  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.
