On universal Banach spaces of density continuum

Abstract

Abstract. We consider the question whether there exists a Banach space X of density continuum such that every Banach space of density not bigger than continuum isomorphically embeds into X (called a universal Banach space of density c). It is well known that ℓ∞/c 0 is such a space if we assume the continuum hypothesis. However, some additional set-theoretic assumption is needed, as we prove in the main result of this paper that it is consistent with the usual axioms of set-theory that there is no universal Banach space of density c. Thus, the problem of the existence of a universal Banach space of density c is undecidable using the usual axioms of set-theory.We also prove that it is consistent that there are universal Banach spaces of density c, but ℓ∞/c 0 is not among them. This relies on the proof of the consistency of the nonexistence of an isomorphic embedding of C([0, c]) into ℓ∞/c 0 .

Some Banach spaces added by a Cohen real

“…This result is implied by our results since, as we now explain, if there is no universal Banach space of a given density θ, then there is no universal Boolean algebra of size θ. This is the case because standard arguments (see Fact 1.1 in [2] imply that every Banach space can be embedded into a Banach space of the same density and the form C(K) for some 0-dimensional space K. Combining that with the Stone representation theorem and further standard observations about preservation of isomorphisms, we obtain that the universality number of Banach spaces under isomorphism is never larger than the universality number of Boolean algebras under the isomorphism of Boolean algebras.…”

“…For part (2), note that if p ∈ P and a i is not in w p , then we can extend p to q which is freely generated by w p ∪ {a i }, except for the equations present in p. Hence the set D i = {p : a i ∈ w p } is dense. Note that if p, q ∈ G satisfy p ∩ A = q ∩ A, then p and q are isomorphic over w p .…”

On automorphisms of the Banach space $\ell _\infty /c_0$

“…The main focus of this paper is another natural question, namely, what is the impact of the combinatorics of ℘(N)/Fin on the automorphisms of ℓ ∞ /c 0 considered as a Banach space 1 , in particular if the Open Coloring Axiom (OCA) or the Proper Forcing Axiom (PFA) can be successfully used in this context. At the moment the situation seems similar to that of the early stage of the research on ℘(N)/Fin and N * : the usual axioms seem too weak to resolve many basic questions about the Banach space ℓ ∞ /c 0 ( [8,7,49,30]), the continuum hypothesis provides some answers leaving a chaotic picture full of pathological objects obtained using transfinite induction ( [18,9]) and there is a hope (based, for example, on e.g., [17]) that alternative axioms like OCA, OCA+MA, PFA etc., would provide an elegant structural theory of automorphisms of ℓ ∞ /c 0 . This hope is not only based on the case of ℘(N)/Fin but some other cases as well ( [48,35]).…”

An isometrically universal Banach space induced by a non-universal Boolean algebra

“…the introduction to [2] or [14]) that the first two statements below are equivalent (by the Stone duality) and that they imply the third one:…”

“…The consistency of the failure has been proved by A. Dow and K. P. Hart for Boolean algebras and compact spaces (see [7]) and by S. Shelah and A. Usvyatsov for isometric embeddings (see [16]). Actually one can even prove the nonexistence of a universal Banach space in B 2 ω for isomorphic embeddings (see [2,3]). Whether the equivalence of all the three statements above can be proved without additional assumptions is the main question that motivates our research in this paper.…”