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The Baum–Connes conjecture localised at the unit element of a discrete group

Part of: Fiber spaces and bundles Partial differential equations on manifolds; differential operators Higher algebraic $K$-theory $K$-theory and operator algebras Selfadjoint operator algebras

Published online by Cambridge University Press:  14 January 2021

Paolo Antonini ,
Sara Azzali  and
Georges Skandalis
Paolo Antonini
Affiliation:
Scuola Internazionale Superiore di Studi Avanzati, via Bonomea, 265, 34136Trieste, [email protected]
Sara Azzali
Affiliation:
Fachbereich Mathematik, Universität Hamburg, Bundesstrasse 55, 20146Hamburg, [email protected]
Georges Skandalis
Affiliation:
Université de Paris and Sorbonne Université, CNRS, IMJ-PRG, F-75006Paris, [email protected]
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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 $\Gamma$.

Keywords

assembly mapsNovikov conjectureequivariant Kasparov theoryclassifying spaces

MSC classification

Primary: 19K35: Kasparov theory ($KK$-theory) 46L80: $K$-theory and operator algebras (including cyclic theory)
Secondary: 58J22: Exotic index theories 55R40: Homology of classifying spaces, characteristic classes 19D55: $K$-theory and homology; cyclic homology and cohomology
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 12 , December 2020 , pp. 2536 - 2559
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Copyright
© The Author(s) 2021

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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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