Positive elements of left amenable Lau algebras

Abstract

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.