On 2-local diameter-preserving maps between C ( X )-spaces
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Positivity
On 2-local diameter-preserving maps between C(X)-spaces A. Jiménez-Vargas1
· Fereshteh Sady2
Received: 14 April 2020 / Accepted: 20 October 2020 © Springer Nature Switzerland AG 2020
Abstract The 2-locality problem of diameter-preserving maps between C(X )-spaces is addressed in this paper. For any compact Hausdorff space X with at least three points, we give an example of a 2-local diameter-preserving map on C(X ) which is not linear. However, we show that for first countable compact Hausdorff spaces X and Y , every 2-local diameter-preserving map from C(X ) to C(Y ) is linear and surjective up to constants in some sense. This fact yields the 2-algebraic reflexivity of isometries with respect to the diameter norms on the quotient spaces. Keywords 2-local map · Diameter-preserving map · Function space · Weighted composition operator Mathematics Subject Classification 46B04 · 47B38
1 Introduction and results Let E and F be Banach spaces and let S be a subset of L(E, F), the space of linear operators from E to F. Let us recall that a linear map T : E → F is a local S-map if for every e ∈ E, there exists a Te ∈ S, depending possibly on e, such that Te (e) = T (e). On the other hand, a map : E → F (which is not assumed to be linear) is called a 2-local S-map if for any e, u ∈ E, there exists a Te,u ∈ S, depending in general on e and u, such that Te,u (e) = (e) and Te,u (u) = (u). Most of the published works on local and 2-local S-maps concern the set S = G(E), the group of surjective linear isometries of E. In this case, the local and 2-
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A. Jiménez-Vargas [email protected] Fereshteh Sady [email protected]
1
Departamento de Matemáticas, Universidad de Almería, 04120 Almerıa, Spain
2
Department of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, 14115-134, Tehran, Iran
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A. Jiménez-Vargas, F. Sady
local G(E)-maps are known as local and 2-local isometries of E, respectively. The main question which one raises is for which Banach spaces, every local isometry is a surjective isometry or, equivalently, which Banach spaces have an algebraically reflexive isometry group. In the 2-local setting, the basic problem is to show that every 2-local isometry is a surjective linear isometry. In [21], Molnár initiated the study of 2-local isometries on operator algebras and proposed to investigate the 2-locality of isometries on function algebras. In this line, Gy˝ory [11] dealt with 2-local isometries on spaces of continuous functions. In [18], Villegas and the first author adapted the Gy˝ory’s technique to analyze the 2-local isometries on Lipschitz algebras. Hatori, Miura, Oka, and Takagi [13] considered 2-local isometries on uniform algebras including certain algebras of holomorphic functions. More recently, Hosseini [15], Hatori and Oi [14] and Li, Peralta, L. Wang and Y.-S. Wang [20] have investigated 2-local isometries of different function algebras such as uniform algebras, Lipschitz algebras, and algebras of continuously differentiable functions. Our aim in this pa
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