A remark on the Kasparov groups Exti(A, B)

Valette, Alain

doi:10.2140/pjm.1983.109.247

Cited by 10 publications

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References 7 publications

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“…We so name the Paschke dual algebra because of Paschke duality, which asserts the existence of group isomorphisms K j (A d B ) ∼ = KK j+1 (A, B) for j = 0, 1. (See [11], [24], [25], [26].) We will show below (Theorem 2.10) that the Paschke dual algebra is also dual in another sense, thus generalizing a remark of Valette ([26]).…”

Section: The Paschke Dual Algebramentioning

confidence: 74%

Remarks On Essential Codimension

Loreaux

2020

Integr. Equ. Oper. Theory

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Section: The Paschke Dual Algebramentioning

confidence: 74%

Remarks On Essential Codimension

Loreaux

2020

Integr. Equ. Oper. Theory

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“…Remark 1.5. Valette established a non-commutative generalization of the classical Paschke duality [Val83] whose statement we briefly recall here. We consider a C * -algebra B with a strictly positive element.…”

Section: Assembly Mapsmentioning

confidence: 99%

Paschke duality and assembly maps

Bunke¹,

Engel²,

Land³

2021

Preprint

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We construct a natural transformation between two versions of G-equivariant K-homology with coefficients in a G-C * -category for a countable discrete group G. Its domain is a coarse geometric K-homology and its target is the usual analytic K-homology. Following classical terminology, we call this transformation the Paschke transformation. We show that under certain finiteness assumptions on a G-space X, the Paschke transformation is an equivalence on X. As an application, we provide a direct comparison of the homotopy theoretic Davis-Lück assembly map with Kasparov's analytic assembly map appearing in the Baum-Connes conjecture. Contents 1 Introduction and statements 2 Constructions with C * -categories 3 G-bornological coarse spaces and KCX G c 4 G-uniform bornological coarse spaces, cones and K G,X C

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“…Notes 3.3.2 We are in fact using the Skandalis (nonunital) version of Paschke-Valette duality, (cf. [Pas81], [Val83], [Ska88], [Hig95]), in which the isomorphism…”

Section: Andmentioning

confidence: 99%

ON THE CLASSIFICATION OF NUCLEAR C^*-ALGEBRAS

Dǎdǎrlat

Eilers

2002

Proc. Lond. Math. Soc.

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We employ results from KK-theory, along with quasidiagonality techniques, to obtain wide-ranging classification results for nuclear C*-algebras. Using a new realization of the Cuntz picture of the Kasparov groups we show that two morphisms inducing equal KK-elements are approximately stably unitarily equivalent. Using K-theory with coefficients to associate a partial KK-element to an approximate morphism, our result is generalized to cover such maps. Conversely, we study the problem of lifting a (positive) partial KK-element to an approximate morphism. These results are employed to obtain classification results for certain classes of quasidiagonal C*-algebras introduced by H. Lin, and to reprove the classification of purely infinite simple nuclear C*-algebras of Kirchberg and Phillips. It is our hope that this work can be the starting point of a unified approach to the classification of nuclear C*-algebras.2000 Mathematical Subject Classification: primary 46L35; secondary 19K14, 19K35, 46L80.

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A remark on the Kasparov groups Extⁱ(A, B)

Cited by 10 publications

References 7 publications

Remarks On Essential Codimension

Remarks On Essential Codimension

Paschke duality and assembly maps

ON THE CLASSIFICATION OF NUCLEAR C^*-ALGEBRAS

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A remark on the Kasparov groups Exti(A, B)

Cited by 10 publications

References 7 publications

Remarks On Essential Codimension

Remarks On Essential Codimension

Paschke duality and assembly maps

ON THE CLASSIFICATION OF NUCLEAR C*-ALGEBRAS

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A remark on the Kasparov groups Extⁱ(A, B)

ON THE CLASSIFICATION OF NUCLEAR C^*-ALGEBRAS