APM  Vol.4 No.4 , April 2014
Some Mappings on Operator Spaces
Author(s) Guimei An*
ABSTRACT

We discuss two types of maps on operator spaces. Firstly, through example we show that there is an isometry on unit sphere of an operator space cannot be extended to be a complete isometry on the whole operator space. Secondly, we give a new characterization for complete isometry by the concept of approximate isometry.


Cite this paper
An, G. (2014) Some Mappings on Operator Spaces. Advances in Pure Mathematics, 4, 98-102. doi: 10.4236/apm.2014.44016.
References
[1]   Effros, E.G. and Ruan, Z-J. (2000) Operator Spaces. In: London Mathematical Society Monographs New Series 23, The Clarendon Press, Oxford University Press, New York.

[2]   An, G., Lee, J.-J. and Ruan, Z.-J. (2010) On p-Approximation Properties for p-Operator Spaces. Journal of Functional Analysis, 259, 933-974. http://dx.doi.org/10.1016/j.jfa.2010.04.007

[3]   Pisier, G. (1990) Completely Bounded Maps between Sets of Banach Space Operators. Indiana University Mathematics Journal, 39, 249-277. http://dx.doi.org/10.1512/iumj.1990.39.39014

[4]   Pisier, G. (2003) An Introduction to the Theory of Operator Spaces. In: London Mathematical Society Lecture Note Series 294, Cambridge University Press, Cambridge.

[5]   Mazur, S. and Ulam, S. (1932) Sur less transformations isométriques d’espaces vectoriels normés. Comptes Rendus de l’Académie des Sciences de Paris, 194, 946-948.

[6]   Mankiewicz, P. (1972) On Extension of Isometries in Normed Linear Spaces. Bulletin de l Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques, et Physiques, 20, 367-371.

[7]   Tingley, D. (1987) Isometries of the Unit Sphere. Geometriae Dedicata, 22, 371-378.
http://dx.doi.org/10.1007/BF00147942

[8]   Ding, G.G. (2003) The Isometric Extension Problem in the Unit Spheres of Type Spaces. Science in China, Series A, 46, 333-338.

[9]   Ding, G.G. (2004) The Representation Theorem of onto Isometric Mappings between Two Unit Spheres of Type Spaces and the Application to Isometric Extension Problem. Acta Mathematica Sinica, English Series, 20, 1089-1094. http://dx.doi.org/10.1007/s10114-004-0447-7

[10]   Ding, G.G. (2004) The Representation of onto Isometric Mappings between Two Spheres of -Type Spaces and the Application on Isometric Extension Problem. Science in China Series A, 47, 722-729.
http://dx.doi.org/10.1360/03ys0049

[11]   Ding, G.G. (2002) The 1-Lipschitz Mapping between the Unit Spheres of Two Hilbert Spaces Can Be Extended to a Real Linear Isometry of the Whole Space. Science in China Series A, 45, 479-483.
http://dx.doi.org/10.1007/BF02872336

[12]   Fang, X.N. and Wang, J.H. (2006) Extension of Isometries between the Unit Spheres of Normed Space E and C( ), Acta Mathematica Sinica, English Series, 22, 1819-1824. http://dx.doi.org/10.1007/s10114-005-0725-z

[13]   Ding, G.G. (2009) On Isometric Extension Problem between Two Unit Spheres. Science in China Series A, 52, 2069-2083. http://dx.doi.org/10.1007/s11425-009-0156-x

[14]   Bourgin, D.G. (1946) Approximate Isometries. Bulletin of the American Mathematical Society, 52, 704-714.
http://dx.doi.org/10.1090/S0002-9904-1946-08638-3

[15]   Bourgin, D.G. (1975) Approximate on Finite Dimensional Banach Spaces. Transactions of the American Mathematical Society, 207, 309-328.

[16]   Figiel, T. (1968) On Non Linear Isometric Embedding of Normed Linear Spaces. Bulletin de l Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques, et Physiques, 16, 185-188.

[17]   Gruber, P.M. (1978) Stability of Isometries. Transactions of the American Mathematical Society, 245, 263-277.
http://dx.doi.org/10.1090/S0002-9947-1978-0511409-2

[18]   Gevirtz, J. (1983) Stability of Isometries on Banach Spaces. Proceedings of the American Mathematical Society, 89, 633-636. http://dx.doi.org/10.1090/S0002-9939-1983-0718987-6

[19]   Ulam, S.M. and Hyers, D.H. (1945) On Approximate Isometries. Bulletin of the American Mathematical Society, 51, 288-292.

[20]   Omladic, M. and Semrl, P. (1995) On Non Linear Perturbations of Isometries. Mathematische Annalen, 303, 617-628.
http://dx.doi.org/10.1007/BF01461008

 
 
Top