## Abstract and Applied Analysis

### The Structure of $\phi$-Module Amenable Banach Algebras

#### Abstract

We study the concept of $\phi$-module amenability of Banach algebras, which are Banach modules over another Banach algebra with compatible actions. Also, we compare the notions of $\phi$-amenability and $\phi$-module amenability of Banach algebras. As a consequence, we show that, if $S$ is an inverse semigroup with finite set $E$ of idempotents and ${l}^{1}(S)$ is a commutative Banach ${l}^{1}(E)$-module, then ${l}^{1}{(S)}^{{\times}{\times}}$ is ${\phi }^{{\times}{\times}}$-module amenable if and only if $S$ is finite, when $\phi \in {\text{H}\text{o}\text{m}}_{{l}^{1}(E)}({l}^{1}(S))$ is an epimorphism. Indeed, we have generalized a well-known result due to Ghahramani et al. (1996).

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 176736, 7 pages.

Dates
First available in Project Euclid: 27 February 2015

https://projecteuclid.org/euclid.aaa/1425049574

Digital Object Identifier
doi:10.1155/2014/176736

Mathematical Reviews number (MathSciNet)
MR3193492

Zentralblatt MATH identifier
07021874

#### Citation

Bami, Mahmood Lashkarizadeh; Valaei, Mohammad; Amini, Massoud. The Structure of $\phi$ -Module Amenable Banach Algebras. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 176736, 7 pages. doi:10.1155/2014/176736. https://projecteuclid.org/euclid.aaa/1425049574

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