In the present paper first we introduce the notion of ${\cal LM}_I^p({\cal A})$, where $\cal A$ is a Banach space, $I$ is an index set and $1\leq p<\infty$. We find necessary and sufficient conditions for which ${\cal LM}_I^p({\cal A})$ is a Banach algebra and investigate amenability of this Banach algebra. Applications to $\ell^p(S)$ ($1\leq p<\infty$), where $S$ is a Brandt semigroup, are also given
