### PFA and complemented subspaces of ℓ_{∞}/c_{0}

#### 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.

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.

#### Full Text:

2. [PDF]DOI: https://doi.org/10.4115/jla.2016.8.2

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Journal of Logic and Analysis ISSN: 1759-9008