# When is multiplication in a Banach algebra open?

## TL;DR

This paper investigates when multiplication in Banach algebras is open, establishing necessary conditions, and providing counterexamples in convolution algebras showing that openness is rare.

## Contribution

It develops the theory of open multiplication in Banach algebras, linking it to topological stable rank, and resolves the openness of convolution in specific semigroup algebras.

## Key findings

- Open multiplication implies topological stable rank 1.
- Convolution in ℓ₁(ℤ) and ℓ₁(ℚ) is not uniformly open.
- Openness of multiplication in operator algebras remains uncertain.

## Abstract

We develop the theory of Banach algebras whose multiplication (regarded as a bilinear map) is open. We demonstrate that such algebras must have topological stable rank 1, however the latter condition is strictly weaker and implies only that products of non-empty open sets have non-empty interior. We then investigate openness of convolution in semigroup algebras resolving in the negative a problem of whether convolution in $\ell_1(\mathbb{N}_0)$ is open. By appealing to ultraproduct techniques, we demonstrate that neither in $\ell_1(\mathbb{Z})$ nor in $\ell_1(\mathbb Q)$ convolution is uniformly open. The problem of openness of multiplication in Banach algebras of bounded operators on Banach spaces and their Calkin algebras is also discussed.