An analytical zero divisor

Question

Let $G$, $\mathbb C[G]$ and $\text{vN}(G)$ be a torsion free group, it's group ring and group von Neumann algebra, resp.. Let $0\neq\alpha\in\mathbb C[G]$ and $0\neq p\neq1$ is a projection in the group von Neumann algebra $\text{vN}(G)$, is it true that $\alpha p\neq0$?

Answer 1

I guess the answer is open in general, since a positive answer to your question implies Kaplansky's zero-divisor conjecture:

Indeed, let $V \subseteq \ell^2(G)$ be the kernel of the left-multiplication with $\alpha$ on $\ell^2(G)$. Note that $V$ is a $\mathrm{vN}(G)$-submodule of $\ell^2(G)$ (I consider $\mathrm{vN}(G)$ to be the algebra of bounded operators which commute with the right action of $G$ on $\ell^2(G)$). Then the orthogonal projection $P \in \mathrm{vN(G)}$ onto $V$ satisfies $\alpha P = 0$. Now $\alpha \neq 0$ shows that $P \neq 1$. Moreover, $P = 0$ exactly if $V = 0$, i.e. exactly if left-multiplication with $\alpha$ on $\ell^2(G)$ is injective. Conclusion: if the answer to your question is yes, then $P=1$ and left-multiplication with $\alpha$ on $\ell^2(G)$ and thus on $\mathbb{C}[G]$ (even on $\mathrm{vN}(G)$) is injective. This means $\alpha$ is a not a left zero-divisor.

This is closely related to the Atiyah conjecture. In fact, your question is equivalent to the question if the von-Neumann rank of every non-zero $\alpha \in \mathbb{C}[G]$ is $\mathrm{rk_{vN}(\alpha)} = 1$. The Atiyah conjecture for $\mathbb{C}[G]$ has been established recently for a large class of torsion-free groups by Andrei Jaikin-Zapirain; see this article.