$K$-theory and homotopies of 2-cocycles on transformation groups

Gillaspy, Elizabeth

doi:10.7900/jot.2014feb14.2033

Cited by 10 publications

(10 citation statements)

References 28 publications

(94 reference statements)

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“…This provides a strong motivation to study methods that enable the computation of K-theory for twisted groupoid C * -algebras. This question has been addressed before in the case of groups by Echterhoff, Lück, Phillips and Walters in [10] and Gillaspy treated the cases of transformation groups, higher-rank graphs and group bundles in a series of articles [16,17,18]. Using the machinery developed by the author in [5] this article presents a unified approach and considerable extension of the above mentioned results.…”

Section: Introductionmentioning

confidence: 94%

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K-theory and homotopies of twists on ample groupoids

Bönicke

2021

J. Noncommut. Geom.

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This paper investigates the K-theory of twisted groupoid C * -algebras. It is shown that a homotopy of twists on an ample groupoid satisfying the Baum-Connes conjecture with coefficients gives rise to an isomorphism between the K-theory groups of the respective twisted groupoid C * -algebras. The results are also interpreted in an inverse semigroup setting and applied to generalized Renault-Deaconu groupoids and P -graph algebras.2010 Mathematics Subject Classification. 46L80, 46L55. Key words and phrases. Twisted groupoid C * -algebra, K-theory. This research was partially supported by Deutsche Forschungsgemeinschaft (SFB 878).

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Section: Introductionmentioning

confidence: 94%

“…[17, Proposition 3.1] If Σ is a continuous homotopy of twists on a compact Hausdorff groupoid G, then the canonical * -homomorphism q t :…”

mentioning

confidence: 99%

K-theory and homotopies of twists on ample groupoids

Bönicke

2021

J. Noncommut. Geom.

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“…Remark 5. 16 (Invariance under homotopies of 2-cocycles) Let us briefly consider the stability of Theorem 5.15 under deformations of the groupoid 2-cocycle σ using results from Gillaspy [17]. Given the groupoid F = R d , we can consider the trivial bundle of groupoids F × [0, 1] equipped with the product topology so that groupoid operations preserve the fibres and such that F × [0, 1] is a locally compact Hausdorff groupoid.…”

Section: The Transversal G Del -Space and Its Localisationmentioning

confidence: 99%

Localised Module Frames and Wannier Bases from Groupoid Morita Equivalences

Bourne

Mesland

2021

J Fourier Anal Appl

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Following the operator algebraic approach to Gabor analysis, we construct frames of translates for the Hilbert space localisation of the Morita equivalence bimodule arising from a groupoid equivalence between Hausdorff groupoids, where one of the groupoids is étale and with a compact unit space. For finitely generated and projective submodules, we show these frames are orthonormal bases if and only if the module is free. We then apply this result to the study of localised Wannier bases of spectral subspaces of Schrödinger operators with atomic potentials supported on (aperiodic) Delone sets. The noncommutative Chern numbers provide a topological obstruction to fast-decaying Wannier bases and we show this result is stable under deformations of the underlying Delone set.

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“…For a standard cocycle θ we can compute the K-theory of A θ by appealing to the following theorem of Gillaspy [10]: Theorem 4 ( [10] Thm. 5.1).…”

Section: The Groupoid C * -Algebra Of λmentioning

confidence: 99%

Gabor frames for quasicrystals, K-theory, and twisted gap labeling

Kreisel

2016

Journal of Functional Analysis

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We study the connection between Gabor frames for quasicrystals, the topology of the hull of a quasicrystal Λ, and the K-theory of the twisted groupoid C * -algebra A σ arising from a quasicrystal. In particular, we construct a finitely generated projective module H Λ over A σ related to time-frequency analysis, and any multiwindow Gabor frame for Λ can be used to construct an idempotent in M N (A σ ) representing H Λ in K 0 (A σ ). We show for lattice subsets in dimension two, this element corresponds to the Bott element in K 0 (A σ ), allowing us to prove a twisted version of Bellissard's gap labeling theorem. Preliminaries Topology of QuasicrystalsThe main objects of our investigation are quasicrystals, so we begin with a review of the topological and dynamical properties of a quasicrystal, as well as properties of the associated operator algebras. We will state the basic definitions and theorems for even dimensional quasicrystals since it will be simplify notation later, however the same definitions and theorems apply in any dimension. We will always think of R 2d ∼ = R d ×R d as time-frequency space, and elements z ∈ R 2d will be written as z = (x, ω) when it is necessary to emphasize this point of view.Definition 1. Let Λ ⊂ R 2d be a discrete set.

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$K$-theory and homotopies of 2-cocycles on transformation groups

Cited by 10 publications

References 28 publications

K-theory and homotopies of twists on ample groupoids

K-theory and homotopies of twists on ample groupoids

Localised Module Frames and Wannier Bases from Groupoid Morita Equivalences

Gabor frames for quasicrystals, K-theory, and twisted gap labeling

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