On the Mazur–Ulam property for the space of Hilbert-space-valued continuous functions

Cueto-Avellaneda, María; Peralta, Antonio M.

doi:10.1016/j.jmaa.2019.06.056

Cited by 28 publications

(18 citation statements)

References 53 publications

(159 reference statements)

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“…Some concrete examples of Banach spaces satisfying the MUp are polyhedral spaces, c 0 (Γ), p (Γ)(1 ≤ p < ∞), ∞ (Γ), C(Ω), (see [5,8,[16][17][18]). This issue is currently a hot topic at the same level as Tingley's Problem (recent articles about the MUp are [19][20][21][22]). Finally, in [21,23] the authors prove that non-strictly convex 2-dimensional Banach spaces and non-smooth 2-dimensional Banach spaces satisfy MUp.…”

Section: Introductionmentioning

confidence: 99%

Geometric Invariants of Surjective Isometries between Unit Spheres

Campos-Jiménez

García-Pacheco

2021

Mathematics

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In this paper we provide new geometric invariants of surjective isometries between unit spheres of Banach spaces. Let X,Y be Banach spaces and let T:SX→SY be a surjective isometry. The most relevant geometric invariants under surjective isometries such as T are known to be the starlike sets, the maximal faces of the unit ball, and the antipodal points (in the finite-dimensional case). Here, new geometric invariants are found, such as almost flat sets, flat sets, starlike compatible sets, and starlike generated sets. Also, in this work, it is proved that if F is a maximal face of the unit ball containing inner points, then T(−F)=−T(F). We also show that if [x,y] is a non-trivial segment contained in the unit sphere such that T([x,y]) is convex, then T is affine on [x,y]. As a consequence, T is affine on every segment that is a maximal face. On the other hand, we introduce a new geometric property called property P, which states that every face of the unit ball is the intersection of all maximal faces containing it. This property has turned out to be, in a implicit way, a very useful tool to show that many Banach spaces enjoy the Mazur-Ulam property. Following this line, in this manuscript it is proved that every reflexive or separable Banach space with dimension greater than or equal to 2 can be equivalently renormed to fail property P.

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Section: Introductionmentioning

confidence: 99%

Geometric Invariants of Surjective Isometries between Unit Spheres

Campos-Jiménez

García-Pacheco

2021

Mathematics

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“…The problem of extending a surjective isometry between the unit spheres of two Banach spaces -named Tingley's problem after the contribution of D. Tingley in [45]is nowadays a trending topic in functional analysis (see a representative sample in the references [6,14,15,19,20,21,22,23,34,35,38,9] and the surveys [47,37]). This isometric extension problem remains open for Banach spaces of dimension bigger than or equal to 3 though.…”

Section: Preliminariesmentioning

confidence: 99%

Exploring new solutions to Tingley's problem for function algebras

Cueto-Avellaneda¹,

Hirota²,

Miura³

et al. 2021

Preprint

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In this note we present two new positive answers to Tingley's problem in certain subspaces of function algebras. In the first result we prove that every surjective isometry between the unit spheres, S(A) and S(B), of two uniformly closed function algebras A and B on locally compact Hausdorff spaces can be extended to a surjective real linear isometry from A onto B. In a second goal we study surjective isometries between the unit spheres of two abelian JB * -triples represented as spaces of continuous functions of the formfor every (λ, t) ∈ T × X}, where X is a (locally compact Hausdorff) principal T-bundle. We establish that every surjective isometry ∆ : S(C T 0 (X)) → S(C T 0 (Y )) admits an extension to a surjective real linear isometry between these two abelian JB * -triples.

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“…Tingley's Problem 1.1 can be equivalently reformulated in terms of the Mazur-Ulam property, introduced by Cheng and Dong [6] and widely used in the the literature, see e.g. [3], [7], [8], [11], [14], [17], [20]. Definition 1.2.…”

Section: Introductionmentioning

confidence: 99%

Every 2-dimensional Banach space has the Mazur-Ulam property

Banakh

2021

Preprint

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On the Mazur–Ulam property for the space of Hilbert-space-valued continuous functions

Abstract: Let K be a compact Hausdorff space and let H be a real or complex Hilbert space with dim(

Cited by 28 publications

References 53 publications

Geometric Invariants of Surjective Isometries between Unit Spheres

Geometric Invariants of Surjective Isometries between Unit Spheres

Exploring new solutions to Tingley's problem for function algebras

Every 2-dimensional Banach space has the Mazur-Ulam property

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