2025/09/26 by Koszmider, Piotr, Rojek, Małgorzata
#FOS: Mathematics #Functional Analysis (math.FA) #Logic (math.LO)
paper · doi:10.48550/arxiv.2509.22376
Answering questions of A. Avilés, F. Cabello Sánchez, J. Castillo, M. González and Y. Moreno we show that the following statements are independent of the usual axioms ZFC with arbitrarily large continuum: for every (some) ω<κ<2ω (1) any linear bounded operator T: c0(κ)→ℓ_∞/c0 extends to any superspace of c0(κ). (2) any isomorphism between any two copies of c0(κ) inside ℓ_∞/c0 extends to an automorphism of ℓ_∞/c0. 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 κ=ω1. In particular, consistently, any two pairwise disjoint families in \wp(\mathbb N)/Fin of the same cardinality ω<κ<2ω can be mapped onto each other by a linear automorphism of ℓ_∞/c0 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 ℓ_∞ which induces an operator on ℓ_∞/c0. 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 c0(κ) into ℓ_∞/c0 they induce and their extensions to ℓ_∞c(κ).