Tight $σ$-filtrations and the BFPP of $C(ω^*)$
Assuming that $\mathcal{P}(ω)$ is tightly $σ$-filtered, we prove that $C(ω^*)$ fails the ball fixed point property. This axiom holds under the continuum hypothesis and in the $\aleph_2$-Cohen model. The proof shows that the Stone space of the countable--cocountable algebra $\operatorname{Cc}(ω_1)$ is a continuous retract of $ω^*$ and uses the equivalence between tight $σ$-filteredness of $\operatorname{Cc}(ω_1)$ and of $\mathcal{P}(ω)$. We also prove that the quotient of a tightly $σ$-filtered Boolean algebra $B$ with the countable separation property by an $\aleph_1$-generated $P$-ideal is a Boolean retract of $B$.
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