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MAZUR–ULAM PROPERTY OF THE SUM OF TWO STRICTLY CONVEX BANACH SPACES

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  11 November 2015

JIAN-ZE LI
JIAN-ZE LI*
Affiliation:
Department of Mathematics, Tianjin University, Tianjin 300072, PR China email [email protected]
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Abstract

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In this article, we study the Mazur–Ulam property of the sum of two strictly convex Banach spaces. We give an equivalent form of the isometric extension problem and two equivalent conditions to decide whether all strictly convex Banach spaces admit the Mazur–Ulam property. We also find necessary and sufficient conditions under which the $\ell ^{1}$-sum and the $\ell ^{\infty }$-sum of two strictly convex Banach spaces admit the Mazur–Ulam property.

Keywords

Mazur–Ulam propertyisometric extension problemstrictly convex

MSC classification

Primary: 46B04: Isometric theory of Banach spaces
Secondary: 46B20: Geometry and structure of normed linear spaces
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 93 , Issue 3 , June 2016 , pp. 473 - 485
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

Cheng, L. X. and Dong, Y. B., ‘On a generalized Mazur–Ulam question: extension of isometries between unit spheres of Banach spaces’, J. Math. Anal. Appl. 377 (2011), 464–470.CrossRefGoogle Scholar
Ding, G. G., ‘The 1-Lipschitz mapping between the unit spheres of two Hilbert spaces can be extended to a real linear isometry of the whole space’, Sci. China Ser. A 45(4) (2002), 479–483.CrossRefGoogle Scholar
Ding, G. G., ‘The isometric extension of the into mapping from the L∞(Γ)-type space to some normed space E’, Illinois J. Math. 51(2) (2007), 445–453.Google Scholar
Ding, G. G., ‘On isometric extension problem between two unit spheres’, Sci. China Ser. A 52(10) (2009), 2069–2083.Google Scholar
Ding, G. G., ‘The isometric extension of an into mapping from the unit sphere S[ℓ (Γ)] to the unit sphere S (E)’, Acta Math. Sci. 29B(3) (2009), 469–479.Google Scholar
Ding, G. G. and Li, J. Z., ‘Sharp corner points and isometric extension problem in Banach spaces’, J. Math. Anal. Appl. 405(1) (2013), 297–309.Google Scholar
Ding, G. G. and Li, J. Z., ‘Isometries between unit spheres of ℓ ∞-sum of strictly convex normed spaces’, Bull. Aust. Math. Soc. 88(3) (2013), 369–375.Google Scholar
Fang, X. N. and Wang, J. H., ‘On linear extension of isometries between the unit spheres’, Acta Math. Sinica (Chin. Ser.) 48 (2005), 1109–1112.Google Scholar
Kadets, V. and Martin, M., ‘Extension of isometries between unit spheres of finite-dimensional polyhedral Banach spaces’, J. Math. Anal. Appl. 396 (2012), 441–447.Google Scholar
Tanaka, R., ‘A further property of spherical isometries’, Bull. Aust. Math. Soc. 90(2) (2014), 304–310.Google Scholar
Tanaka, R., ‘Tingley’s problem on symmetric absolute normalized norms on ℝ2’, Acta Math. Sin. (Engl. Ser.) 30(8) (2014), 1324–1340.Google Scholar
Tingley, D., ‘Isometries of the unit sphere’, Geom. Dedicata 22 (1987), 371–378.CrossRefGoogle Scholar
Wang, R. S. and Orihara, A., ‘Isometries on the ℓ 1 -sum of C 0(Ω, E) type spaces’, J. Math. Sci. Univ. Tokyo 2 (1995), 131–154.Google Scholar
Wang, R. S. and Orihara, A., ‘Isometries between the unit spheres of ℓ 1 -sum of strictly convex normed spaces’, Acta Sci. Natur. Univ. Nankai 35(1) (2002), 38–42.Google Scholar