There can be $C^ *$-embedded dense proper subspaces in $\beta \omega -\omega$

Abstract:

Fine and Gillman have shown that CH implies that if $X$ is a dense proper subspace of ${\omega ^ * } = \beta \omega - \omega$, then $\beta X \ne {\omega ^ * }$. Here it is shown to be consistent with ${\text {MA + c = }}{\omega _2}$ that for every $p \in {\omega ^ * }$ we have $\beta ({\omega ^ * } - \{ p\} ) = {\omega ^ * }$ and also that ${\omega ^ * }$ has a dense subspace $X$ with dense complement such that $\beta X = {\omega ^ * }$.