra in [9]. The next best unders... |

25 |
On the K-theory of the C∗-algebra generated by a projective representation of a torsion-free discrete abelian group
- Elliott
- 1980
(Show Context)
Citation Context ... )) between [ω] and the trivial cocycle [1] by the formula Ω(s, t)(x) = e ix·c(s,t) for x ∈ [0, 1] and s, t ∈ G. As a consequence we get the following corollary. Several special cases of it appear in =-=[24, 53, 11, 43, 44]-=-. (In [45], Mathai states a theorem saying that for any discrete G and any real 2-cocycle ω on G, one has K0(C ∗ r (G, ω)) ∼ = K0(C ∗ r (G)). Unfortunately, the proof given in [45] has a substantial g... |

25 |
Classification of simple C*-algebras with tracial topological rank zero
- Lin
(Show Context)
Citation Context ...of [59]. Given these steps, one uses Theorem 2.6 of [59] to prove that the crossed product has tracial rank zero in the sense of [39, 40], and then Huaxin Lin’s classification theorem (Theorem 5.2 of =-=[41]-=-) to conclude that the crossed product is AF. Our results have been motivated by previous studies of the algebras Aθ ⋊α F and AF θ by several authors. See, for example, [7, 9, 26, 27, 28, 29, 30, 31, ... |

25 | Crossed products by finite cyclic group actions with the tracial Rokhlin property, preprint math.QA/0306410
- Phillips
(Show Context)
Citation Context ...ty. In Section 6, we prove the Universal Coefficient Theorem and put everything together. We also prove Theorem 0.4. This paper contains the main result of Section 10 of the unpublished long preprint =-=[58]-=-, Theorem 0.4 here. Although the three steps, as described at the beginning of the introduction, are the same, the proofs of all three of them differ substantially from the proofs given in [58]. This ... |

24 | Quantum Hall effect on the hyperbolic plane
- Carey, Hannabuss, et al.
- 1998
(Show Context)
Citation Context ...with coefficients A. See Section 9 of [3], where it is called KG ∗ (EG; A). As there, we let µ: K top ∗ (G; A) → K∗(A⋊α,r G) be the assembly map. Versions of the following definition have appeared in =-=[45, 11]-=-. Definition 1.2. Let [ω] ∈ H2 (G, T). Then the twisted topological K-theory of G with respect to [ω] is defined to be the topological K-theory K top ∗ (G; ω) = (G; Kω). The twisted assembly map for G... |

23 |
The structure of the irrational rotation -algebra
- Elliott, Evans
- 1993
(Show Context)
Citation Context ...he groups generated as in (0.2). Suppose θ ̸∈ Q. Then Aθ ⋊α F is an AF algebra. Proof. We know that Aθ is a simple separable unital C*-algebra with tracial rank zero, say by the Elliott-Evans Theorem =-=[25]-=-. (See Theorem 3.5 of [60].) By Corollary 5.11, we may apply Corollary 1.6 and Theorem 2.6 of [59] to conclude that Aθ⋊αF is a simple separable unital C*-algebra with tracial rank zero. Corollary 6.2 ... |

19 |
Appendix to O. Bratteli’s paper on “Crossed products of UHF algebras
- Rosenberg
- 1979
(Show Context)
Citation Context ...equence, Aθ⋊αZk is isomorphic to Aθ ′ ⋊α Zl if and only if k = l and θ ′ = ±θ mod Z. If θ ∈ R�Q then the fixed point algebra A F θ is Morita equivalent to Aθ⋊αF. (This follows from the Proposition in =-=[66]-=-.) Thus, as a direct corollary of Theorem 0.1 we get: Corollary 0.2 (Corollary 6.5). Let α : F → Aut(Aθ) be as above with F = Z2, Z3, Z4, Z6. Then the fixed point algebras AF θ are AF for all θ ∈ R � ... |

