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.
Let $UC(K)$ denote the Banach space of all bounded uniformly continuous functions on a hypergroup $K$. The main results of this article concern on the $alpha$-amenability of $UC(K)$ and quotients and products of hypergroups. It is also shown that a S
turm-Liouville hypergroup with a positive index is $alpha$-amenable if and only if $alpha=1$.
Let $K$ be a commutative hypergroup and $alphain hat{K}$. We show that $K$ is $alpha$-amenable with the unique $alpha$-mean $m_alpha$ if and only if $m_alphain L^1(K)cap L^2(K)$ and $alpha$ is isolated in $hat{K}$. In contrast to the case of amenable
noncompact locally compact groups, examples of polynomial hypergroups with unique $alpha$-means ($alpha ot=1$) are given. Further examples emphasize that the $alpha$-amenability of hypergroups depends heavily on the asymptotic behavior of Haar measures and characters.
