Document Type : Research Paper

Authors

• Hamidreza Alihoseini
• Davood Alimohammadi

Arak University

Abstract

‎Let $(A,\Vert \cdot \Vert )$ be a real Banach algebra‎. ‎In this paper we first introduce left and right $\phi$-amenability of $A$ and discuss the relation between left (right‎, ‎respectively) $\phi$-menability and $\bar{\phi }$-amenability of $A$ for $\phi \in \mathrm{△}(A)\cup \{0\}$ where $\bar{\phi }\in \mathrm{△}(A)$ is the conjugate of $\phi$‎. ‎Next we show that $A$ is left (right‎, ‎respectively) $\phi$-amenable if and only if‎ ‎$A_{\mathbb{C}}$ is left (right‎, ‎respectively) ${\phi }_{\mathbb{C}}$-amenable‎, ‎where $A_{\mathbb{C}}$ is a suitable complexification of $A$ and ${\phi }_{\mathbb{C}}\in \mathrm{△}(A_{\mathbb{C}})$ is the induced character by‎ ‎$\phi$ on $A_{\mathbb{C}}$‎. ‎In continue‎, ‎we give a hereditary property for 0-amenability of $A$‎. ‎We also study relations between the injectivity of Banach left $A$-modules and right $\phi$-amenability of $A$‎. ‎Finally‎, ‎we characterize the left character amenability of certain real Banach algebras‎.

Keywords

• Banach algebra‎
• ‎Character amenable‎
• ‎Complexification‎
• ‎Banach left module‎
• ‎injectivity‎