In the present paper, we deal with a large class of
Banach algebras known as Lau algebras.
It is well-known that if ${\frak A}$ is a left amenable Lau
algebra, then any $f\in {\frak A}$ such that $|fg|=|f|g$ for all
$g\in {\frak A}$ with $g\geq 0$ is a scalar multiple of a positive
element in ${\frak A}$. We show that this result remains
valid for the group algebra $\ell^1(G)$ of any,
not necessarily amenable, discrete group $G$.
We also give an example which shows that the result is, in general,
not true without the hypothesis of left amenability of ${\frak A}$.
This resolves negatively an open problem raised by F. Ghahramani and A. T. Lau.