The Banach-Stone property and the weak Banach-Stone property in three-dimensional spaces
HTML articles powered by AMS MathViewer
- by Michael Cambern PDF
- Proc. Amer. Math. Soc. 67 (1977), 55-61 Request permission
Abstract:
Let X and Y be compact Hausdorff spaces, E a Banach space, and $C(X,E)$ the space of continuous functions on X to E. E has the weak Banach-Stone property if, whenever $C(X,E)$ and $C(Y,E)$ are isometric, then X and Y are homeomorphic. E has the Banach-Stone property if the descriptive as well as the topological conclusions of the Banach-Stone theorem for scalar functions remain valid in the case of isometries of $C(X,E)$ onto $C(Y,E)$. These two properties were first studied by M. Jerison, and it we later shown that every space E found by Jerison to have the weak Banach-Stone property actually has the Banach-Stone property, thus raising the question of whether the two properties are distinct. Here we characterize all three-dimensional spaces with the weak Banach-Stone property, and, in so doing, show the properties to be distinct.References
- Michael Cambern, On mappings of spaces of functions with values in a Banach space, Duke Math. J. 42 (1975), 91–98. MR 370173
- Michael Cambern, Reflexive spaces with the Banach-Stone property, Rev. Roumaine Math. Pures Appl. 23 (1978), no. 7, 1005–1010. MR 509598
- N. Dinculeanu, Vector measures, Hochschulbücher für Mathematik, Band 64, VEB Deutscher Verlag der Wissenschaften, Berlin, 1966. MR 0206189
- Meyer Jerison, The space of bounded maps into a Banach space, Ann. of Math. (2) 52 (1950), 309–327. MR 36942, DOI 10.2307/1969472
- S. B. Myers, Banach spaces of continuous functions, Ann. of Math. (2) 49 (1948), 132–140. MR 23000, DOI 10.2307/1969119
- K. Sundaresan, Spaces of continuous functions into a Banach space, Studia Math. 48 (1973), 15–22. MR 331042, DOI 10.4064/sm-48-1-15-22
Additional Information
- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 67 (1977), 55-61
- MSC: Primary 46E40
- DOI: https://doi.org/10.1090/S0002-9939-1977-0461126-7
- MathSciNet review: 0461126