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 $A^{astast}$. 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.