Isomorphisms of C(K,E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {C}}(K, E)$$\end{document} Spaces and Height of K

Let K1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_1$$\end{document}, K2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_2$$\end{document} be compact Hausdorff spaces and E1,E2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_1, E_2$$\end{document} be Banach spaces not containing a copy of c0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_0$$\end{document}. We establish lower estimates of the Banach–Mazur distance between the spaces of continuous functions C(K1,E1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {C}}(K_1, E_1)$$\end{document} and C(K2,E2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {C}}(K_2, E_2)$$\end{document} based on the ordinals ht(K1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ht(K_1)$$\end{document}, ht(K2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ht(K_2)$$\end{document}, which are new even for the case of spaces of real-valued functions on ordinal intervals. As a corollary we deduce that C(K1,E1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {C}}(K_1, E_1)$$\end{document} and C(K2,E2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {C}}(K_2, E_2)$$\end{document} are not isomorphic if ht(K1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ht(K_1)$$\end{document} is substantially different from ht(K2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ht(K_2)$$\end{document}.


Introduction
We first recall several notions. To start with, for a compact space K and a Banach space E, let C(K, E) denote the space of all continuous E-valued functions endowed with the sup-norm. We write C(K) for C(K, R). All compact spaces are assumed to be Hausdorff.
Next we recall that the Banach-Mazur distance of Banach spaces E 1 , E 2 is defined to be the infimum of T T −1 over the set of all isomorphisms T : E 1 → E 2 and is denoted by d BM (E 1 , E 2 ).
Further, the derivative of a topological space S is defined recursively as follows. The set S (1) is the set of accumulation points of S, and for an ordinal α > 1, let S (α) = (S (β) ) (1) , if α = β + 1, and S (α) = β<α S (β) , if α is a limit ordinal. Moreover, let S (0) = S. The topological space S is called scattered if there exists an ordinal α such that S (α) is empty, and minimal such α is called the height of S (or the Cantor-Bendixon index of S) and is denoted by ht(S). If S is not scattered, then we define ht(S) to be ∞. We use the convention that α < ∞ for each ordinal α. If K is a scattered compact space, see e.g. the important paper [11]. Other than that, there have been proved better estimates based on the properties of derived sets of the compact spaces. Firstly, in [17], Gordon proved that if K 1 , K 2 are compact spaces such that d BM (C(K 1 ), C(K 2 )) < 3, then all derivatives of K 1 and K 2 have the same cardinality. This result was generalized to the case of vector-valued functions in [5,Theorem 1.5] and [14,Theorem 1.7]. Next, it was proved in [7, Theorem 1.2] that if K is a compact space with K (n) nonempty for some n ∈ N and F is a compact space with F (2) = ∅, then d BM (C(K), C(F )) ≥ 2n − 1. Moreover, if K (n) > F (1) , then d BM (C(K), C(F )) ≥ 2n+1. In [3,Theorem 1.1] it has been showed that if Γ is an infinite discrete space, E is a Banach space not containing an isomorphic copy of c 0 and T : is an into isomorphism, then for each n ∈ N, if K (n) is nonempty, then T T −1 ≥ 2n + 1. 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.
The purpose of this paper is to show that for Banach spaces E 1 and E 2 not containing an isomorphic copy of c 0 , the condition Γ(ht(K 1 )) = Γ(ht(K 2 )) remains necessary for the spaces C(K 1 , E 1 ) and C(K 2 , E 2 ) to be isomorphic, and also, that the known distance estimates between spaces of continuous functions on compact spaces of finite height can be extended to compact spaces of arbitrary height. More precisely, we have the following result. Theorem 1.1. Let K 1 , K 2 be infinite compact spaces, E be a Banach space containing no copy of c 0 and T : C(K 1 ) → C(K 2 , E) be an isomorphic embedding. If n, k ∈ N, n > k, and α is an ordinal such that ht(K 2 ) ≤ ω α (k + 1) and ht(K 1 ) > ω α n, The following corollary is immediate. Corollary 1.2. Let K 1 , K 2 be infinite compact spaces, E 1 , E 2 be Banach spaces containing no copy of c 0 . If n, k ∈ N, n > k, and α is an ordinal such that In particular, if C(K 1 , E 1 ) is isomorphic to C(K 2 , E 2 ), then Γ(ht(K 1 )) = Γ(ht(K 2 )).

