Products of $\omega^*$ images

Let $\omega^*$ be the \u{C}ech-Stone remainder $\beta\omega \setminus \omega$. We show that there exists a large class $\cal O$ of images of $\omega^*$ such that whenever $\cal S$ is a subset of $\cal O$ of cardinality at most the continuum, then $\omega^* \times \prod {\cal S}$ is again an image of $\omega^*$. The class $\cal O$ contains all separable compact spaces, all compact spaces of weight at most $\omega_1$ and all perfectly normal compact spaces.