  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.
