PFA and complemented subspaces of ℓ∞/c0

Abstract

The Banach space ∞ /c 0 is isomorphic to the linear space of continuous functions on N * with the supremum norm, C(N * ). Similarly, the canonical representation of the ∞ sum of ∞ /c 0 is the Banach space of continuous functions on the closure of any non-compact cozero subset of N * . It is important to determine if there is a continuous linear lifting of this Banach space to a complemented subset of C(N * ). We show that PFA implies there is no such lifting.2010 Mathematics Subject Classification 54A25 (primary); 03E35 (secondary)

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]).…”

“…The are many basic problems concerning the automorphisms of ℓ ∞ /c 0 left open, some of them are listed in Section 7. A breakthrough in developing the methods of direct applications of PFA in the space ℓ ∞ /c 0 which was recently obtained by A. Dow in [17] may be especially useful in attacking these problems. The structure of the paper is as follows:…”

“…In this section we mention some open problems and some observations related to them. This should not be considered as a full list of urgent open problems concerning the Banach space ℓ ∞ /c 0 , for example we do not touch problems related to the primariness of ℓ ∞ /c 0 (see [18], [17], [28]) subspaces of ℓ ∞ /c 0 ( [7], [29], [49], [30]), ℓ ∞ -sums ( [18], [17], [8]) or extensions of operators on ℓ ∞ /c 0 ([9], [4]).…”

“…The existence of such an operator is a weak version of the existence of a retraction from N * onto F . Alan Dow in [17] developed new methods (which may be quite useful in the above context) proving that PFA implies that the cozero sets do not have the linear extension property. The last couple of problems is related to possible applications of canonizations of embeddings.…”

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

“…At the same time, while well-known that the algebra of clopen subsets of the cozero set in N * does not embed into ℘(N)/F in under OCA (see [17]), its is still unknown if the corresponding Banach space ℓ ∞ (ℓ ∞ /c 0 ) can be embedded under OCA into ℓ ∞ /c 0 (cf. [4,8]). …”