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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.
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. If, on the other hand, the heights of the compact spaces K 1 , K 2 are infinite, we do not have any information for the above question, since the lower estimate that we get from Theorem 1.1 still corresponds to some kind of a ratio of the heights instead of their difference. It seems possible that the optimal lower estimate of the distance is 3 in such a case, and that the distance starts to grow only when the difference between the heights of K 1 and K 2 is more significant.
We further note that the above assumption that c 0 does not embed in E 1 , E 2 cannot be plainly removed, which follows for example from the fact that but Γ(ht([0, ω])) = Γ(2) = ω = ω 2 = Γ(ω + 1) = Γ(ht([0, ω ω ])). Further, notice that Theorem 1.1 together with the classical isomorphic classification due to Bessaga, Pe lczyński, and Milutin yields that for compact metric spaces K 1 , K 2 and a Banach space E not containing an isomorphic copy of c 0 , C(K 1 , E) is isomorphic to C(K 2 , E) if and only if C(K 1 ) is isomorphic to C(K 2 ). This gives a strengthening of the results from [13] and [15]. It is worth to note that this is no longer true when the compact spaces K 1 , K 2 are not metrizable, see [16,Remark 1.2].
Finally, a typographical note: the symbol denotes the end of a proof, while, in nested proofs, we use for the end of the inner proof.

The Proofs
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. x ∈ L 1 such that f = 1 on an open neighbourhood of x, f = 0 on K 1 \ U , and such that T f(y) < ε for each y ∈ L 2 .
Now, we claim that one of the functions f n has the desired properties. Assuming the contrary, there exists ε > 0 such that for each n ∈ N, there exists y ∈ L 2 such that T f n (y) ≥ ε. Then, since L 2 is finite, passing to a subsequence we may fix a point y 0 ∈ L 2 satisfying that T f n (y 0 ) ≥ ε for each n ∈ N. Now, by the classical characterization of the Banach spaces containing c 0 , see [19,Theorem 6.7], to finish the proof it is enough to show that the series To show this, we consider the evaluation mapping φ : Clearly, φ(y, e * ) = e * . Now, we fix e * ∈ E * , and let T * be the adjoint of 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]).
be an isomorphic embedding and let L 2 be a subset of K 2 . Let n, k ∈ N, n > k, and let ε > 0 be given. Suppose that there exist functions g 1 , . . . , g n ∈ C(K 1 , [0, 1]) and x ∈ K 1 such that g 1 (x) = . . . = g n (x) = 1, and such that for each y ∈ L 2 , the set Proof. The function h is defined simply as h = 1 n n = 1. If y ∈ L 2 is arbitrary, then there exist n − k functions g i1 , . . . , g i n−k satisfying that for each j = 1, . . . , n − k, T g ij (y) < ε. Thus To find the function f we first consider the function g = 2 and hence there exists y ∈ K 2 such that T g(y) ≥ 2n−k T −1 . Next, there exist indices i 1 , . . . , i n−k ∈ {1, . . . n} such that for each j = 1, . . . , n − k, T g ij (y) < ε. We denote and we check that this function has the desired properties. Firstly, since and hence f = 1. On the other hand, which finishes the proof. We notice that the formula in (b) of the previous result provide the following lower bound for the norm of T : T ≥ 2n−k k (1 − ε), whenever T −1 = 1.
Next, we are going to state some elementary results on scattered derivatives. We stress that in the following lemma we consider the respective sets L as topological spaces with the topology inherited from the compact space K. Proof. In order to get the assertion (a), it is enough to prove that for each ordinal α, 2 . Let us prove it by using a transfinite induction argument. The case α = 0 is trivial. Next we consider the case α = 1. Thus, we suppose that x is an accumulation point of L 1 ∪ L 2 . We find a net (x λ ) λ∈Λ ⊆ L 1 ∪ L 2 converging to x. Passing to a subnet, we may suppose that (x λ ) λ∈Λ ∪ {x} ⊆ L 1 . Thus x ∈ L (1) 1 , which finishes the proof for α = 1. Now, we suppose that the statement holds for an ordinal α ≥ 1. Then we have Further, for a limit ordinal α we have which finishes the proof of (a).

