Ultrapowers of Banach algebras and modules

The Arens products are the standard way of extending the product from a Banach algebra $\text{mc}A$ to its bidual $\text{mc}{A}^{\mathrm{\prime }\mathrm{\prime }}$. Ultrapowers provide another method which is more symmetric, but one that in general will only give a bilinear map, which may not be associative. We show that if $\text{mc}A$ is Arens regular, then there is at least one way to use an ultrapower to recover the Arens product, a result previously known for C${}^{\ast }$-algebras. Our main tool is a Principle of Local Reflexivity result for modules and algebras.