PFA and complemented subspaces of ℓ∞/c0

Abstract

The Banach space $\ell_\infty/c_0$ is isomorphic
to the linear space of continuous functions on $\mathbb N^*$ with the
supremum norm, $C(\mathbb N^*)$. Similarly, the canonical
representation of the
 $\ell_\infty$ sum of $\ell_\infty/c_0$ is the Banach space of
 continuous
functions on the closure of any non-compact cozero subset of $\mathbb
N^*$.  It is important to determine if there is a continuous
linear lifting of this Banach space to a complemented
subset of $C(\mathbb N^*)$. We show that PFA implies there is no such
lifting.