Generalized notions of character amenability

TL;DR

This paper introduces and explores generalized notions of character amenability in Banach algebras, establishing equivalences among various approximate and uniform properties, and clarifying their relationships especially in commutative cases.

Contribution

It defines new concepts of character contractibility and approximate character amenability, and proves their equivalence under certain conditions, advancing the theoretical understanding of Banach algebra properties.

Findings

Approximate character amenability equals approximate character contractibility.

Uniform approximate character amenability equals character amenability.

In commutative Banach algebras, character contractibility equals contractibility.

Abstract

In this paper the concepts of character contractibility, approximate character amenability (contractibility) and uniform approximate character amenability (contractibility) are introduced. We are concerned with the relations among the generalized concepts of character amenability for Banach algebra. We prove that approximate character amenability and approximate character contractibility are the same properties, as are uniform approximate character amenability and character amenability, as are uniform approximate character contractibility and character contractibility. For commutative Banach algebra, we prove that character contractibility and contractibility are the same properties. Moreover, general theory for those concepts is developed.