Forcing properties of Boolean algebras of the type $\mathcal P(\omega )/\mathcal I$

We study forcing properties of the Boolean algebras $\mathcal {P}(\omega )/\mathcal {I}$, where $\mathcal {I}$ is a Borel ideal on $\omega $. We show (Theorem 2.12) that (under a large cardinal hypothesis) $\mathcal {P}(\omega )/\mathcal {I}$ does not add reals if and only if it has a dense $\sigma $-closed subset. For analytic P-ideals $\mathcal {I}$ we show (Theorem 3.3) that either $\mathcal {P}(\omega )/\mathcal {I}$ is $\omega ^{\omega }$-bounding or it is not proper. We also investigate the existence of completely separable $\mathcal {I}$-MAD families.