Premium
Isomorphisms of C 0 ( K , X ) spaces with large distortion
Author(s) -
Medina Galego Elói,
Porto da Silva André Luis
Publication year - 2019
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201800038
Subject(s) - mathematics , hausdorff space , banach space , isomorphism (crystallography) , regular polygon , distortion (music) , pure mathematics , constant (computer programming) , compact space , hausdorff distance , combinatorics , discrete mathematics , mathematical analysis , geometry , amplifier , programming language , chemistry , cmos , electronic engineering , computer science , crystal structure , engineering , crystallography
Let K and S be locally compact Hausdorff spaces and let X be a strictly convex Banach space of finite dimension at least 2. In this paper, we prove that if there exists an isomorphism T fromC 0 ( K , X )ontoC 0 ( S , X )satisfying∥ T ∥T − 1= λ ( X ) , then K and S are homeomorphic. Here λ ( X ) denotes the Schäffer constant of X . Even for the classical cases X = ℓ p n , 1 < p < ∞ and n ≥ 2 , this result is the X ‐valued Banach–Stone theorem via isomorphism with the largest distortion that is known so far, namely λ ( X ) = min { 2 1 / p , 2 1 − 1 / p } . On the other hand, it is well known that this result is not true for X = R , even though K and S are compact Hausdorff spaces.