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.