Character amenability of real Banach algebras

‎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 $\varphi\in\triangle(A)\cup\{0\}$ where $\overline{\varphi}\in\triangle(A)$ 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 $\varphi_{\mathbb{C}}\in\triangle(A_{\mathbb{C}})$ 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.