On Character Amenability of Banach Algebras

Associated to a nonzero homomorphism $\varphi$ of a Banach algebra $A$, we regard special functionals, say $m_\varphi$, on certain subspaces of $A^\ast$ which provide equivalent statements to the existence of a bounded right approximate identity in the corresponding maximal ideal in $A$. For instance, applying a fixed point theorem yields an equivalent statement to the existence of a $m_\varphi$ on $A^\ast$; and, in addition we expatiate the case that if a functional $m_\varphi$ is unique, then $m_\varphi$ belongs to the topological center of the bidual algebra $A^{\ast\ast}$. An example of a function algebra, surprisingly, contradicts a conjecture that a Banach algebra $A$ is amenable if $A$ is $\varphi$-amenable in every character $\varphi$ and if functionals $m_\varphi$ associated to the characters $\varphi$ are uniformly bounded. Aforementioned are also elaborated on the direct sum of two given Banach algebras.