The Bost conjecture and proper Banach algebras
type: Item
instance of: scholarly article
title: The Bost conjecture and proper Banach algebras
zbMATH Open document ID: 1267.19004
DOI: 10.4171/JNCG/113
author: Walther Paravicini
published in: Journal of Noncommutative Geometry
publication date: 2013-04-19T00:00:00Z
review text: Let \(\mathcal{G}\) be a locally compact Hausdorff groupoid that is equipped with a Haar system and for which there exists a locally compact classifying space \(\underline{E}\mathcal{G}\) for proper actions of \(\mathcal{G}\). Using the bivariant \(KK\)-theory for Banach algebras defined by \textit{V. Lafforgue} [Invent. Math. 149, No.~1, 1--95 (2002; Zbl 1084.19003)], one can define for each nonnegative integer \(n\) and any \(\mathcal{G}\)-Banach algebra \(B\) an abelian group \(K_n^{\text{top, ban}}(\mathcal{G}, B)\). For an unconditional completion (this term is defined in the paper under review) \(\mathcal{A}(\mathcal{G}, B)\) of \(\mathcal{C}_c(\mathcal{G}, B)\), the author defines a Bost assembly map \(\mu_{\mathcal{A}}^{\mathcal{B}}\) from \(K_n^{\text{top, ban}}(\mathcal{G}, B)\) to \(K_n(\mathcal{A}(\mathcal{G}, B))\). The Banach algebraic version of the Bost conjecture asserts that \(\mu_{\mathcal{A}}^{\mathcal{B}}\) is an isomorphism. In this paper, the author defines a notion of a proper \(\mathcal{G}\)-Banach algebra and proves that under mild conditions on \(\mathcal{A}\), the Bost assembly map \(\mu_{\mathcal{A}}^{\mathcal{B}}\) is split surjective for any non-degenerate proper \(\mathcal{G}\)-Banach algebra \(B\). Furthermore, the splitting is natural in \(B\). The author proves the result by first establishing it when \(\mathcal{G}\) is proper and \(\underline{E}\mathcal{G}\slash\mathcal{G}\) is compact; to do this, he employs a generalized Green-Julg theorem that appears in another paper of his [J. Noncommut. Geom. 7, No. 1, 149--190 (2013; Zbl 1273.43003)].
reviewed by: Efton Park
Mathematics Subject Classification ID: 19K35
Mathematics Subject Classification ID: 43A99
Mathematics Subject Classification ID: 22A22
Mathematics Subject Classification ID: 22D15
Mathematics Subject Classification ID: 46L80
zbMATH DE Number: 6155608
zbMATH Keywords: Bost conjecture
zbMATH Keywords: Banach algebra
zbMATH Keywords: topological groupoid
MaRDI profile type: Publication
Wikidata QID: Q122930164
cites work: Q3545869
cites work: Q4317850
cites work: Shapiro's lemma for topological \(K\)-theory of groups
cites work: Bundles of topological vector spaces and their duality
cites work: Q4956084
cites work: Groups acting properly on ``bolic'' spaces and the Novikov conjecture
full work available at URL: https://doi.org/10.4171/jncg/113
OpenAlex ID: W2165773222
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description: scientific article; zbMATH DE number 6155608