We deal with locally convex algebras in which operate the algebra $H(\ell^\infty)$ of all entire functions with infinitely many variables. These algebras are shown to be exactly the bornological inductive limits of Fr\'echet locally m-convex ones. In the Mackey-complete commutative case, the operation of $H(\ell^\infty)$ is shown to be equivalent to that of the algebra $H(\C)$ of all entire functions over $\C$. We finally provide an example of a commutative locally convex algebra with continuous multiplication admitting no locally m-convex algebra topology at all, but in which $H(\ell^\infty)$ operates.