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Two new examples of Banach spaces with a plastic unit ball
Abstract
We prove that Banach spaces ℓ1 ⊕2 R and X ⊕∞ Y , with strictly convex X and Y , have plastic unit balls (we call a metric space plastic if every non-expansive bijection from this space onto itself is an isometry).
... -The space c of all convergent sequences of real numbers ( [7]); and the space c 0 if one weakens the definition of plasticity by considering only non-expansive bijections with a continuous inverse ( [7]). -Any ℓ ∞ -direct sum of two strictly convex spaces ( [4]). ...
... -The space ℓ 1 ⊕ 2 R ( [4]). ...
We show that if K is a compact metrizable space with finitely many accumulation points, then the closed unit ball of C(K) is a plastic metric space, which means that any non-expansive bijection from onto itself is in fact an isometry. We also show that if K is a zero-dimensional compact Hausdorff space with a dense set of isolated points, then any non-expansive homeomorphism of is an isometry.
... There are a number of relatively recent particular results, devoted to these problems, see Angosto et al. [7], Haller et al. [8], Kadets andd Zavarzina [9], Leo [10], and Zavarzina [11]. There exists also a circle of problems connected with plasticity property of the unit balls. ...
The study deals with plastic and non-plastic sub-spaces A of the real-line ℝ with the usual Euclidean metric d . It investigates non-expansive bijections, proves properties of such maps, and demonstrates their relevance by hands of examples. Finally, it is shown that the plasticity property of a sub-space A contains at least two complementary questions, a purely geometric and a topological one. Both contribute essential aspects to the plasticity property and get more critical in higher dimensions and more abstract metric spaces.
It is known that if any function acting from precompact metric space to itself increases the distance between some pair of points then it must decrease distance between some other pair of points. We show that this is not the case for quasi-metric spaces. After that, we present some sufficient conditions under which the previous property holds true for hereditarily precompact quasi-metric spaces.
Extending recent results by Cascales, Kadets, Orihuela and Wingler (2016), Kadets and Zavarzina (2017), and Zavarzina (2017) we demonstrate that for every Banach space X and every collection of strictly convex Banach spaces every non-expansive bijection from the unit ball of X to the unit ball of sum of by is an isometry.
In the recent paper by Cascales, Kadets, Orihuela and Wingler it is shown that for every strictly convex Banach space X every non-expansive bijection is an isometry. We extend this result to the space , which is not strictly convex.
http://vestnik-math.univer.kharkov.ua/Vestnik-KhNU-83-2016-kadets.pdf
https://rdcu.be/6Cru
Let X be a strictly convex Banach space, and let be its unit ball. Then every non-expansive bijection is an isometry.
We prove the plasticity of the unit ball of c. That is, we show that every non-expansive bijection from the unit ball of c onto itself is an isometry. We also demonstrate a slightly weaker property for the unit ball of c0 – we prove that a non-expansive bijection is an isometry, provided that it has a continuous inverse.
We extend the result of B. Cascales et al. about expand-contract plasticity of the unit ball of strictly convex Banach space to those spaces whose unit sphere is the union of all its finite-dimensional polyhedral extreme subsets. We also extend the definition of expand-contract plasticity to uniform spaces and generalize the theorem on expand-contract plasticity of totally bounded metric spaces to this new setting.
In this paper we examine the properties of EC-plastic metric spaces, spaces which have the property that any noncontractive bijection from the space onto itself must be an isometry.
- W Hurewicz
- Dehnungen
- Verkürzungen
- Fund Isometrien
H. Freudenthal and W. Hurewicz, Dehnungen, Verkürzungen, Isometrien, Fund. Math.
26 (1936), 120-122.