Character amenability of real Banach algebras

Abstract

‎Let $ (A,| cdot |) $ be a real Banach algebra‎. ‎In this paper we first introduce left and right $varphi$-amenability of $A$ and discuss the relation between left (right‎, ‎respectively) $varphi$-menability and $overline{varphi}$-amenability of $A$ for $varphiintriangle(A)cup{0}$ where $overline{varphi}intriangle(A)$ is the conjugate of $varphi$‎. ‎Next we show that $A$ is left (right‎, ‎respectively) $varphi$-amenable if and only if‎ ‎$A_{mathbb{C}}$ is left (right‎, ‎respectively) $varphi_{mathbb{C}}$-amenable‎, ‎where $A_{mathbb{C}}$ is a suitable complexification of $ A $ and $varphi_{mathbb{C}}intriangle(A_{mathbb{C}})$ is the induced character by‎ ‎$varphi$ 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 $varphi$-amenability of $A$‎. ‎Finally‎, ‎we characterize the left character amenability of certain real Banach algebras‎.