Non-Archimedean $C^ {\#}(X)$
HTML articles powered by AMS MathViewer
- by Jesús M. Domínguez PDF
- Proc. Amer. Math. Soc. 97 (1986), 525-530 Request permission
Abstract:
Let $E$ be a nonarchimedean rank-one valued field, and $X$ an ultraregular topological space. We consider the Gelfand subalgebra ${C^\# }(X,E)$ of the algebra of all $E$-valued continuous functions on $X$, and the algebra $F(X,E)$ consisting of those $E$-valued continuous functions $f$ for which there exists a compact set $K \subset X$ such that $f(X - K)$ is finite. We obtain some characterizations of ${C^\# }(X,E)$, analogous to those obtained in the real case, which we use to find conditions that imply the equality ${C^\# }(X,E) = F(X,E)$ holds.References
- George Bachman, Edward Beckenstein, Lawrence Narici, and Seth Warner, Rings of continuous functions with values in a topological field, Trans. Amer. Math. Soc. 204 (1975), 91–112. MR 402687, DOI 10.1090/S0002-9947-1975-0402687-6
- Choo Eng-Ung, Note on a subring of $C^{\ast } (X)$, Canad. Math. Bull. 18 (1975), no. 2, 177–179. MR 388325, DOI 10.4153/CMB-1975-035-4
- Jesús M. Domínguez, The Gel′fand subalgebra of the ring of continuous functions with values in a non-Archimedean valued field, Rev. Mat. Hisp.-Amer. (4) 42 (1982), no. 4-6, 133–138 (Spanish). MR 791385
- Jesús M. Domínguez, The Gel′fand subalgebra of real or non-Archimedean valued continuous functions, Proc. Amer. Math. Soc. 90 (1984), no. 1, 145–148. MR 722433, DOI 10.1090/S0002-9939-1984-0722433-7 —, Note on two subrings of $C(X)$, preprint.
- Robert L. Ellis, Extending continuous functions on zero-dimensional spaces, Math. Ann. 186 (1970), 114–122. MR 261565, DOI 10.1007/BF01350686
- Leonard Gillman and Meyer Jerison, Rings of continuous functions, Graduate Texts in Mathematics, No. 43, Springer-Verlag, New York-Heidelberg, 1976. Reprint of the 1960 edition. MR 0407579
- Melvin Henriksen, An algebraic characterization of the Freudenthal compactification for a class of rimcompact spaces, Topology Proc. 2 (1977), no. 1, 169–178 (1978). MR 540604
- J. R. Isbell, Uniform spaces, Mathematical Surveys, No. 12, American Mathematical Society, Providence, R.I., 1964. MR 0170323, DOI 10.1090/surv/012
- L. D. Nel and D. Riordan, Note on a subalgebra of $C(X)$, Canad. Math. Bull. 15 (1972), 607–608. MR 317274, DOI 10.4153/CMB-1972-108-4
- Krzysztof Nowiński, Closed mappings and the Freudenthal compactification, Fund. Math. 76 (1972), no. 1, 71–83. MR 324628, DOI 10.4064/fm-76-1-71-83
- Niel Shilkret, Non-Archimedean Gelfand theory, Pacific J. Math. 32 (1970), 541–550. MR 257752, DOI 10.2140/pjm.1970.32.541
- Oscar Stefani and Alessandra Zanardo, Un’osservazione su una sottoalgebra di $C(X)$, Rend. Sem. Mat. Univ. Padova 53 (1975), 327–328. MR 410672
- A. C. M. van Rooij, Non-Archimedean functional analysis, Monographs and Textbooks in Pure and Applied Mathematics, vol. 51, Marcel Dekker, Inc., New York, 1978. MR 512894
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 97 (1986), 525-530
- MSC: Primary 54C40; Secondary 46H10, 46P05
- DOI: https://doi.org/10.1090/S0002-9939-1986-0840640-1
- MathSciNet review: 840640