19 |
On factor representations and the C∗-algebra of canonical commutation relations
- Slawny
- 1972
(Show Context)
Citation Context ...1. Recall that a real skew symmetric d × d matrix Θ is nondegenerate if whenever x ∈ Z d satisfies exp(2πi〈x, Θy〉) = 1 for all y ∈ Z d , then x = 0. This definition is essentially from Section 1.1 of =-=[69]-=-. It is well known that the noncommutative torus AΘ (defined after Corollary 1.13) is simple if and only if Θ is nondegenerate; the main part is Theorem 3.7 of [69], and the complete statement is Theo... |

18 |
Permanence properties of the Baum-Connes conjecture
- Chabert, Echterhoff
(Show Context)
Citation Context ...C*-algebras follows directly from Lemma 2.1. For the isomorphisms on K-theory, first note that, because SL2(Z) satisfies the BaumConnes Conjecture with coefficients, Theorem 2.5 and Corollary 3.14 of =-=[13]-=- imply that Z2 ⋊ H does too. Then use Corollary 1.13 and the fact that [˜ωθ] is real in the sense of Definition 1.12. □ Remark 2.3. In our application, to the classification of crossed products Aθ⋊α,r... |

18 | Every simple higher dimensional noncommutative torus is an at algebra
- Phillips
- 2006
(Show Context)
Citation Context ...d in the introduction to [28]. Theorem 6.6. Let Θ be a nondegenerate real d × d skew symmetric matrix. Let ϕ: Z2 → Aut(AΘ) be the flip action. Then AΘ ⋊ϕ Z2 is an AF algebra. Proof. By Theorem 3.5 of =-=[60]-=-, we know that AΘ is a simple separable unital C*algebra with tracial rank zero. By Corollary 5.12, we may apply Corollary 1.6 and Theorem 2.6 of [59] to conclude that AΘ ⋊ϕ Z2 is a simple separable u... |

17 | Furstenberg transformations on irrational rotation algebras, Ergod
- Osaka, Phillips
(Show Context)
Citation Context ...or all τ ∈ T(A), it is easy to show that the eg satisfy (1), (2), and (3). Now we prove that if (1), (2), and (3) hold, then α has the tracial Rokhlin property. We follow the proof of Theorem 2.14 of =-=[51]-=- (but note that the finite set there is called F). We describe the choices and constructions carefully, but omit many of the details in the verification of the estimates. We verify the condition of32... |

16 | Crossed products by C0(X)-actions
- Echterhoff, Williams
- 1998
(Show Context)
Citation Context ...topy corresponding to the inverse Ω of Ω, then there is an isomorphism ( C([0, 1]) ⋊Ω,r G ) ⊗ K ∼ = C ( [0, 1], K ) ⋊α,r G. For the full twisted crossed products, this follows from Proposition 4.6 of =-=[22]-=- (which is a special case of Theorem 3.4 of [52]). Remark 3.12 and the proof of Theorem 3.11 of [52] imply that this is correct for the reduced crossed products as well. With ωx = Ω( · , · )(x) for x ... |

15 |
Twisted equivariant KK-theory and the Baum-Connes conjecture for group extensions
- Chabert, Echterhoff
(Show Context)
Citation Context ...phism for all compact subgroups L of G and for all real finite dimensional representation spaces V of L. Let V be a real finite dimensional representation space of L. Lemma 7.7(2) and Lemma 7.7(1) of =-=[12]-=-, in order, give isomorphisms KK L n(C0(V ), D) ∼ = −→ KK L n (C0(V ) ⊗C0(V ), C0(V ) ⊗D) ∼ = −→ KK L n (C, C0(V ) ⊗D) for any D, which respect Kasparov products. To prove that (1.2) is an isomorphism... |

15 | Going-down functors, the Künneth formula, and the Baum-Connes conjecture
- Chabert, Echterhoff, et al.
(Show Context)
Citation Context ...cit basis for K0(Aθ ⋊α F). Step (2) (the Universal Coefficient Theorem) will be obtained from the BaumConnes conjecture by slightly extending ideas first used in Section 10.2 of [71] and Lemma 5.4 of =-=[15]-=-. Step (3) (the tracial Rokhlin property) is obtained via Theorem 5.5, according to which an action of a finite group on a simple separable unital nuclear C*-algebra with unique tracial state has the ... |

