Tingley's problem for $p$-Schatten von Neumann classes

Fernández–Polo, Francisco J.; Jordá, Enrique; Peralta, Antonio M.

doi:10.4171/jst/313

Cited by 11 publications

(7 citation statements)

References 30 publications

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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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Section: Preliminariesmentioning

confidence: 99%

Exploring new solutions to Tingley's problem for function algebras

Cueto-Avellaneda¹,

Hirota²,

Miura³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Finally, we shall describe ∆ 0 in terms of p and Φ. To achieve this goal, we shall employ the equalities obtained in ( 14), (16), and the definition of the exponential in a Jordan algebra. According to this, given an arbitrary h ∈ M sa , we have…”

Section: Lemma 33 Applied To ∆ and The Family {Umentioning

confidence: 99%

“…To the best of our knowledge, Tingley's problem remains open even for two dimensional spaces (see [5] where it is solved for non-strictly convex two dimensional spaces). A full machinery has been developed in the different partial positive solutions to Tingley's problem in the case of classical Banach spaces, C * -and operator algebras and JB * -triples (see, for example the references [2,5,8,10,11,15,16,18,19,20,30,32,35,36,38,39,43] and the surveys [47,37]).…”

Section: Introductionmentioning

confidence: 99%

Can one identify two unital JB*-algebras by the metric spaces determined by their sets of unitaries?

Cueto-Avellaneda

Peralta

2021

Linear and Multilinear Algebra

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Let M and N be two unital JB * -algebras and let U (M ) and U (N ) denote the sets of all unitaries in M and N , respectively. We prove that the following statements are equivalent:(a) M and N are isometrically isomorphic as (complex) Banach spaces;(b) M and N are isometrically isomorphic as real Banach spaces; (c) There exists a surjective isometry ∆ : U (M ) → U (N ). We actually establish a more general statement asserting that, under some mild extra conditions, for each surjective isometry ∆ : U (M ) → U (N ) we can find a surjective real linear isometry Ψ : M → N which coincides with ∆ on the subset e iMsa . If we assume that M and N are JBW * -algebras, then every surjective isometry ∆ : U (M ) → U (N ) admits a (unique) extension to a surjective real linear isometry from M onto N . This is an extension of the Hatori-Molnár theorem to the setting of JB * -algebras.

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“…Please see [20] and the survey [5] for classical Banach spaces, and see the survey [16] which contains a good description of non-commutative operator algebras. For some of the most recent papers, one can see [1,3,4,7,12,15,17].…”

Section: Introductionmentioning

confidence: 99%

Phase-isometries on the unit sphere of C(K)

Tan

Gao

2020

Ann. Funct. Anal.

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We say that a map T ∶ S X → S Y between the unit spheres of two real normed-spaces X and Y is a phase-isometry if it satisfies for all x, y ∈ S X . In the present paper, we show that there is a phase function ∶ S X → {−1, 1} such that ⋅ T is an isometry which can be extended a real linear isometry on X whenever T is surjective, X = C(K) and Y is an arbitrary Banach space. Additionally, if T is a surjective phase-isometry between the unit spheres of C(K) and C( ) , where K and are compact Hausdorff spaces, we prove that there are a homeomorphism ∶ → K and a continuous unimodular function h on such that for all f ∈ S C(K) . This also can be seen as a Banach-Stone type representation for phase-isometries in C(K) spaces.

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Tingley's problem for $p$-Schatten von Neumann classes

Cited by 11 publications

References 30 publications

Exploring new solutions to Tingley's problem for function algebras

Exploring new solutions to Tingley's problem for function algebras

Can one identify two unital JB*-algebras by the metric spaces determined by their sets of unitaries?

Phase-isometries on the unit sphere of C(K)

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