On Character Amenability of Banach Algebras

TL;DR

This paper explores the concept of character amenability in Banach algebras, linking it to special functionals, approximate identities, and topological centers, with examples challenging existing conjectures.

Contribution

It introduces new characterizations of character amenability using special functionals and provides counterexamples to a conjecture relating amenability and $ extit{ ext{φ}}$-amenability.

Findings

Existence of special functionals $m_ extit{ ext{φ}}$ relates to approximate identities.

Unique $m_ extit{ ext{φ}}$ belongs to the topological center of $A^{**}$.

An example contradicts the conjecture that $ extit{ ext{φ}}$-amenability implies amenability.

Abstract

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.