# On a question related to bounded approximate identities of ideals in Banach algebras

## TL;DR

This paper explores the relationship between bounded approximate identities and multiplier algebras in Banach algebras, providing examples and conditions that clarify when ideals have bounded approximate identities.

## Contribution

It presents a counterexample of a Banach algebra ideal with a multiplier algebra equal to the algebra but lacking a bounded approximate identity, and establishes a necessary condition for such ideals.

## Key findings

- Counterexample of an ideal without bounded approximate identity
- Necessary condition for ideals with approximate identities
- Fourier algebra density result in harmonic analysis

## Abstract

In this paper we give an example of a Banach algebra $A$ and a closed ideal $I$ of $A$ such that the multiplier algebra of $I$ is equal to $A$ but $I$ does not have any bounded approximate identity. In the case that $I$ has an approximate identity, we give a necessary condition on $I$ for which $A=\mathcal{M}(I)$, where $\mathcal{M}(I)$ denotes the multiplier algebra of $I$. Finally, as a corollary of our results, we show that the Fourier algebra of an amenable group is strictly dense in the Fourier-Stieltjes algebra.