Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-05T04:25:19.724Z Has data issue: false hasContentIssue false
  • English
  • Français

LINEAR SURJECTIVE ISOMETRIES BETWEEN VECTOR-VALUED FUNCTION SPACES

Part of: Linear function spaces and their duals Special properties

Published online by Cambridge University Press:  27 January 2016

KAZUHIRO KAWAMURA
KAZUHIRO KAWAMURA*
Affiliation:
Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove some Banach–Stone type theorems for linear isometries of vector-valued continuous function spaces, by making use of the extreme point method.

Keywords

isometryreal-linearityweighted composition operatordimension theory

MSC classification

Primary: 46E15: Banach spaces of continuous, differentiable or analytic functions
Secondary: 54F45: Dimension theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 100 , Issue 3 , June 2016 , pp. 349 - 373
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

Al-Halees, H. and Fleming, R. J., ‘Extreme point methods and Banach–Stone theorem’, J. Aust. Math. Soc. 75 (2003), 125–143.CrossRefGoogle Scholar
Araujo, J. and Font, J. J., ‘Linear isometries between subspaces of continuous functions’, Trans. Amer. Math. Soc. 349 (1997), 413–428.Google Scholar
Behrends, E., M-Structures and the Banach–Stone Theorem, Lecture Notes in Mathematics, 736 (Springer, Berlin, 1979).Google Scholar
Bing, R. H., ‘A homogeneous indecomposable plane continuum’, Duke Math. J. 15 (1948), 729–742.Google Scholar
Botelho, F., Fleming, R. J. and Jamison, J. E., ‘Extreme points and isometries on vector-valued Lipschitz spaces’, J. Math. Anal. Appl. 381 (2011), 821–832.Google Scholar
Botelho, F. and Jamison, J. E., ‘Surjective isometries on spaces of vector valued continuous and Lipschitz functions’, Positivity 17 (2013), 395–497.Google Scholar
Botelho, F., Jamison, J. E. and Zheng, V., ‘Isometries of spaces of vector valued Lipschitz functions’, Positivity 17 (2013), 47–65.Google Scholar
Brosowski, B. and Deutsch, F., ‘On some geometric properties of suns’, J. Approx. Theory 10 (1974), 245–267.Google Scholar
Cambern, M., ‘Isometries of certain Banach algebras’, Studia Math. 25 (1964–1965), 217–225.CrossRefGoogle Scholar
Cambern, M., ‘A Holsztynski theorem for spaces of continuous vector-valued functions’, Studia Math. 63 (1978), 213–217.Google Scholar
Caralambous, M. G. and Krzempek, J., ‘Rigid continua and transfinite inductive dimension’, Topol. Appl. 157 (2010), 1690–1702.Google Scholar
Conway, J. B., A Course in Functional Analysis, 2nd edn, Graduate Texts in Mathematics, 96 (Springer, New York, 1990).Google Scholar
Cook, H., ‘Continua which admit only the identity mapping onto non-degenerate subcontinua’, Fund. Math. 60 (1967), 241–249.Google Scholar
Dunford, N. and Schwarz, J., Linear Operators: General Theory, Vol. 1 (Interscience, New York, 1958).Google Scholar
Engelking, R., Theory of Dimensions, Finite and Infinite, Sigma Series in Pure Mathematics, 10 (Helderman Verlag, Lemgo, 1995).Google Scholar
Fleming, R. J. and Jamison, J. E., Isometries on Banach Spaces: Function Spaces, Monographs and Surveys in Pure and Applied Mathematics, 129, Vol. 1 (Chapman & Hall/CRC, Boca Raton, 2003).Google Scholar
Fleming, R. J. and Jamison, J. E., Isometries on Banach Spaces: Vector-Valued Function Spaces, Monographs and Surveys in Pure and Applied Mathematics 138, Vol. 2 (Chapman & Hall/CRC, Boca Raton, 2008).Google Scholar
Font, J. J., ‘Linear isometries between certain subspaces of continuous vector-valued functions’, Illinois J. Math. 42 (1998), 389–397.Google Scholar
Hatori, O. and Miura, T., ‘Real linear isometries between function algebras. II’, Cent. Eur. J. Math. 11 (2013), 1838–1842.Google Scholar
Holsztynski, W., ‘Continuous mappings induced by isometries of spaces of continuous functions’, Studia Math. 26 (1966), 133–136.Google Scholar
Jamshidi, A. and Sady, F., ‘Real-linear isometries between certain subspaces of continuous functions’, Cent. Eur. J. Math. 11 (2013), 2034–2043.Google Scholar
Jarosz, K. and Pathak, V. D., ‘Isometries between function spaces’, Trans. Amer. Math. Soc. 305 (1988), 193–206.Google Scholar
Jarosz, K. and Pathak, V. D., ‘Isometries and small bound isomorphisms of function spaces’, in: Function Spaces, Edwardsville Il 1990, Lecture Notes in Pure and Applied Mathematics, 136 (Marcel Dekker, New York, 1992), 241–271.Google Scholar
Jeang, J.-S. and Wong, N.-C., ‘On the Banach–Stone problem’, Studia Math. 155 (2003), 95–105.CrossRefGoogle Scholar
Jerison, M., ‘The space of bounded maps into Banach space’, Ann. of Math. (2) 52 (1950), 309–327.CrossRefGoogle Scholar
Jiménez-Vargas, A. and Villegas-Vallecillos, M., ‘Linear isometries between spaces of vector-valued Lipschitz functions,’, Proc. Amer. Math. Soc. 137 (2009), 1381–1388.Google Scholar
Koshimizu, H., Miura, T., Takagi, H. and Takahasi, S.-E., ‘Real-linear isometries between subspaces of continuous functions’, J. Math. Anal. Appl. 413 (2014), 229–241.CrossRefGoogle Scholar
Lau, K.-S., ‘A representation theorem for isometries of C (X, E)’, Pacific J. Math. 60 (1975), 229–233.Google Scholar
Lima, A. and Olsen, G., ‘Extreme points in duals of complex operator spaces’, Proc. Amer. Math. Soc. 94 (1985), 437–440.Google Scholar
Liu, R., ‘On extension of isometries between unit spheres of L∞(Γ) type spaces’, J. Math. Anal. Appl. 333 (2007), 959–970.Google Scholar
Megginson, R., An Introduction to Banach Space Theory, Graduate Texts in Mathematics, 183 (Springer, New York, 1998).Google Scholar
Miura, T., ‘Real-linear isometries between function algebras’, Cent. Eur. J. Math. 9 (2011), 778–788.Google Scholar
Miura, T., ‘Surjective isometries between function spaces’, Contemp. Math. 645 (2015), 231–239.CrossRefGoogle Scholar
Novinger, W., ‘Linear isometries of subspaces of spaces of continuous functions’, Studia Math. 53 (1975), 273–276.Google Scholar
Ruess, W. M. and Stegall, C. P., ‘Extreme points in duals of operator spaces’, Math. Ann. 261 (1982), 535–546.CrossRefGoogle Scholar
Sundaresan, K., ‘Spaces of continuous functions into Banach spaces’, Studia Math. 48 (1973), 15–22.CrossRefGoogle Scholar
Väisälä, J., ‘A proof of the Mazur–Ulam theorem’, Amer. Math. Monthly 110 (2003), 633–635.Google Scholar