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The following proposition, which essentially contains the proof of Theorem 1.1, is inspired by the approach of Bessaga and Pe lczyński [1] and by [20 (c) If n, k ∈ N, n > k, and α is an ordinal such that then for each ε > 0 there exist a function f ∈ C(K 1 , [0, 1]) of norm 1 such that Proof. Throughout the proof, for a function g ∈ C(K 1 ) and ε > 0 we will use { T g ≥ ε} as a shortcut for {y ∈ K 2 : T g(y) ≥ ε}. Further, for the sake of clarity we denote by A(α), B(α) and C(α) the statements that the assertions (a), (b) and (c) respectively hold for each compact set L 2 ⊆ K 2 with Γ(ht(L 2 )) ≤ ω α (or for each compact space K 2 satisfying Γ(ht(K 2 )) ≤ ω α in the case of (c)). We proceed to prove simultaneously (a), (b) and (c) by transfinite induction on α based on the following scheme.
The statement B(0) follows again from Lemma 2.1, in almost the same way. Indeed, if Γ(ht(L 2 )) ≤ ω 0 = 1, then ht(L 2 ) = 1, and hence L 2 is again finite. On the other hand, the assumption that ht(L 1 ) > ω α n for some ordinal α implies that L 1 is infinite. Hence for any k, n ∈ N and ε > 0 it is possible to find the function h satisfying the estimate from (b).
Similarly we get the assertion C(0), since, in fact, its assumptions cannot be satisfied. From the assumptions it follows as above that K 1 is infinite and K 2 is finite. Thus we see again from Lemma 2.1 that there can not be any isomorphic embedding from C(K 1 ) into C (K 2 , E). Now, assume that A(α) and B(α) hold. We want to prove B(α + 1). Thus we pick a compact set L 2 ⊆ K 2 satisfying that Γ(ht(L 2 )) = ω α+1 . This means that there exists k ∈ N such that ht(L 2 ) ≤ ω α (k + 1). Let a set L 1 ⊆ K 1 satisfies ht(L 1 ) > ω α n for some n > k, U be an open set containing L 1 , and fix ε > 0.
Next, we know that we can find an open neighbourhood V of x i contained in U such that g i = 1 on V . Since x i ∈ V ∩ L (ω α (n−i)) 1 , using Lemma 2.3(b) and (c) on the sets L (ω α (n−i−1)) 1 and V we deduce that ht(V ∩ L (ω α (n−i−1)) 1 ) > ω α . Thus, by A(α), there exist a function g i+1 and a point x i+1 ∈ V ∩ L (ω α (n−i−1)) 1 such that g i+1 = 1 on an open neighbourhood of x i+1 , g i+1 = 0 on K 1 \V and T g i+1 (y) < ε for each y ∈ N . Then g i+1 ≤ g i . Thus, to finish the proof of the claim, it is now enough to check that for each s = 1, . . . , i + 1, the set M i+1 s is empty. To this end, for each s = 0, . . . , i we have Hence, if we just shift the index s by 1, we get that for each s = 1, . . . , i + 1, But now it is simple to check that M i+1 s can be written as the union of M i s and the above set. Thus for each s = 1, . . . i + 1, the set M i+1 s is empty, which finishes the induction step and the proof of the claim. Now, we use the above claim to obtain the functions g 1 , . . . , g n . Hence, we in particular know that the set is empty. Thus, we can use Lemma 2.2(a) to obtain a linear combination h of the functions g 1 , . . . , g n which satisfies h = 1 and Moreover, since each of the functions g i is constant 1 on an open neighbourhood of the point x n ∈ L (0) 1 = L 1 , so is h. Finally, since each of the functions g i satisfies g i = 0 on K 1 \U , so does the function h, which finishes the proof of B(α + 1).
Moreover, in the case when L 1 = K 1 and L 2 = K 2 , if we assume that C(α) holds instead of B(α), an application of Lemma 2.2(b) proves C(α + 1).
Hence, by B(α + 1), there exists a function h ∈ C(K 1 , [0, 1]) of norm 1 and a point x ∈ L 1 such that h = 1 on an open neighbourhood of x, h = 0 on K 1 \ U and which proves A(α + 1). Finally, notice that the limit steps are trivial. This follows from the fact that, since ht(L 2 ) is a successor ordinal for each compact set L 2 ⊆ K 2 , if Γ(ht(L 2 )) ≤ ω α for some limit ordinal α, then there exists β < α such that Γ(ht(L 2 )) ≤ ω β . The proof is finished.
The proof of our main result now follows promptly from Proposition 2.4.
Proof of Theorem 1.1. We recall that the lower bound 3 is known, see, e.g. [20,Theorem 1.1]. The bound 2n−k k follows immediately from Proposition 2.4(c), and the "in particular" statement can be easily deduced from this estimate, or alternatively, if follows directly from Proposition 2.4(a).