- - - Arak University
- - - Arak University

‎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\{۰\} 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 ۰-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.

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