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Local isometries of $\mathcal{L}(X,C(K))$


Author: T. S. S. R. K. Rao
Journal: Proc. Amer. Math. Soc. 133 (2005), 2729-2732
MSC (2000): Primary 47L05, 46B20
DOI: https://doi.org/10.1090/S0002-9939-05-07832-9
Published electronically: March 22, 2005
MathSciNet review: 2146220
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Abstract: In this paper we study the structure of local isometries on $\mathcal{L}(X,C(K))$. We show that when $K$ is first countable and $X$ is uniformly convex and the group of isometries of $X^\ast$ is algebraically reflexive, the range of a local isometry contains all compact operators. When $X$ is also uniformly smooth and the group of isometries of $X^\ast$ is algebraically reflexive, we show that a local isometry whose adjoint preserves extreme points is a $C(K)$-module map.


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  • 1. Arias de Reyna, J. Diestel, V. Lomonosov, L. Rodriguez-Piazza, Some observations about the space of weakly continuous functions from a compact space into a Banach space, Quaestiones Math. 15 (1992) 415-425. MR 1201299 (94b:46055)
  • 2. E. Behrends, M-structure and the Banach-Stone theorem, Springer Lecture Notes in Math., no. 736, Springer, Berlin, 1979. MR 0547509 (81b:46002)
  • 3. F. Cabello Sanchez and L. Molnár, Reflexivity of the isometry group of some classical spaces, Rev. Mat. Iberoamericana 18 (2002) 409-430. MR 1949834 (2003j:47046)
  • 4. M. Cambern and P. Greim, Mappings of continuous functions on hyper-Stonean spaces, Acta Univ. Carolinae, Math. Phys. 28 (1987) 31-40. MR 0932737 (89f:46081)
  • 5. Michael Cambern and Krzysztof Jarosz, Isometries of spaces of weak$^\ast$ continuous functions, Proc. Amer. Math. Soc., 106 (1989) 707-712. MR 0968623 (90e:46031)
  • 6. Richard J. Fleming and James E. Jamison, Isometries on Banach spaces: function spaces, Monographs and Surveys in Pure and Applied Mathematics, 129, Chapman and Hall-CRC, Boca Raton, 2003. MR 1957004 (2004j:46030)
  • 7. Krzysztof Jarosz and T. S. S. R. K. Rao, Local isometries of function spaces, Math. Z., 243 (2003) 449-469. MR 1970012 (2003m:46036)
  • 8. R. V. Kadison, Isometries of operator algebras, Ann. Math. 54 (1951) 325-338. MR 0043392 (13:256a)
  • 9. A. T. M. Lau and P. F. Mah, Quasinormal structure for certain spaces of operators on a Hilbert space, Pacific J. Math.,121 (1986) 109-118. MR 0815037 (87f:47065)
  • 10. L. Molnár, The set of automorphisms of $ B(H)$ is topologically reflexive in $B( B(H))$, Studia Math. 122 (1997) 183-193. MR 1432168 (98e:47068)
  • 11. L. Molnár and B. Zalar, On local automorphisms of group algebras of compact groups, Proc. Amer. Math. Soc. 128 (2000) 93-99. MR 1637412 (2000f:43002)
  • 12. L. Molnár, Preserver Problems on Algebraic Structures of Linear Operators and on Function Spaces, Dissertation for the D.Sc degree of the Hungarian Academy of Sciences, 2003.
  • 13. T. S. S. R. K. Rao, A note on the extreme points of $WC(K,X)^*_1,$ J. Ramanujan Math. Soc. 9 (1994) 215-219. MR 1308414 (95j:46039)
  • 14. T. S. S. R. K. Rao, Spaces with the Namioka-Phelps property have trivial $L$-structure, Archiv der Math., 62 (1994) 65-68. MR 1249587 (94m:46015)
  • 15. T. S. S. R. K. Rao, Local surjective isometries of function spaces, Expo. Math., 18 (2000) 285-296. MR 1788324 (2001k:46042)
  • 16. T. S. S. R. K. Rao,Weakly continuous functions of Baire class 1, Extracta Mathematicae, 15 (2000) 207-212. MR 1792989 (2001h:54023)

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Additional Information

T. S. S. R. K. Rao
Affiliation: Statistics and Mathematics Unit, Indian Statistical Institute, R. V. College P.O., Bangalore 560059, India
Email: tss@isibang.ac.in

DOI: https://doi.org/10.1090/S0002-9939-05-07832-9
Keywords: Isometries
Received by editor(s): March 16, 2004
Received by editor(s) in revised form: May 12, 2004
Published electronically: March 22, 2005
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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