The Bost conjecture, open subgroups and groups acting on trees

Paravicini, Walther

doi:10.1017/is009010006jkt074

Cited by 3 publications

(8 citation statements)

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“…We have A Y (G ⋉ Y, B) ∼ = A(G, ρ * B) for all G ⋉ Y -Banach algebras B. In [Par09a], it is shown that the following diagram is commutative Finally, the following result was shown in [Par07], the proof being somewhat technical (Proposition 8.3.26):…”

Section: The Forgetful Mapmentioning

confidence: 97%

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A generalised Green–Julg theorem for proper groupoids and Banach algebras

Paravicini¹

2013

J. Noncommut. Geom.

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The Green-Julg theorem states that K G 0 (B) ∼ = K 0 (L 1 (G, B)) for every compact group G and every G-C * -algebra B. We formulate a generalisation of this result to proper groupoids and Banach algebras and deduce that the Bost assembly map is surjective for proper Banach algebras. Definition of RKK banG (C 0 (X); A, B) Definition 1.5. Let A and B be G-C 0 (X)-Banach algebras. Then the class E ban G (C 0 (X); A, B) is defined to be the class of pairs (E, T ) such that E is a non-degenerate graded G-C 0 (X)-Banach A-B-pair and, if we forget the C 0 (X)-structure, the pair (E, T ) is an element of E ban G (A, B). Note that T in the definition is automatically C 0 (X)-linear because E is non-degenerate.The constructions one usually performs with KK ban -cycles are obviously compatible with the additional C 0 (X)-structure, so we can form the sum of KK ban -cycles and take their pushout along homomorphisms of G-C 0 (X)-Banach algebras. We also have a C 0 (X)-linear notion of morphisms of KK ban -cycles, giving us a C 0 (X)-linear version of isomorphisms of KK ban -cycles. Hence also the notion of homotopy makes sense in the C 0 (X)-setting so we can formulate the following definition:Definition 1.6. The class of all homotopy classes of elements of E ban G (C 0 (X); A, B) is denoted by RKK ban G (C 0 (X); A, B). The sum of cycles induces a law of composition on RKK ban G (C 0 (X); A, B) making it an abelian group.The fact that the composition on RKK ban G (C 0 (X); A, B) has inverses can be proved just as in the case without the C 0 (X)-structure, i.e., Lemme 1.2.5 of [Laf02] and its proof are compatible with the additional C 0 (X)-module action. There is an obvious forgetful group homomorphism RKK ban G (C 0 (X); A, B) → KK ban G (A, B) .

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Section: The Forgetful Mapmentioning

confidence: 97%

“…We give a short summary of a pushforward construction introduced in [Par07], Section 8.3; see also [Par09a], Section 1. Let Y be a locally compact Hausdorff left G-space with anchor map ρ and let B be a G ⋉ Y -Banach algebra.…”

Section: The Forgetful Mapmentioning

confidence: 99%

A generalised Green–Julg theorem for proper groupoids and Banach algebras

Paravicini¹

2013

J. Noncommut. Geom.

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“…We thus have to take the push-forward C X instead, which is a field over G .0/ , but to make the notation more familiar, we nevertheless write C 0 .X/. See Section 2.1 for a sketch of the push-forward construction, and for more details, consult [Par09b] or [Par07].…”

Section: The Bost Conjecturementioning

confidence: 99%

“…/ denotes the action of KK ban on K-theory. See [Par09b] or [Par07] and to some extent [Laf06] for more details.…”

Section: The Bost Conjecturementioning

confidence: 99%

“…We give a brief summary of the push-forward construction introduced in [Par07], Section 8.3; see also [Par09b], Section 1 and compare [CEOO03]. Let Y be a locally compact Hausdorff left G -space with anchor map and let B be a G Ë Y -Banach algebra (we use the letter Y instead of Z to emphasise the fact that we do not assume that Y is proper).…”

Section: The Push-forward Constructionmentioning

confidence: 99%

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