14 |
Crossed products with continuous trace
- Echterhoff
- 1996
(Show Context)
Citation Context ...morphisms are given by the map Φ: L 1 (G, ω) ⊙ K → L 1 (G, K) determined by (1.1) Φ(f ⊗ k)(s) = f(s)kV (s) ∗ for f ∈ L 1 (G, ω), k ∈ K, and s ∈ G. (See for example Theorem 1.4.15 and Example 1.1.4 of =-=[19]-=-, but this is easily proved directly.) Conversely, if α: G → Aut(K) is any action of G on K, then by choosing a Borel section c: PU → U we obtain a Borel map V α = c ◦ α: G → U such that αs = Ad(V α (... |

14 | L-theory of the semi-direct product of the discrete 3-dimensional Heisenberg group by Z/4
- K-
- 2005
(Show Context)
Citation Context ...∗ (Mj)) ⊕ l j=0 bj −−−−−→ K0(C ∗ (G)) p◦µ−1 −−−−→ K0(G\EG) → 0. To get these, apply Theorem 5.1(a) and Remark 5.2 of [18] to the extension 1 → Z2 → G → F → 1, or apply the more general Theorem 1.6 of =-=[42]-=-, taking there K = {1} and G = Q. (Theorem 1.6 of [42] actually gives a long exact sequence, which breaks up into short exact sequences after tensoring with Q. However, every group appearing in the lo... |

13 |
Comparison theory for simple C∗-algebras, pages 21–54 in: Operator Algebras and Applications
- Blackadar
- 1988
(Show Context)
Citation Context ... for projections. Recall that, if A is a unital C*-algebra, then we say that the order on projections over A is determined by traces if Blackadar’s Second Fundamental Comparability Question (1.3.1 in =-=[4]-=-) holds for all matrix algebras over A. That is, whenever n ∈ N and p, q ∈ Mn(A) are projections such that τ(p) < τ(q) for all tracial states τ on A, then p � q.CROSSED PRODUCTS OF IRRATIONAL ROTATIO... |

13 |
The structure of higher-dimensional noncommutative tori and metric Diophantine approximation
- Boca
- 1997
(Show Context)
Citation Context ...enote the flip action. Then K0(AΘ ⋊α Z2) ∼ = Z 3·2d−1 and K1(AΘ ⋊α Z2) = {0}. If AΘ is simple, then AΘ ⋊α Z2 and the fixed point algebra A Z2 Θ are AF algebras. This result generalizes Theorem 3.1 of =-=[6]-=- and completely answers a question raised in the introduction of [28]. In Section 1, we give the necessary background on twisted group algebras and the proof of Theorem 0.3. The realization of the cro... |

13 | Locally inner actions on C0(X)-algebras
- Echterhoff, Williams
(Show Context)
Citation Context ...3 of [54].) It is easily seen that exterior equivalent actions have isomorphic crossed products and isomorphic reduced crossed products, with isomorphisms given on L 1 (G, A) by f ↦→ f · v. Following =-=[21]-=-, if X is any locally compact space we denote by EG(X) the set of all exterior equivalence classes of C0(X)-linear actions of G on C0(X, K), that is, actions α: G → Aut ( C0(X, K) ) such that αs(f · g... |

13 | Finite cyclic group actions with the tracial Rokhlin property - Phillips |

12 | Raeburn: Twisted Crossed Products of - Packer, I - 1990 |

12 |
Holomorphic bundles on 2-dimensional noncommutative toric orbifolds
- Polishchuk
- 2006
(Show Context)
Citation Context ...on theorem (Theorem 5.2 of [41]) to conclude that the crossed product is AF. Our results have been motivated by previous studies of the algebras Aθ ⋊α F and AF θ by several authors. See, for example, =-=[7, 9, 26, 27, 28, 29, 30, 31, 32, 38, 62, 72, 73, 74]-=-. The most was known about the crossed product by the flip: its (unordered) K-theory has been computed in [38], and the crossed product has been proved to be an AF algebra in [9]. The next best unders... |

