On two refinements of the bounded weak approximate identities

TL;DR

This paper introduces two new refinements of bounded approximate identities in commutative Banach algebras, focusing on convergence and boundedness via the Gel'fand transform, bridging existing concepts.

Contribution

It defines a new type of approximate identity called c-w approximate identity and introduces the notion of weakly bounded nets, expanding the theory of approximate identities.

Findings

Defined c-w approximate identity with Gel'fand transform convergence

Introduced weakly bounded nets in the context of Banach algebras

Bridged the gap between bounded and weak approximate identities

Abstract

Let $A$ be a commutative Banach algebra with non-empty character space $\Delta(A)$. In this paper, we change the concepts of convergence and boundedness in the classical notion of bounded approximate identity. This work give us a new kind of approximate identity between bounded approximate identity and bounded weak approximate identity. More precisely, a net $\{e_{\alpha}\}$ in $A$ is a \emph{c-w approximate identity} if for each $a\in A$, the Gel'fand transform of $e_{\alpha}a$ tends to the Gel'fand transform of $a$ in the compact-open topology and we say $\{e_{\alpha}\}$ is \emph{weakly bounded} if the image of $\{e_{\alpha}\}$ under the Gel'fand transform is bounded in $C_{0}(\Delta(A))$.