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On separating maps between locally compact spaces. (English) Zbl 0805.46049
A linear map $H$ defined from a subalgebra $A$ of $C\sb 0(T)$ into a subalgebra $B$ of $C\sb 0(S)$ is said to be separating or disjointness preserving if $x\cdot y\equiv 0$ implies $Hx\cdot Hy\equiv 0$ for all $x,y\in A$. The authors show that a separating bijection $H$ is automatically continuous (indeed, a weighted composition map) and induces a homeomorphism between the locally compact spaces $T$ and $S$. If $A$ and $B$ are the continuous functions on $T$ and $S$, respectively, with compact support, then a similar result for a separating injection is obtained. This result is applied to generalize to functions with compact support a well-kown theorem by Holsztyński about linear into isometries between $C(T)$ and $C(S)$ with $T$ and $S$ compact spaces.
Reviewer: J.J.Font (Castellón)

46H40Automatic continuity
46J10Banach algebras of continuous functions, function algebras
47B38Operators on function spaces (general)
Full Text: DOI
[1] Y. Abramovich, Multiplicative representation of disjointness preserving operators. Indag. Math.45, 265-279 (1983). · Zbl 0527.47025
[2] Y. Abramovich, A. I. Veksler andA. V. Koldunov, On operators preserving disjointness. Soviet Math. Dokl.248, 1033-1036 (1979). · Zbl 0445.46017
[3] Y.Abramovich, E. L.Arenson and A. K.Kitover, BanachC(K)-modules and operators preserving disjointness. Pitman Res. Notes Math. Ser.277 (1993). · Zbl 0795.47024
[4] E. Albrecht andM. Neumann, Automatic continuity for operators of local type. In: LNM975, 342-355. Berlin-Heidelberg-New York 1983.
[5] J.Araujo, E.Beckenstein and L.Narici, On biseparating maps between realcompact spaces. To appear in J. Math. Anal. Appl. · Zbl 0911.54014
[6] J. Araujo andJ. Martínez-Maurica, The nonarchimedean Banach-Stone theorem. In: LNM1454, 64-79. Berlin-Heidelberg-New York 1990.
[7] E. Beckenstein andL. Narici, Automatic continuity of certain linear isomorphism. Acad Roy. Belg. Bull. Cl. Sci. (5)73, 191-200 (1987). · Zbl 0664.46079
[8] E. Beckenstein, L. Narici andR. Todd, Automatic continuity of certain linear maps on spaces of continuous functions. Manuscripta Math.62, 257-275 (1988). · Zbl 0666.46018 · doi:10.1007/BF01246833
[9] W. A. Feldman andJ. F. Porter, Operators on Banach lattices as weighted compositions. J. London Math. Soc.33, 149-156 (1986). · Zbl 0564.46016 · doi:10.1112/jlms/s2-33.1.149
[10] L.Gillman and M.Jerison, Rings of continuous functions. Princeton, N.J. 1960. · Zbl 0093.30001
[11] W. Holszty?ski, Continuous mappings induced by isometries of spaces of continuous functions. Studia Math.26, 133-136 (1966).
[12] K. Jarosz, Automatic continuity of separating linear isomorphisms. Canad. Math. Bull.33, 139-144 (1990). · Zbl 0714.46040 · doi:10.4153/CMB-1990-024-2
[13] B. de Pagter, A note on disjointness preserving operators. Proc. Amer. Math. Soc.90, 543-549 (1984). · Zbl 0541.47032