11 |
Twisted index theory on good orbifolds I
- Marcolli, Mathai
- 1999
(Show Context)
Citation Context ... )) between [ω] and the trivial cocycle [1] by the formula Ω(s, t)(x) = e ix·c(s,t) for x ∈ [0, 1] and s, t ∈ G. As a consequence we get the following corollary. Several special cases of it appear in =-=[24, 53, 11, 43, 44]-=-. (In [45], Mathai states a theorem saying that for any discrete G and any real 2-cocycle ω on G, one has K0(C ∗ r (G, ω)) ∼ = K0(C ∗ r (G)). Unfortunately, the proof given in [45] has a substantial g... |

8 |
The p-chain spectral sequence
- Davis, Lück
- 2003
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Citation Context ...t the K-groups of Aθ ⋊α F are isomorphic to the K-groups of the untwisted group algebra C∗ r(Z2 ⋊F) ∼ = C(T2 )⋊F, which we compute (together with an explicit basis) by using some general methods from =-=[18]-=-. Theorem 0.3 (Theorem 1.9). Suppose that ω0, ω1 ∈ Z 2 (G, T) are homotopic Borel 2-cocycles on the second countable locally compact group G, that is, there exists aCROSSED PRODUCTS OF IRRATIONAL ROT... |

8 |
The Connes spectrum for actions of abelian groups on continuous-trace algebras, Ergod
- Hurder, Olesen, et al.
- 1986
(Show Context)
Citation Context ... homotopy between classes in H 2 (G, T) ∼ = EG(pt) used in Theorem 0.3. Indeed, let Ω: G × G → C ( [0, 1], T ) be a 2-cocycle with evaluations ωx = Ω( · , · )(x) for x ∈ [0, 1]. By Proposition 3.1 of =-=[34]-=-, and with Lωx being the regular ωx-representation on K(L 2 (G)), the formula αs(g)(x) = Lωx(s)g(x)Lωx(s) ∗ , for g ∈ C ( [0, 1], K ) , defines a continuous action α: G → Aut ( C ( [0, 1], K )) . Thus... |

7 |
The structure of the Brauer group and crossed products of C0(X)-linear group actions on C0(X,K
- Echterhoff, Nest
(Show Context)
Citation Context ...us, if [α0], [α1] ∈ EG(pt) correspond to [ω0], [ω1] ∈ H 2 (G, T) under the correspondence described in Proposition 1.1, we see that [α0] and [α1] are homotopic classes in EG(pt). Using the results of =-=[21, 20]-=- one can show that, conversely, homotopy of classes in EG(pt) implies homotopy of the corresponding cocycles for a very large class of groups. This class includes all almost connected groups by Theore... |

7 |
characters of Fourier modules
- Chern
(Show Context)
Citation Context ...on theorem (Theorem 5.2 of [41]) to conclude that the crossed product is AF. Our results have been motivated by previous studies of the algebras Aθ ⋊α F and AF θ by several authors. See, for example, =-=[7, 9, 26, 27, 28, 29, 30, 31, 32, 38, 62, 72, 73, 74]-=-. The most was known about the crossed product by the flip: its (unordered) K-theory has been computed in [38], and the crossed product has been proved to be an AF algebra in [9]. The next best unders... |

6 |
On the K-theory of the symmetrized non-commutative torus
- Kumjian
- 1990
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Citation Context |

6 | K-theory of Non-Commutative Spheres Arising from the Fourier Automorphism - Walters |

