×

Nonexpansive bijections between unit balls of Banach spaces. (English) Zbl 1400.46013

Summary: It is known that if \(M\) is a finite-dimensional Banach space, or a strictly convex space, or the space \(\ell_{1}\), then every nonexpansive bijection \(F: B_{M}\to B_{M}\) of its unit ball \(B_{M}\) is an isometry. We extend these results to nonexpansive bijections \(F: B_{E}\to B_{M}\) between unit balls of two different Banach spaces. Namely, if \(E\) is an arbitrary Banach space and \(M\) is finite-dimensional or strictly convex, or the space \(\ell_{1}\), then every nonexpansive bijection \(F: B_{E}\to B_{M}\) is an isometry.

MSC:

46B20 Geometry and structure of normed linear spaces
46B04 Isometric theory of Banach spaces
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
PDF BibTeX XML Cite
Full Text: DOI arXiv Euclid

References:

[1] L. E. J. Brouwer, Beweis der Invarianz des n-dimensionalen Gebiets, Math. Ann. 71 (1912) 305-315. · JFM 42.0418.01
[2] B. Cascales, V. Kadets, J. Orihuela, E. J. Wingler, Plasticity of the unit ball of a strictly convex Banach space, Rev. R. Acad. Cienc. Exactas Fiís. Nat. Ser. A Math. RACSAM 110 (2016), no. 2, 723-727. · Zbl 1362.46011
[3] G. Ding, On isometric extension problem between two unit spheres, Sci. China Ser. A 52 (2009), 2069-2083. · Zbl 1190.46013
[4] V. M. Kadets, A course in Functional Analysis, Khar’kovskii Natsional’ny Universitet Imeni V. N. Karazina, Kharkiv, 2006.
[5] V. Kadets and M. Martín, Extension of isometries between unit spheres of finite-dimensional polyhedral Banach spaces, J. Math. Anal. Appl. 386 (2012), no. 2, 441-447. · Zbl 1258.46004
[6] V. Kadets and O. Zavarzina, Plasticity of the unit ball of \(ℓ_{1}\), Visn. Hark. Nac. Univ. Im. V. N. Karazina, Ser.: Mat. Prikl. Mat. Mech. 83 (2017) 4-9. · Zbl 1374.46005
[7] P. Mankiewicz, On extension of isometries in normed linear spaces, Bull. Acad. Polon. Sci., Sér. Sci. Math. Astronom. Phys. 20 (1972), 367-371. · Zbl 0234.46019
[8] S. A. Naimpally, Z. Piotrowski, and E. J. Wingler, Plasticity in metric spaces, J. Math. Anal. Appl. 313 (2006), no. 1, 38-48. · Zbl 1083.54016
[9] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1997. · Zbl 0932.90001
[10] D. Tan, X. Huang, R. Liu, Generalized-lush spaces and the Mazur-Ulam property, Stud. Math. 219 (2013), no. 2, 139-153. · Zbl 1296.46009
[11] D. Tingley, Isometries of the unit sphere, Geom. Dedicata 22 (1987), no. 3, 371-378. · Zbl 0615.51005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.