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NONEXPANSIVE MAPPINGS ON THE UNIT SPHERES OF SOME BANACH SPACES

Part of: Topological linear spaces and related structures Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  19 June 2009

DONG-NI TAN
DONG-NI TAN*
Affiliation:
School of Mathematical Sciences, Nankai University, Tianjin 300071, PR China (email: [email protected])
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Abstract

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We characterize surjective nonexpansive mappings between unit spheres of ℒ∞(Γ)-type spaces. We show that such mappings turn out to be isometries and can be extended to linear isometries on the whole space ℒ∞(Γ).

Keywords

nonexpansive mappingisometric extensionℒ∞(Γ)-type spaces

MSC classification

Secondary: 46A22: Theorems of Hahn-Banach type; extension and lifting of functionals and operators 46B20: Geometry and structure of normed linear spaces
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 80 , Issue 1 , August 2009 , pp. 139 - 146
Copyright
Copyright © Australian Mathematical Society 2009

Footnotes

This work was partially supported by Research Foundation for Doctor Programme (20070055010) and National Natural Science Foundation of China (10571090).

References

[1] An, G. M., ‘Isometries on unit sphere of (ℓ β n)’, J. Math. Anal. Appl. 301 (2005), 249–254.Google Scholar
[2] Benyamini, Y. and Lindenstrauss, J., Geometric Nonlinear Functional Analysis, Vol. I, Colloquium Publications, Vol. 48 (American Mathematical Society, Providence, RI, 2000).Google Scholar
[3] Brown, T. A. and Comfort, W. W., ‘New method for expansion and contraction maps in uniform spaces’, Proc. Amer. Math. Soc. 11 (1960), 483–486.Google Scholar
[4] Ding, G. G., ‘The 1-Lipschitz mapping between the unit spheres of two Hilbert spaces can be extended to a real linear linear isometry of the whole space’, Sci. China Ser. A 45(4) (2002), 479–483.CrossRefGoogle Scholar
[5] Ding, G. G., ‘The isometric extension problem in the unit spheres of ℓ p(Γ)(p>1) type spaces’, Sci. China Ser. A 46(3) (2003), 333–338.CrossRefGoogle Scholar
[6] Ding, G. G., ‘The representation theorem of onto isometric mapping between two unit spheres of ℓ 1(Γ) type spaces and the application on isometric extension problem’, Acta Math. Sin. (Engl. Ser.) 20(6) (2004), 1089–1094.Google Scholar
[7] Ding, G. G., ‘The representation of onto isometric mappings between two spheres of ℓ ∞-type spaces and the application on isometric extension problem’, Sci. China Ser. A 34(2) (2004), 157–164 (in Chinese); 47 (5) (2004), 722–729 (in English).Google Scholar
[8] Ding, G. G., ‘On the extension of isometries between unit spheres of E and C(Ω)’, Acta Math. Sin. (Engl. Ser.) 19(4) (2003), 793–800.Google Scholar
[9] Ding, G. G., ‘The isometric extension of the into mapping from the ℒ∞(Γ)-type space to some normed space E’, Illinois J. Math. 52(2) (2007), 445–453.Google Scholar
[10] Freudenthal, H. and Hurewicz, W., ‘Dehnungen, Verkurzungen, Isometrien’, Fund. Math. 26 (1936), 120–122.Google Scholar
[11] Liu, R., ‘On extension of isometries between the unit spheres of ℒ∞-type space and some Banach space E’, J. Math. Anal. Appl. 333 (2007), 959–970.Google Scholar
[12] Mankiewicz, P., ‘On extension of isometries in normed linear spaces’, Bull. Acad. Polon. Sci. 20 (1972), 367–371.Google Scholar
[13] Rhodes, F., ‘A generalization of isometries to uniform spaces’, Proc. Cambridge Philos. Soc. 52 (1956), 399–405.Google Scholar
[14] Tingley, D., ‘Isometries of the unit sphere’, Geom. Dedicata 22 (1987), 371–378.Google Scholar
[15] Wang, J., ‘On extension of isometries between unit spheres of AL p-spaces (1<p<∞)’, Proc. Amer. Math. Soc. 132(10) (2004), 2899–2909.Google Scholar
[16] Yang, X. Z., ‘On extension of isometries between unit spheres of L p(μ) and L p(ν,H) ( 1<p≠2,H is a Hilbert space)’, J. Math. Anal. Appl. 323 (2006), 985–992.Google Scholar