On a generalized Mazur–Ulam question: Extension of isometries between unit spheres of Banach spaces

Cheng, Lixin; Dong, Yunbai

doi:10.1016/j.jmaa.2010.11.025

Cited by 68 publications

(58 citation statements)

References 10 publications

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“…Following [1], we shall say that a Banach space Z satisfies the Mazur-Ulam property if every surjective isometry from the unit sphere of Z to the unit sphere of any Banach space Y admits a (unique) extension to a surjective real linear isometry from Z onto Y . A pioneering contribution due to G.G.…”

Section: Introductionmentioning

confidence: 99%

The Mazur–Ulam property for the space of complex null sequences

Jiménez-Vargas

Campoy

Peralta

et al. 2018

Linear and Multilinear Algebra

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Given an infinite set Γ, we prove that the space of complex null sequences c 0 (Γ) satisfies the Mazur-Ulam property, that is, for each Banach space X, every surjective isometry from the unit sphere of c 0 (Γ) onto the unit sphere of X admits a (unique) extension to a surjective real linear isometry from c 0 (Γ) to X. We also prove that the same conclusion holds for the finite dimensional space ℓ m ∞ .2010 Mathematics Subject Classification. Primary 46B20, 46A22, 46B04, 46B25.

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Section: Introductionmentioning

confidence: 99%

The Mazur–Ulam property for the space of complex null sequences

Jiménez-Vargas

Campoy

Peralta

et al. 2018

Linear and Multilinear Algebra

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“…Up to now, most of the studies on Tingley's problem are based on a good and appropriate knowledge of the geometric properties of the involved spaces. This is because the most general geometric conclusion which can be derived from the existence of a surjective isometry between the unit spheres of two Banach spaces is the following result, which was originally established by L. Cheng and Y. Dong [5], and later rediscovered by R. Tanaka [50,49]. From now on, given a normed space X, the symbol B X will stand for the closed unit ball of X.…”

Section: Geometric Backgroundmentioning

confidence: 99%

A survey on Tingley’s problem for operator algebras

Peralta¹

2018

Acta Sci. Math.

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We survey the most recent results on extension of isometries between special subsets of the unit spheres of C * -algebras, von Neumann algebras, trace class operators, preduals of von Neumann algebras, and p-Schattenvon Neumann spaces, with special interest on Tingley's problem.Problem 1.1. Let X and Y be two Banach spaces whose unit spheres are denoted by S(X) and S(Y ), respectively. Let S 1 and S 2 be two subsets of S(X) and S(Y ), respectively. Suppose ∆ : S 1 → S 2 is a surjective isometry. Does ∆ extend to a real linear isometry from X onto Y ?Henceforth, we shall write T for the unit sphere of C. The complex conjugation on T cannot be extended to a complex linear isometry on C. So, in the case of complex Banach spaces, a complex linear extension is simply hopeless for all cases. Similar constrains will appear in subsequent results.These problems, whose origins are in geometry, are nowadays a central topic for those researchers working on preservers. If in Problem 1.1 we consider S 1 = S(X) and S 2

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“…As in recent contributions studying Tingley's problem on C * -algebras and von Neumann algebras, our arguments are based on an useful result due to L. Cheng, Y. Dong and R. Tanaka, which asserts that every surjective isometry between the unit spheres of two Banach spaces X and Y maps maximal proper (norm closed) face of the unit ball of X to maximal proper (norm closed) face of the unit ball of Y (see [6,Lemma 5.1], [27,Lemma 3.5] and [28,Lemma 3.3]). …”

Section: A Jbwmentioning

confidence: 99%

Tingley's problem through the facial structure of an atomic JBW⁎-triple

Fernández–Polo

Peralta

2017

Journal of Mathematical Analysis and Applications

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Abstract. We prove that every surjective isometry between the unit spheres of two atomic JBW * -triples E and B admits a unit extension to a surjective real linear isometry from E into B. This result constitutes a new positive answer to Tignley's problem in the Jordan setting. IntroductionIn a recent contribution we establish that every surjective isometry between the unit spheres of two B(H)-spaces extends uniquely to a surjective complex linear or conjugate linear surjective isometry between the corresponding spaces (see [15]). This result constitutes a positive answer to Tingley's isometric extension problem [31] in the setting of B(H)-spaces and atomic von Neumann algebras. Solutions to Tingley's problem for compact operators, compact C * -algebras and weakly compact JB * -triples have been previously obtained in [26,14]. For additional information on the historic background and the state of the art of Tingley's problem the reader is referred to the introduction of [15] and to the monograph [32].Problems in C * -algebras, von Neumann algebras and operator algebras are often considered in the context of Banach Jordan algebras and Jordan triple systems. Such studies widen the scope and often introduce new ideas and techniques not present in the associative case. The class of JB * -triples have a rich interaction with Banach Space Theory. The spaces in this class enjoy a unique geometry which makes more interesting the study of certain geometric problems in a wider setting. This paper is devoted to extend the recent results in [15] to the context of atomic JBW * -triples (i.e. ℓ ∞ -sums of Cartan factors).We recall that a JB * -triple is a complex Banach space E which can be equipped with a continuous triple product {., ., .} : E × E × E → E, which is symmetric and linear in the first and third variables, conjugate linear in the second variable and satisfies the following axioms , a) is an hermitian operator with non-negative spectrum; (c) L(a, a) = a 2 .2010 Mathematics Subject Classification. Primary 47B49, Secondary 46A22, 46B20, 46B04, 46A16, 46E40.

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On a generalized Mazur–Ulam question: Extension of isometries between unit spheres of Banach spaces

Cited by 68 publications

References 10 publications

The Mazur–Ulam property for the space of complex null sequences

The Mazur–Ulam property for the space of complex null sequences

A survey on Tingley’s problem for operator algebras

Tingley's problem through the facial structure of an atomic JBW⁎-triple

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