Variable Lebesgue algebra on a Locally Compact group

TL;DR

This paper investigates the properties of variable Lebesgue algebras on locally compact groups, focusing on conditions for the existence of identities and approximate identities, and characterizing left ideals.

Contribution

It provides necessary and sufficient conditions for the existence of identities and approximate identities in variable Lebesgue algebras on locally compact groups.

Findings

If the algebra has bounded exponent, it contains a left approximate identity.

A closed linear subspace is a left ideal iff it is left translation invariant.

Characterization of when the algebra has an identity.

Abstract

For a locally compact group $H$ with a left Haar measure, we study variable Lebesgue algebra $\mathcal{L}^{p(\cdot)}(H)$ with respect to a convolution. We show that if $\mathcal{L}^{p(\cdot)}(H)$ has bounded exponent, then it contains a left approximate identity. We also prove a necessary and sufficient condition for $\mathcal{L}^{p(\cdot)}(H)$ to have an identity. We observe that a closed linear subspace of $\mathcal{L}^{p(\cdot)}(H)$ is a left ideal if and only if it is left translation invariant.