Pere Ara
(Universitat Autónoma de Barcelona)
March 19, 2021
Organized by Americas GNCG Seminar.

Crossed products and the Atiyah problem

It was shown by Austin in 2013 that the question by Atiyah about the rationality of $\ell^2$-Betti numbers has a negative answer. However the problem of determining the exact set of real numbers appearing as $\ell^2$-Betti numbers from a given group $G$ is widely open. In particular, this is an open question for the lamplighter group $L=\mathbb Z_2 \wr \mathbb Z$, which was the first known counterexample to the Strong Atiyah Conjecture, stating that all $\ell^2$-Betti numbers arising from a group $G$ belong to the subgroup $\sum_{H\le G, H \text{ finite}} \frac{1}{|H|} \mathbb Z$ of $\mathbb R$.

I will review some recent progress on this question, obtained in joint work with Joan Claramunt and Ken Goodearl. I will recall some of the basic techniques in our approach, which involve the consideration of Sylvester matrix rank functions on certain crossed products, and their associated $*$-regular envelopes.

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