E-theory and KK-theory for groups which act properly and isometrically on Hilbert space

Higson, Nigel; Kasparov, Gennadi

doi:10.1007/s002220000118

Cited by 316 publications

(345 citation statements)

References 17 publications

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“…By (2.4), (2.10) and the same argument as in [Roe02], µ X,G is isomorphic to the Real Baum-Connes assembly map with coefficient in K(V) R , which is an isomorphism since G has the Haagerup property [HK97].…”

Section: 32mentioning

confidence: 89%

Controlled Topological Phases and Bulk-edge Correspondence

Kubota

2016

Commun. Math. Phys.

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Abstract. In this paper, we introduce a variation of the notion of topological phase reflecting metric structure of the position space. This framework contains not only periodic and non-periodic systems with symmetries in Kitaev's periodic table but also topological crystalline insulators. We also define the bulk and edge indices as invariants taking values in the twisted equivariant K-groups of Roe algebras as generalizations of existing invariants such as the Hall conductance or the Kane-Mele Z2-invariant. As a consequence, we obtain a new mathematical proof of the bulk-edge correspondence by using the coarse MayerVietoris exact sequence.

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Section: 32mentioning

confidence: 89%

Controlled Topological Phases and Bulk-edge Correspondence

Kubota

2016

Commun. Math. Phys.

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“…This work is inspired by Gromov's deep questions concerning uniform embedding into Hilbert space ( [10], [11]) and by the remarkable work of Higson and Kasparov on the Baum-Connes conjecture [14].…”

Section: Proposition 27)mentioning

confidence: 99%

The coarse Baum–Connes conjecture for spaces which admit a uniform embedding into Hilbert space

2000

Invent. math.

479

565

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“…Other examples include free groups (via Haagerup's original paper [38]), finitely generated Coxeter groups, SL.2; Z/, SO.1; n/ and S U.1; n/; moreover the Haagerup property is preserved by free products. Striking applications include Higson and Kasparov's proof of the Baum-Connes conjecture in the presence of the Haagerup property [40], and Popa's deformation-rigidity approach to structural properties of type II 1 factors [62,63]. We refer to [19] for the equivalence of the formulations above, examples and applications.…”

Section: Introductionmentioning

confidence: 99%

The Haagerup property for locally compact quantum groups

Daws

Fima

Skalski

et al. 2014

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. The Haagerup property for locally compact groups is generalised to the context of locally compact quantum groups, with several equivalent characterisations in terms of the unitary representations and positive-definite functions established. In particular it is shown that a locally compact quantum group G has the Haagerup property if and only if its mixing representations are dense in the space of all unitary representations. For discrete G we characterise the Haagerup property by the existence of a symmetric proper conditionally negative functional on the dual quantum group O G; by the existence of a real proper cocycle on G, and further, if G is also unimodular we show that the Haagerup property is a von Neumann property of G. This extends results of Akemann, Walter, Bekka, Cherix, Valette, and Jolissaint to the quantum setting and provides a connection to the recent work of Brannan. We use these characterisations to show that the Haagerup property is preserved under free products of discrete quantum groups.

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E-theory and KK-theory for groups which act properly and isometrically on Hilbert space

Cited by 316 publications

References 17 publications

Controlled Topological Phases and Bulk-edge Correspondence

Controlled Topological Phases and Bulk-edge Correspondence

The coarse Baum–Connes conjecture for spaces which admit a uniform embedding into Hilbert space

The Haagerup property for locally compact quantum groups

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