Going-down functors, the Künneth formula, and the Baum-Connes conjecture
type: Item
instance of: scholarly article
title: Going-down functors, the Künneth formula, and the Baum-Connes conjecture
zbMATH Open document ID: 1063.46056
author: Hervé Oyono-Oyono
author: Jérôme Chabert
author: Siegfried Echterhoff
published in: Geometric and Functional Analysis. GAFA
publication date: 2004-12-13T00:00:00Z
review text: The paper begins with an extension of the result of \textit{C.~Schochet} [Pac. J. Math. 98, 443--458 (1982; Zbl 0439.46043)] on the Künneth formula for \(C^*\)-algebras. Let \(\mathcal N\) be the class of all \(C^*\)-algebras \(A\) that satisfy the \(K\)-theory Künneth formula with any \(C^*\)-algebra \(B\). For a locally compact group \(G\), an equivariant analogue \({\mathcal N}_G\) of \(\mathcal N\) is defined as the class of all \(G\)-algebras \(A\) that satisfy the (mixed) Künneth formula \centerline{\(0\to K_*^{\text{top}}(G;A)\otimes K_*(B)\to K_*^{\text{top}}(G;A\otimes B)\to {\text{Tor}}(K_*^{\text{top}}(G;A),K_*(B))\to 0 \)} for any \(C^*\)-algebra \(B\) with trivial \(G\)-action, where \(\otimes\) denotes the minimal tensor product. It is shown that \(A\in{\mathcal N}_G\) iff \(A\rtimes_r G\in{\mathcal N}\) and that if \(A\) satisfies \noindent \((*)\)\qquad \(A\rtimes K\in{\mathcal N}\) for all compact subgroups \(K\subset G,\) \noindent then \(A\in{\mathcal N}_G\). In particular, \({\mathcal N}_G\) contains all type I \(G\)-algebras. Then the connection between the Künneth formula and the Baum-Connes conjecture for \(G\) with coefficients is studied. Let \(A\) satisfy \((*)\). Suppose further that \(G\) satisfies the Baum-Connes conjecture with coefficients in \(A\otimes B\) for every \(C^*\)-algebra \(B\) with trivial \(G\)-action. Then \(A\rtimes_r G\in{\mathcal N}\). As a corollary, it is shown that if \(G\) satisfies the Baum-Connes conjecture with trivial coefficients, then it satisfies this conjecture with any coefficient \(C^*\)-algebra \(B\in{\mathcal N}\) with trivial \(G\)-action. Using this corollary, the authors prove the following permanence result: if the groups \(G_1\) and \(G_2\) satisfy the Baum-Connes conjecture with trivial coefficients and if \(C^*_r(G_1)\) or \(C^*_r(G_2)\) belong to \(\mathcal N\), then \(G_1\times G_2\) satisfies the same conjecture. Some other applications are presented.
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 19K99
zbMATH DE Number: 2121853
zbMATH Keywords: Künneth formula
zbMATH Keywords: \(K\)-theory
zbMATH Keywords: Baum-Connes conjecture
reviewed by: Vladimir M. Manuilov
MaRDI profile type: Publication
full work available at URL: s00039-004-0467-6
OpenAlex ID: W2035470509
Wikidata QID: Q123326262
DOI: 10.1007/S00039-004-0467-6
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description: scientific article; zbMATH DE number 2121853