## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 13, Number 2 (2006), 319-324.

### Positive elements of left amenable Lau algebras

B. Mohammadzadeh and R. Nasr-Isfahani

#### 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.

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 13, Number 2 (2006), 319-324.

**Dates**

First available in Project Euclid: 19 May 2006

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1148059466

**Digital Object Identifier**

doi:10.36045/bbms/1148059466

**Mathematical Reviews number (MathSciNet)**

MR2259910

**Zentralblatt MATH identifier**

1166.46026

**Subjects**

Primary: 46H05: General theory of topological algebras 43A07: Means on groups, semigroups, etc.; amenable groups

Secondary: 43A20: $L^1$-algebras on groups, semigroups, etc.

**Keywords**

Absolute value Lau algebra left amenability positive element

#### Citation

Mohammadzadeh, B.; Nasr-Isfahani, R. Positive elements of left amenable Lau algebras. Bull. Belg. Math. Soc. Simon Stevin 13 (2006), no. 2, 319--324. doi:10.36045/bbms/1148059466. https://projecteuclid.org/euclid.bbms/1148059466