Higher Kazhdan projections, $\ell_2$-Betti numbers and Baum–Connes conjectures

Kang Li; Piotr Nowak; Sanaz Pooya

doi:10.4171/jncg/529

JournalsjncgVol. 18, No. 1pp. 313–336

Higher Kazhdan projections, $ℓ_{2}$ -Betti numbers and Baum–Connes conjectures

Kang Li
Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany
Piotr Nowak
Institute of Mathematics of the Polish Academy of Sciences, Warszawa, Poland
Sanaz Pooya
Stockholm University, Sweden

$Higher Kazhdan projections, $\ell_2$-Betti numbers and Baum–Connes conjectures cover$

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Abstract

We introduce higher-dimensional analogs of Kazhdan projections in matrix algebras over group $C^{*}$ -algebras and Roe algebras. These projections are constructed in the framework of cohomology with coefficients in unitary representations and in certain cases give rise to non-trivial $K$ -theory classes. We apply the higher Kazhdan projections to establish a relation between $ℓ_{2}$ -Betti numbers of a group and surjectivity of different Baum–Connes type assembly maps.

Cite this article

Kang Li, Piotr Nowak, Sanaz Pooya, Higher Kazhdan projections, $ℓ_{2}$ -Betti numbers and Baum–Connes conjectures. J. Noncommut. Geom. 18 (2024), no. 1, pp. 313–336

DOI 10.4171/JNCG/529