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

TL;DR

This paper introduces higher-dimensional Kazhdan projections linked to $ ext{l}_2$-Betti numbers and explores their implications for Baum-Connes conjectures, revealing new connections between group cohomology, $K$-theory, and operator algebras.

Contribution

It develops higher-dimensional Kazhdan projections in group $C^*$-algebras and Roe algebras, and relates these to $ ext{l}_2$-Betti numbers and Baum-Connes conjecture surjectivity.

Findings

Higher Kazhdan projections can produce non-trivial $K$-theory classes.

Established a relation between $ ext{l}_2$-Betti numbers and Baum-Connes assembly maps.

Provided new tools for studying the Baum-Connes conjecture via cohomological methods.

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 $\ell_2$-Betti numbers of a group and surjectivity of different Baum-Connes type assembly maps.