On Character Amenability of Banach Algebras

Abstract

Associated to a nonzero homomorphism $\phi$ of a Banach algebra $A$, we regard special functionals, say $m_{\phi }$, 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 $m_{\phi }$ on $A^{\ast }$; and, in addition we expatiate the case that if a functional $m_{\phi }$ is unique, then $m_{\phi }$ 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 $\phi$-amenable in every character $\phi$ and if functionals $m_{\phi }$ associated to the characters $\phi$ are uniformly bounded. Aforementioned are also elaborated on the direct sum of two given Banach algebras.

Keywords: Banach algebra, topological center, amenability