A generalized Banach-Stone theorem for C-0(K, X) spaces via the modulus of convexity of X

作者：Cidral Fabian C; Cortes Vinicius M; Galego Eloi M^*

来源：Journal of Mathematical Analysis and Applications, 2017, 450(1): 12-20.

DOI：10.1016/j.jmaa.2017.01.009

• Generalization of the Banach Stone theorem James constant
• C-0 (K, X) spaces
• Moduli of convexity of a Banach space and its dua l

Suppose that K and S are locally compact Hausdorff spaces, X is a Banach space and A is a real number satisfying
2 lambda(1 - delta(x)-(2/lambda)) ) < r,
where delta(x).. denotes the modulus of convexity of the dual space X* and r is the unique real root of the polynomial p(x) = 2x(3) - x(2) -I-8x + 20. We prove that if T is an isomorphism from C-0 (K, X) onto Co (5, X) with
parallel to R parallel to parallel to T-1 parallel to <=lambda,
then K is homeomorphic to S. This improves an earlier generalization of the Banach Stone theorem via geometric properties of X* due to Jarosz, the case where K and S are compact and r is exchanged by the smaller number 4/3. In the process of the proof we establish a connection between the moduli of convexity of X and X* which has independent interest. Namely, put C = r/4. Then C is approximately 0.41883 and for all Banach space X and epsilon is an element of (0,2], we have
1 - C epsilon < delta(X) (epsilon)double right arrow 1 - 1/2 epsilon < delta(X).

• 出版日期2017-6-1