C(K) spaces which cannot be uniformly embedded into c₀(Γ )

Abstract

Abstract. We give two examples of scattered compact spaces K such that C(K) is not uniformly homeomorphic to any subset of c 0 (Γ ) for any set Γ . The first one is [0, ω 1 ] and hence it has the smallest possible cardinality, the other one has the smallest possible height ω 0 + 1.Foreword. We present two results of Jan Pelant. He presented them at seminars in the Mathematical Institute of Czech Academy of Sciences during the last two years. The example described in Theorem 4.1 below is quite recent. Unfortunately, Jan Pelant died before the results were prepared for publication. We decided to reconstruct them using his hand-written notes.1. Introduction. We give two examples of scattered compact spaces K such that C(K) is not uniformly homeomorphic to any subset of c 0 (Γ ) for any set Γ . This contributes to the study of nonlinear embeddings of (real) Banach spaces into other ones. This investigation is related to the study of the question which topological (or metric) properties enable one to reconstruct the linear structure of a Banach space.The well known Mazur-Ulam theorem says that the existence of an isometry of two Banach spaces implies their linear isometry. On the other hand, the result of H. Toruńczyk [17] (of M. I. Kadec [10] for separable spaces) shows that homeomorphism of two infinite-dimensional Banach spaces gives no information about their linear structure. This is the reason why uniform

Cauchy filters from Pelant's games

“…Thus we get Theorem 1.1 in [9]: The Banach space C([0, ω 1 ]) is not uniformly homeomorphic to a subset of c 0 (Γ) for any Γ. This and related results are discussed in more detail in [9].…”

“…It also incorporates elements of the proof of Th. 1.1 in [9]. It should be noted that [6, Th.17] deals with more general point characters of uniformities; the point-finite version that I prove here is a special case.…”

“…Pelant [8] proved that X has a uniformity basis consisting of point-finite covers if and only if X is uniformly homeomorphic to a subset of c 0 (Γ) for some index set Γ. Thus we get Theorem 1.1 in [9]: The Banach space C([0, ω 1 ]) is not uniformly homeomorphic to a subset of c 0 (Γ) for any Γ. This and related results are discussed in more detail in [9].…”

“…Pelant [4][7] and Ščepin [10] constructed metric spaces X for which the answer is negative. In a later paper, Pelant, Holický and Kalenda [9] prove that the answer is no for X = C([0, ω 1 ]); this immediately follows also from the main theorem in section 2.…”

“…Although using a different terminology, the proof of Theorem 8 here is a modification of Pelant's construction in [5] and [6]. The fact that we can take X = C([0, ω 1 ]) is not surprising in view of [9,Thm 1.1].…”