5 |
Cubic algebras
- Farsi, Watling
- 1993
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5 |
The Atiyah-Segal completion theorem for C*-algebras. KTheory
- Phillips
(Show Context)
Citation Context ...ted assembly map suggests that it is actually useful to consider homotopies for actions on K. The first part of the following definition is a special case, in different language, of Definition 3.1 of =-=[57]-=-. Definition 1.3. A homotopy of actions on K is a C ( [0, 1] ) -linear action of G on C ( [0, 1], K ) . A homotopy between two classes [β0], [β1 ( ) ] ∈ EG(pt) is a class 0 1 [β] ∈ EG [0, 1] with eval... |

4 |
K-Theory of twisted group C∗-algebras and positive scalar curvature
- Mathai
- 1999
(Show Context)
Citation Context ...with coefficients A. See Section 9 of [3], where it is called KG ∗ (EG; A). As there, we let µ: K top ∗ (G; A) → K∗(A⋊α,r G) be the assembly map. Versions of the following definition have appeared in =-=[45, 11]-=-. Definition 1.2. Let [ω] ∈ H2 (G, T). Then the twisted topological K-theory of G with respect to [ω] is defined to be the topological K-theory K top ∗ (G; ω) = (G; Kω). The twisted assembly map for G... |

4 |
On the structure of twisted group C∗-algebras
- Packer, Raeburn
- 1992
(Show Context)
Citation Context ... )) between [ω] and the trivial cocycle [1] by the formula Ω(s, t)(x) = e ix·c(s,t) for x ∈ [0, 1] and s, t ∈ G. As a consequence we get the following corollary. Several special cases of it appear in =-=[24, 53, 11, 43, 44]-=-. (In [45], Mathai states a theorem saying that for any discrete G and any real 2-cocycle ω on G, one has K0(C ∗ r (G, ω)) ∼ = K0(C ∗ r (G)). Unfortunately, the proof given in [45] has a substantial g... |

4 | The AF structure of non commutative toroidal Z/4Z orbifolds
- Walters
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3 |
Projective representations of space groups
- Backhouse
- 1970
(Show Context)
Citation Context ...re homotopic Borel 2-cocycles on the second countable locally compact group G, that is, there exists aCROSSED PRODUCTS OF IRRATIONAL ROTATION ALGEBRAS BY FINITE GROUPS 3 Borel 2-cocycle Ω ∈ Z2 (G, C(=-=[0, 1]-=-, T)) such that ωj = Ω( · , · )(j) for j = 0, 1. Suppose further that G satisfies the Baum-Connes conjecture with coefficients. (If G is amenable or a-T-menable, this is automatic by Theorem 1.1 of [3... |

3 |
Symmetrized non-commutative tori
- Farsi, Watling
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3 | Periodic integral transforms and C∗-algebras
- Walters
(Show Context)
Citation Context ...K0(C ∗ (Z 2 ⋊ F)). So the theorem remains true if we replace it by any other section ˜K0(C ∗ (Mj)) → K0(C ∗ (Mj)). We give two proofs of Theorem 3.5, one depending on the constructions of Sam Walters =-=[72, 75]-=- and some computations from Section 4 below, and the other more topological, although it has much in common with the first. Before doing so, however, we should describe what one can get from known res... |

2 |
Fixed point subalgebras of the rotation algebra
- Farsi, Watling
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2 |
Abstract characterizations of fixed point subalgebras of the rotation algebra
- Farsi, Watling
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2 |
C*-algebras of dynamical systems on the non-commutative torus
- Farsi, Watling
(Show Context)
Citation Context ...stands for the cyclic group of order m) generated by the matrices ( ( ) −1 0 −1 −1 (for Z3), (0.2) 0 −1 ( 0 −1 1 0 ) ) (for Z2), (for Z4), and 1 0 ( 0 −1 1 1 ) (for Z6). We refer to Proposition 21 of =-=[32]-=- for a proof that one obtains essentially all interesting actions of the cyclic groups Z2, Z3, Z4, Z6 on Aθ by restricting the action α of SL2(Z) to these subgroups. The main results of this paper cul... |