On the Extension Problems of Isometric and Nonexpansive Mappings

Yang, Xiuzhong; Zhao, Xiaopeng

doi:10.1007/978-1-4939-1106-6_24

Cited by 25 publications

(22 citation statements)

References 45 publications

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“…Tingley's problem also admits a positive solution in the case of finite dimensional polyhedral Banach spaces (see [18]). The reader is referred to the surveys [12] and [35] for additional details.…”

Section: Introductionmentioning

confidence: 99%

On the extension of isometries between the unit spheres of a C*-algebra and 𝐵(𝐻)

Fernández–Polo

Peralta

2018

Trans. Amer. Math. Soc. Ser. B

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Abstract. Given two complex Hilbert spaces H and K, let S(B(H)) and S(B(K)) denote the unit spheres of the C * -algebras B(H) and B(K) of all bounded linear operators on H and K, respectively. We prove that every surjective isometry f : S(B(K)) → S(B(H)) admits an extension to a surjective complex linear or conjugate linear isometry T : B(K) → B(H).This provides a positive answer to Tingley's problem in the setting of B(H) spaces.

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Section: Introductionmentioning

confidence: 99%

On the extension of isometries between the unit spheres of a C*-algebra and 𝐵(𝐻)

Fernández–Polo

Peralta

2018

Trans. Amer. Math. Soc. Ser. B

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“…Wang, and R. Liu, among others, have been overview in full detail in the excellent surveys published by G.G. Ding [12] and X. Yang and X. Zhao [55].…”

Section: Introductionmentioning

confidence: 99%

“…A reborn interest on the problems concerning extension of isometries between subsets of the unit spheres of two operator algebras has been materialized in a fruitful series of recent papers dealing with Tingley's problem and related questions for certain operator algebras, which have been published during the short interval determined by the last three years. The abundance of new results for operator algebras motivates and justifies the writing of this survey with the aim of completing and updating the surveys [12,55], and providing a recent state of the art of these problems. The real "avalanche" of recent achievements provides enough material to write a new and detailed survey on this topic.…”

Section: Introductionmentioning

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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Section: Introductionmentioning

confidence: 99%

“…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.…”

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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 the Extension Problems of Isometric and Nonexpansive Mappings

Cited by 25 publications

References 45 publications

On the extension of isometries between the unit spheres of a C*-algebra and 𝐵(𝐻)

On the extension of isometries between the unit spheres of a C*-algebra and 𝐵(𝐻)

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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