A necessary condition for zero divisors in complex group algebra of torsion-free groups (Q6619560)

Let \(\mathbb C G\) be the complex group algebra of a torsion-free group \(G\) and \(\alpha = \sum_{g\in G} \alpha_g g \in \mathbb C G\). The element \(\alpha\) is a zero divisor if there exists \(0 \neq \beta \in \mathbb C G\) such that \(\alpha \beta = 0\).\N\NThe authors prove the following result.\N\NIf \(\alpha\) is a non-zero zero divisor element of the complex group algebra \(\mathbb C G\), then\N\[\N2{\sum _{g\in G} \vert \alpha_g \vert}^2 <\left(\sum _{g\in G} \vert \alpha_g \vert\right)^2.\N\]