ℕ-compactness and weighted composition maps

Araujo, Jesús

doi:10.1090/S0002-9939-01-06135-4

$\mathbb {N}$-compactness and weighted composition maps
HTML articles powered by AMS MathViewer

by Jesús Araujo PDF

Proc. Amer. Math. Soc. 130 (2002), 1225-1234 Request permission

Abstract:

In this paper we study some conditions on (not necessarily continuous) linear maps $T$ between spaces of real- or complex-valued continuous functions $C (X)$ and $C (Y)$ which allow us to describe them as weighted composition maps. This description depends strongly on the topology in $X$; namely, it can be given when $X$ is $\mathbb {N}$-compact, but cannot in general if some kind of connectedness on $X$ is assumed. Finally we also give an infimum-preserving version of the Banach-Stone theorem. The results are also proved for spaces of bounded continuous functions when $\mathbb {K}$ is a field endowed with a nonarchimedean valuation and it is not locally compact.

References

Jesús Araujo, $\textbf {N}$-compactness and automatic continuity in ultrametric spaces of bounded continuous functions, Proc. Amer. Math. Soc. 127 (1999), no. 8, 2489–2496. MR 1487354, DOI 10.1090/S0002-9939-99-04781-4
J. Araujo, E. Beckenstein, and L. Narici, Biseparating maps and homeomorphic real-compactifications, J. Math. Anal. Appl. 192 (1995), no. 1, 258–265. MR 1329423, DOI 10.1006/jmaa.1995.1170
J. Araújo, E. Beckenstein, and L. Narici, When is a separating map biseparating?, Arch. Math. (Basel) 67 (1996), no. 5, 395–407. MR 1411994, DOI 10.1007/BF01189099
Jesús Araujo and J. Martínez-Maurica, The non-Archimedean Banach-Stone theorem, $p$-adic analysis (Trento, 1989) Lecture Notes in Math., vol. 1454, Springer, Berlin, 1990, pp. 64–79. MR 1094847, DOI 10.1007/BFb0091134
Edward Beckenstein and Lawrence Narici, A non-Archimedean Stone-Banach theorem, Proc. Amer. Math. Soc. 100 (1987), no. 2, 242–246. MR 884460, DOI 10.1090/S0002-9939-1987-0884460-1
Ryszard Engelking, General topology, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from the Polish by the author. MR 1039321
Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199, DOI 10.1007/978-1-4615-7819-2
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