Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A Banach-Stone theorem for Riesz isomorphisms of Banach lattices


Authors: Jin Xi Chen, Zi Li Chen and Ngai-Ching Wong
Journal: Proc. Amer. Math. Soc. 136 (2008), 3869-3874
MSC (2000): Primary 46E40; Secondary 46B42, 47B65
DOI: https://doi.org/10.1090/S0002-9939-08-09582-8
Published electronically: June 24, 2008
MathSciNet review: 2425726
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ X$ and $ Y$ be compact Hausdorff spaces, and $ E$, $ F$ be Banach lattices. Let $ C(X,E)$ denote the Banach lattice of all continuous $ E$-valued functions on $ X$ equipped with the pointwise ordering and the sup norm. We prove that if there exists a Riesz isomorphism $ \Phi: C(X,E)\rightarrow C(Y,F)$ such that $ \Phi f$ is non-vanishing on $ Y$ if and only if $ f$ is non-vanishing on $ X$, then $ X$ is homeomorphic to $ Y$, and $ E$ is Riesz isomorphic to $ F$. In this case, $ \Phi$ can be written as a weighted composition operator: $ \Phi f(y)=\Pi(y)(f(\varphi(y)))$, where $ \varphi$ is a homeomorphism from $ Y$ onto $ X$, and $ \Pi(y)$ is a Riesz isomorphism from $ E$ onto $ F$ for every $ y$ in $ Y$. This generalizes some known results obtained recently.


References [Enhancements On Off] (What's this?)

  • 1. Y.A. Abramovich and C.D. Aliprantis, An Invitation to Operator Theory, Graduate Studies in Mathematics 50, American Mathematical Society, Providence, RI, 2002. MR 2003h:47072
  • 2. C.D. Aliprantis and O. Burkinshaw, Positive Operators, Pure and Applied Mathematics 119, Academic Press, New York, 1985. MR 87h:47086
  • 3. E. Behrends, How to obtain vector-valued Banach-Stone theorems by using $ M$-structure methods, Math. Ann. 261 (1982), 387-398. MR 84c:46012
  • 4. E. Behrends and M. Cambern, An isomorphic Banach-Stone theorem, Studia Math. 90 (1988), 15-26. MR 89h:46020
  • 5. J. Cao, I. Reilly and H. Xiong, A lattice-valued Banach-Stone theorem, Acta Math. Hungar. 98 (2003), 103-110. MR 2003m:46028
  • 6. J.B. Conway, A Course in Functional Analysis, Second edition, Graduate Texts in Mathematics 96, Springer, New York, 1990. MR 91e:46001
  • 7. Z. Ercan and S. Önal, Banach-Stone theorem for Banach lattice valued continuous functions, Proc. Amer. Math. Soc. 135 (2007), 2827-2829. MR 2008a:46038
  • 8. M. Fabian et al., Functional Analysis and Infinite-Dimensional Geometry, CMS Books in Mathematics 8, Springer, New York, 2001. MR 2002f:46001
  • 9. H.-L. Gau, J.-S. Jeang and N.-C. Wong, Biseparating linear maps between continuous vector-valued function spaces, J. Aust. Math. Soc. 74 (2003), 101-109. MR 2003j:47041
  • 10. S. Hernandez, E. Beckenstein and L. Narici, Banach-Stone theorems and separating maps, Manuscr. Math. 86 (1995), 409-416. MR 95m:46054
  • 11. W. Holsztyński, Continuous mappings induced by isometries of spaces of continuous function, Studia Math. 26 (1966), 133-136. MR 33:1711
  • 12. J.-S. Jeang and N.-C. Wong, On the Banach-Stone problem, Studia Math. 155 (2003), 95-105. MR 2004a:46039
  • 13. M. Jerison, The space of bounded maps into a Banach space, Ann. of Math.(2) 52 (1950), 309-327. MR 12:188c
  • 14. E. de Jonge and A. van Rooij, Introduction to Riesz Spaces, Mathematical Centre Tracts 78, Amsterdam, 1977. MR 57:13439
  • 15. J.-H. Liu and N.-C. Wong, Local automorphisms of operator algebras, Taiwanese J. Math. 11 (2007), 611-619. MR 2340153
  • 16. X. Miao, J. Cao and H. Xiong, Banach-Stone theorems and Riesz algebras, J. Math. Anal. Appl. 313 (2006), 177-183. MR 2006m:46030

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46E40, 46B42, 47B65

Retrieve articles in all journals with MSC (2000): 46E40, 46B42, 47B65


Additional Information

Jin Xi Chen
Affiliation: Department of Mathematics, Southwest Jiaotong University, Chengdu 610031, People’s Republic of China
Email: jinxichen@home.swjtu.edu.cn

Zi Li Chen
Affiliation: Department of Mathematics, Southwest Jiaotong University, Chengdu 610031, People’s Republic of China
Email: zlchen@home.swjtu.edu.cn

Ngai-Ching Wong
Affiliation: Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan
Email: wong@math.nsysu.edu.tw

DOI: https://doi.org/10.1090/S0002-9939-08-09582-8
Keywords: Banach lattice, Banach-Stone theorem, Riesz isomorphism, weighted composition operator
Received by editor(s): June 1, 2007
Published electronically: June 24, 2008
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society