## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 22, Number 4 (2015), 655-668.

### On $\phi$-ergodic property of Banach modules

#### Abstract

Let ${\mathcal A}$ be a Banach algebra and let $\phi$ be a non-zero character on ${\mathcal A}$. We introduce the notion of $\phi$-ergodic property for a Banach right ${\mathcal A}$-module $X$. This concept considerably generalizes the existence of $\phi$-means of norm one on ${\mathcal A}^*$. We also show that the $\phi$-ergodic property of $X$ is related to some other properties such as a Hahn-Banach type extension property and the existence of $\phi$-means of norm one on a certain subspace of ${\mathcal A}^*$. Finally, we give some characterizations for $\phi$-amenability of a Banach algebra in terms of its closed ideals.

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 22, Number 4 (2015), 655-668.

**Dates**

First available in Project Euclid: 18 November 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1447856065

**Mathematical Reviews number (MathSciNet)**

MR3429177

**Zentralblatt MATH identifier**

1338.46059

**Subjects**

Primary: 46H20: Structure, classification of topological algebras 46H25: Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)

Secondary: 16E40: (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.)

**Keywords**

Banach module $\phi$-ergodic property $\phi$-amenability Lau algebra $\phi$-mean

#### Citation

Nemati, Mehdi. On $\phi$-ergodic property of Banach modules. Bull. Belg. Math. Soc. Simon Stevin 22 (2015), no. 4, 655--668. https://projecteuclid.org/euclid.bbms/1447856065