Variable Lebesgue algebra on a Locally Compact group

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.