Fibrations with noncommutative fibers

Echterhoff, Siegfried; Nest, Ryszard; Oyono‐Oyono, Hervé

doi:10.4171/jncg/41

Cited by 22 publications

(28 citation statements)

References 19 publications

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“…Since T is the crossed product of A by a spectrum-fixing Z d -action, by Remark 2.3 Part (1) of Ref. [15], T is a K fibration as well. The existence of a K-theory bundle and a monodromy map follow from Prop.…”

Section: Proofmentioning

confidence: 94%

Topological T-duality and T-folds

Bouwknegt¹,

Pande²

2009

Advances in Theoretical and Mathematical Physics

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We explicitly construct the C * −algebras arising in the formalism of Topological T-duality due to Mathai and Rosenberg from string-theoretic data in several key examples. We construct a continuous-trace algebra with an action of R d unique up to exterior equivalence from the data of a smooth T d -equivariant gerbe on a trivial bundle X = W × T d . We argue that the 'noncommutative T-duals' of Mathai and Rosenberg [7], should be identified with the nongeometric backgrounds well-known in string theory. We also argue that the C * -algebra A ⋊ α| Z d Z d should be identified with the T-folds of Hull [1, 2] which geometrize these backgrounds.We identify the charge group of D-branes on T-fold backgrounds in the C * -algebraic formalism of Topological T-duality. We also study D-branes on T-fold backgrounds. We show that the K-theory bundles of Ref. [14] give a natural description of these objects.

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Section: Proofmentioning

confidence: 94%

Topological T-duality and T-folds

Bouwknegt¹,

Pande²

2009

Advances in Theoretical and Mathematical Physics

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“…In [9] we had then computed the K-theory (in fact in a bivariant setting) of this crossed product using a particular feature ("independence", see below) of Ω P together with the following "descent to compact subgroups" principle taken from [13], [5].…”

Section: Introductionmentioning

confidence: 99%

On the K-Theory of Crossed Products by Automorphic Semigroup Actions

Cuntz¹,

Echterhoff²,

Li³

2013

The Quarterly Journal of Mathematics

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Let P be a semigroup that admits an embedding into a group G. Assume that the embedding satisfies the Toeplitz condition of [24] and that the Baum-Connes conjecture holds for G. We prove a formula describing the Ktheory of the reduced crossed product A ⋊α,r P by any automorphic action of P . This formula is obtained as a consequence of a result on the K-theory of crossed products for special actions of G on totally disconnected spaces. We apply our result to various examples including left Ore semigroups and quasilattice ordered semigroups. We also use the results to show that for certain semigroups P , including the ax + b-semigroup R ⋊ R × for a Dedekind domain R, the K-theory of the left and right regular semigroup C*-algebras C * λ (P ) and C * ρ (P ) coincide, although the structure of these algebras can be very different. Preliminaries on totally disconnected spacesRecall that a locally compact Hausdorff space Ω is totally disconnected if and only if its topology has a basis of compact open subsets. The corresponding algebras C 0 (Ω) of continuous functions which vanish at infinity are precisely the commutative AF-Algebras. In what follows, if V ⊆ Ω, then 1 V : Ω → C denotes the characteristic function of V . Definition 2.1. Let Ω be a totally disconnected locally compact Hausdorff space and let V be a family of compact open subsets in Ω. Moreover, let U c (Ω) denote the set of all compact open subsets of Ω. Then we say that V is a generating family of the compact open sets of Ω if U c (Ω) coincides with the smallest family U of compact open sets in Ω which contains V and which is closed under finite intersections, finite unions, and under taking differences U W with U, W ∈ U . Lemma 2.2. Suppose that V is a family of compact open sets in the totally disconnected space Ω. Then the following are equivalent (1) The set {1 V : V ∈ V} generates C 0 (Ω) as a C*-algebra. (2) The set V generates U c (Ω) in the sense of Definition 2.1. Moreover, if V is closed under taking finite intersections, then (1) and (2) are equivalent to(3) span{1 V : V ∈ V} is a dense subalgebra of C 0 (Ω) containing span{1 U : U ∈ U c (Ω)}.

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“…Bunke and Schick in [8] and the noncommutative geometry approach which we used in [20,21]. This grew out of a larger project, still in progress, of trying to identify a good larger category of "generalized bundles" that incorporates the bundles of [14,15] and is well-adapted to the noncommutative geometry approach to topological T-duality.…”

Section: Introductionmentioning

confidence: 99%

T-duality for circle bundles via noncommutative geometry

Mathai

Rosenberg

2014

Advances in Theoretical and Mathematical Physics

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Recently Baraglia showed how topological T-duality can be extended to apply not only to principal circle bundles, but also to non-principal circle bundles. We show that his results can also be recovered via two other methods: the homotopy-theoretic approach of Bunke and Schick, and the noncommutative geometry approach which we previously used for principal torus bundles. This work has several interesting byproducts, including a study of the Ktheory of crossed products by O(2) = Isom(R), the universal cover of O(2), and some interesting facts about equivariant K-theory for Z/2. In the final section of this paper, some of these results are extended to the case of bundles with singular fibers, or in other words, non-free O(2)-actions.

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Fibrations with noncommutative fibers

Cited by 22 publications

References 19 publications

Topological T-duality and T-folds

Topological T-duality and T-folds

On the K-Theory of Crossed Products by Automorphic Semigroup Actions

T-duality for circle bundles via noncommutative geometry

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