Vol. 297, No. 1, 2018

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ISSN: 0030-8730
An Amir–Cambern theorem for quasi-isometries of $C_{0}(K, X)$ spaces

Elói Medina Galego and André Luis Porto da Silva

Vol. 297 (2018), No. 1, 87–100
Abstract

Let X be a finite-dimensional Banach space. We prove that if K and S are locally compact Hausdorff spaces and there exists a bijective map T : C0(K,X) C0(S,X) such that

1 Mf g L T(f) T(g) Mf g + L,

for every f,g C0(K,X) then K and S are homeomorphic, whenever L 0 and 1 M2 < S(X), where S(X) denotes the Schäffer constant of X.

This nonlinear vector-valued extension of the Amir–Cambern theorem via quasi-isometries T with large M was previously unknown even for the classical pn spaces, 1 < p < , p2 and n 2.

Keywords
vector-valued Amir–Cambern theorem, $C_{0}(K, X)$ spaces, finite-dimensional uniformly non-square spaces, quasi-isometry, Schäffer constant
Mathematical Subject Classification 2010
Primary: 46B03, 46E15
Secondary: 46B25, 46E40
Milestones
Received: 6 January 2018
Revised: 12 April 2018
Accepted: 20 April 2018
Published: 7 October 2018
Authors
Elói Medina Galego
Department of Mathematics, IME
University of São Paulo
S ao Paulo
Brazil
Department of Mathematics, IME
University of São Paulo
Rua do Matão 1010
05508-090 São Paulo-
Brazil
André Luis Porto da Silva
Department of Mathematics, IME
University of São Paulo
São Paulo
Brazil