## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 18, Number 3 (2011), 415-429.

### Approximate weak amenability of Banach algebras

G. H. Esslamzadeh and B. Shojaee

#### Abstract

In this paper we deal with four generalized notions of amenability which are called approximate, approximate weak, approximate cyclic and approximate $n$-weak amenability. The first two were introduced and studied by Ghahramani and Loy in [9]. We introduce the third and fourth ones and we show by means of some examples, their distinction with their classic analogs. Our main result is that under some mild conditions on a given Banach algebra $\mathcal{A}$, if its second dual $\mathcal{A}^{**}$ is $(2n-1)$-weakly [respectively approximately/ approximately weakly/ approximately $n$-weakly] amenable, then so is $\mathcal{A}$. Also if $\mathcal{A}$ is approximately $(n+2)$-weakly amenable, then it is approximately $n$-weakly amenable. Moreover we show the relationship between approximate trace extension property and approximate weak [respectively cyclic] amenability. This answers question 9.1 of [9] for approximate weak and cyclic amenability.

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 18, Number 3 (2011), 415-429.

**Dates**

First available in Project Euclid: 17 August 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1313604448

**Digital Object Identifier**

doi:10.36045/bbms/1313604448

**Mathematical Reviews number (MathSciNet)**

MR2883138

**Zentralblatt MATH identifier**

1230.46046

**Subjects**

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

Secondary: 46H35: Topological algebras of operators [See mainly 47Lxx]

**Keywords**

Approximately inner derivation Approximately weakly amenable Approximately $n$-weakly amenable Approximately amenable Approximate trace extension property

#### Citation

Esslamzadeh, G. H.; Shojaee, B. Approximate weak amenability of Banach algebras. Bull. Belg. Math. Soc. Simon Stevin 18 (2011), no. 3, 415--429. doi:10.36045/bbms/1313604448. https://projecteuclid.org/euclid.bbms/1313604448