Variable Lebesgue algebra on a Locally Compact group

For a locally compact group $H$ with a left Haar measure, we study the variable Lebesgue algebra $\mathcal{L}^{p(\cdot)}(H)$ with respect to convolution. We show that if $\mathcal{L}^{p(\cdot)}(H)$ has a 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.