On Character Contractibility of Real Banach Algebras

‎In this paper‎, ‎we introduce and study $\varphi$-contractibility and character contractibility of real Banach algebras‎. ‎We first show that if $ (A,\| \cdot \|) $ is a real Banach algebra with nonempty character space $\triangle(A)$ then $A$ is $\varphi$-contractibile if and only if $A$ is $\overline{\varphi}$-contractible‎, ‎where $\varphi\in\triangle(A)$ and $\overline{\varphi}$ is the conjugate of $\varphi$‎. ‎We next prove that if $ (A,\| \cdot \|) $ is a real Banach algebra and $A_{\mathbb{C}}$ is a suitable complexification of $ A $‎, ‎then $A$ is character contractibile if and only if $A_{\mathbb{C}}$ is character contractibile‎. ‎We also give a hereditary property for 0-contractibility of $A$‎. ‎In continue‎, ‎for a real Banach algebra $A$ we study relations between the projectivity of Banach left $A$-modules and left $\varphi$-contractibility of $A$‎. ‎Finally‎, ‎we investigate the left character contractibility of certain real Banach algebras‎.