Positive elements of left amenable Lau algebras

In the present paper, we deal with a large class of Banach algebras known as Lau algebras. It is well-known that if U is a left amenable Lau algebra, then any f is an element of U such that vertical bar fg vertical bar = vertical bar f vertical bar g for all g is an element of U with g >= 0 is a scalar multiple of a positive element in U. We show that this result remains valid for the group algebra l(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 U. This resolves negatively an open problem raised by F. Ghahramani and A. T. Lau.