On the Banach-Mazur distance between continuous function spaces with scattered boundaries

Abstract

We study the dependence of the Banach-Mazur distance between two subspaces of vector-valued continuous functions on the scattered structure of their boundaries. In the spirit of a result of Gordon [16], we show that the constant 2 appearing in the Amir-Cambern theorem may be replaced by 3 for some class of subspaces. This we achieve by showing that the Banach-Mazur distance of two function spaces is at least 3, if the height of the set of weak peak points of one of the spaces is larger than the height of a closed boundary of the second space. Next we show that this estimate can be improved, if the considered heights are finite and significantly different. As a corollary, we obtain new results even for the case of C(K, E) spaces.

Isomorphisms of $${\mathcal {C}}(K, E)$$ Spaces and Height of K

“…Similar results for isomorphisms with range in C 0 (Γ, E) spaces were proven before in [4,6]. These estimates were extended in [20] to the case of two spaces K 1 and K 2 of finite height.…”

“…It is natural to ask what is the optimal estimate of the Banach-Mazur distance of two spaces C(K 1 ), C(K 2 ) when the heights of K 1 and K 2 are very close to each other, for example, when they differ by an integer. In the case when the heights of K 1 and K 2 are finite, we see from Theorem 1.1 (and it has been already proved in [20]) that the best lower bound of the distance known so far is comparable to the ratio of the heights of K 1 and K 2 . In the case when K 1 , K 2 are ordinal intervals and one of them is [0, ω], there has been found in [7] also the upper bound of the distance which is quite close to this lower bound.…”

“…We note that we will only need to prove that the lower bound 2n−k k is true, since the bound 3 is known, see [14,Theorem 1.8] or [20,Theorem 1.1]. Actually, the lower bound 3 holds whenever the heights of the considered compact spaces are not the same, as mentioned above.…”

“…We start with the following lemma, which is essentially known in a slightly different formulation (see, e.g. [14,Proposition 2.3] or [20,Lemma 2.1]). Even though the proof is not complicated, we decided to include it for the convenience of the reader.…”

“…Further, the following important lemma is based on an idea of Gordon [17, Lemma 2.2], which has been improved and generalized subsequently by several authors ([3, Lemma 2.1], [5], [7], and [20,Lemma 4.3]). Lemma 2.2.…”