Weak forms of Banach-Stone theorem for C-0(K, X) spaces via the alpha th derivatives of K

作者：Galego Eloi Medina^*; Rincon Villamizar Michael A

来源：Bulletin des Sciences Mathematiques, 2015, 139(8): 880-891.

DOI：10.1016/j.bulsci.2015.04.002

• C-0(K) spaces
• C-0 (K, X) spaces
• Banach-Stone theorem
• Banach spaces without copy of c(0)

Let X be a Banach space and S be a locally compact Hausdorff space. By C-0 (S, X) we will stand the Banach space of all continuous X-valued functions on S endowed with the supremum norm.
Suppose that C-0(S, X) contains a copy of some C-0(K) space with K infinite. Does it follow that the cardinality of the alpha th derivative of K is less than or equal to the ath derivative of S, for every ordinal number alpha? In general the answer is no, even when alpha = 0.
In the present paper we prove that the answer is yes whenever X contains no copy of c(0) and alpha = 0. Moreover, in the case where alpha > 0 and the ath derivative of S is infinite, we show that the existence an isomorphism from C-0 (K) into C-0 (S, X) with distortion parallel to T parallel to parallel to T-1 parallel to strictly less than 3 provides also a positive answer to this question.
As a consequence, we improve a classical Cengiz theorem and a recent result on isomorphisms between spaces of vector-valued continuous functions by obtaining two weak forms of Banach-Stone theorem for C-0(S, X) spaces via the ath derivatives of S.

• 出版日期2015-12