When amenable groups have real rank zero $C^*$-algebras?

Abstract: For every torsion free, discrete and amenable group $G$, the Kadison-Kaplansky conjecture has been verified, so $C^*(G)$ has no nontrivial projections. On the other hand, every torsion element $g\in G$, of order $n$, gives rise to a projection $\frac{1+g+...+g^{n-1}}{n}\in C^*(G)$. Actually, if $G$ is locally finite, then $C^*(G)$ is an AF-algebra, so it has an abundance of projections. So, it is natural to ask what happens when the group has both torsion and non-torsion elements. A result on this direction came from Scarparo, who showed that for every discrete, infinite, finitely generated elementary amenable group, $C^*(G)$ cannot have real rank zero. In this talk, we will explain why if $G$ is discrete, amenable and $C^*(G)$ has real rank zero, then all elementary amenable normal subgroups with finite Hirsch length must be locally finite.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

( paper | slides | video )

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Noncommutative geometry in NYC

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