Tight $\sigma$-filtrations and the BFPP of $C({\omega }^{\ast })$

Assuming that $\mathcal{P}(\omega )$ is tightly $\sigma$-filtered, we prove that $C({\omega }^{\ast })$ fails the ball fixed point property. This axiom holds under the continuum hypothesis and in the ${\mathrm{\aleph }}_2$-Cohen model. The proof shows that the Stone space of the countable–cocountable algebra $\mathrm{Cc}({\omega }_1)$ is a continuous retract of ${\omega }^{\ast }$ and uses the equivalence between tight $\sigma$-filteredness of $\mathrm{Cc}({\omega }_1)$ and of $\mathcal{P}(\omega )$. We also prove that the quotient of a tightly $\sigma$-filtered Boolean algebra $B$ with the countable separation property by an ${\mathrm{\aleph }}_1$-generated $P$-ideal is a Boolean retract of $B$.