Almost disjoint families and some automorphic and injective properties of $\ell_\infty/c_0$

Abstract

Answering questions of A. Avil\'es, F. Cabello S\'anchez, J. Castillo, M. Gonz\'alez and Y. Moreno we show that the following statements are independent of the usual axioms ZFC with arbitrarily large continuum: for every (some) $\omega<\kappa<2^\omega$ (1) any linear bounded operator $T: c_0(\kappa)\rightarrow\ell_\infty/c_0$ extends to any superspace of $c_0(\kappa)$. (2) any isomorphism between any two copies of $c_0(\kappa)$ inside $\ell_\infty/c_0$ extends to an automorphism of $\ell_\infty/c_0$. This contrasts with Boolean, Banach algebraic or isometric levels, where the objects known as Hausdorff gap and Luzin gap witness the failure in ZFC of the corresponding properties for the corresponding structures already at the first uncountable cardinal $\kappa=\omega_1$. In particular, consistently, any two pairwise disjoint families in $\wp(\mathbb N)/Fin$ of the same cardinality $\omega<\kappa<2^\omega$ can be mapped onto each other by a linear automorphism of $\ell_\infty/c_0$ regardless of their different combinatorial, algebraic or topological positions in $\wp(\mathbb N)/Fin$. Our positive consistency results use a restricted version of Martin's axiom for a partial order that adds an infinite block diagonal matrix of an operator on $\ell_\infty$ which induces an operator on $\ell_\infty/c_0$. The construction of its finite blocks relies on a lemma of Bourgain and Tzafriri on finite dimensional Banach spaces. Our negative consistency results rely on an analysis of almost disjoint families of $\mathbb N$, the embeddings of $c_0(\kappa)$ into $\ell_\infty/c_0$ they induce and their extensions to $\ell_\infty^c(\kappa)$.