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.
Citation
B. Mohammadzadeh. R. Nasr-Isfahani. "Positive elements of left amenable Lau algebras." Bull. Belg. Math. Soc. Simon Stevin 13 (2) 319 - 324, June 2006. https://doi.org/10.36045/bbms/1148059466
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