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The Baum–Connes conjecture localised at the unit element of a discrete group
Published online by Cambridge University Press: 14 January 2021
Abstract
We construct a Baum–Connes assembly map localised at the unit element of a discrete group $\Gamma$. This morphism, called
$\mu _\tau$, is defined in
$KK$-theory with coefficients in
$\mathbb {R}$ by means of the action of the idempotent
$[\tau ]\in KK_{\mathbin {{\mathbb {R}}}}^\Gamma (\mathbb {C},\mathbb {C})$ canonically associated to the group trace of
$\Gamma$. We show that the corresponding
$\tau$-Baum–Connes conjecture is weaker than the classical version, but still implies the strong Novikov conjecture. The right-hand side of
$\mu _\tau$ is functorial with respect to the group
$\Gamma$.
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- Research Article
- Information
- Copyright
- © The Author(s) 2021
Footnotes
Paolo Antonini was partially supported by the grant H2020-MSCA-RISE-2015-691246-QUANTUM DYNAMICS; Sara Azzali acknowledges support by the DFG grant Secondary invariants for foliations within the Priority Programme SPP 2026 ‘Geometry at Infinity’.
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