Kaplansky Theorem for completely regular spaces

Li, Lei; Wong, Ngai-Ching

doi:10.1090/S0002-9939-2014-11889-2

Kaplansky Theorem for completely regular spaces
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by Lei Li and Ngai-Ching Wong

Proc. Amer. Math. Soc. 142 (2014), 1381-1389

DOI: https://doi.org/10.1090/S0002-9939-2014-11889-2

Published electronically: January 30, 2014

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Abstract:

Let $X, Y$ be realcompact spaces or completely regular spaces consisting of $G_\delta$-points. Let $\phi$ be a linear bijective map from $C(X)$ (resp. $C^b(X)$) onto $C(Y)$ (resp. $C^b(Y)$). We show that if $\phi$ preserves nonvanishing functions, that is, \[ f(x)\neq 0,\forall x\in X, \quad \Longleftrightarrow \quad \phi (f)(y)\neq 0, \forall y\in Y, \] then $\phi$ is a weighted composition operator \[ \phi (f)=\phi (1)\cdot f\circ \tau , \] arising from a homeomorphism $\tau$ from $Y$ onto $X$. This result is applied also to other nice function spaces, e.g., uniformly or Lipschitz continuous functions on metric spaces.

References

Jesús Araujo, Separating maps and linear isometries between some spaces of continuous functions, J. Math. Anal. Appl. 226 (1998), no. 1, 23–39. MR 1646465, DOI 10.1006/jmaa.1998.6031
Jesús Araujo, Realcompactness and Banach-Stone theorems, Bull. Belg. Math. Soc. Simon Stevin 11 (2004), no. 2, 247–258. MR 2080425
Jesús Araujo and Luis Dubarbie, Biseparating maps between Lipschitz function spaces, J. Math. Anal. Appl. 357 (2009), no. 1, 191–200. MR 2526819, DOI 10.1016/j.jmaa.2009.03.065
Bernard Aupetit, A primer on spectral theory, Universitext, Springer-Verlag, New York, 1991. MR 1083349, DOI 10.1007/978-1-4612-3048-9
Bernard Aupetit, Spectrum-preserving linear mappings between Banach algebras or Jordan-Banach algebras, J. London Math. Soc. (2) 62 (2000), no. 3, 917–924. MR 1794294, DOI 10.1112/S0024610700001514
Luis Dubarbie, Maps preserving common zeros between subspaces of vector-valued continuous functions, Positivity 14 (2010), no. 4, 695–703. MR 2741327, DOI 10.1007/s11117-010-0046-z
M. Isabel Garrido and Jesús A. Jaramillo, A Banach-Stone theorem for uniformly continuous functions, Monatsh. Math. 131 (2000), no. 3, 189–192. MR 1801746, DOI 10.1007/s006050070008
M. I. Garrido and J. A. Jaramillo, Homomorphisms on function lattices, Monatsh. Math. 141 (2004), no. 2, 127–146. MR 2037989, DOI 10.1007/s00605-002-0011-4
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
Andrew M. Gleason, A characterization of maximal ideals, J. Analyse Math. 19 (1967), 171–172. MR 213878, DOI 10.1007/BF02788714
Krzysztof Jarosz, Automatic continuity of separating linear isomorphisms, Canad. Math. Bull. 33 (1990), no. 2, 139–144. MR 1060366, DOI 10.4153/CMB-1990-024-2
Jyh-Shyang Jeang and Ngai-Ching Wong, Weighted composition operators of $C_0(X)$’s, J. Math. Anal. Appl. 201 (1996), no. 3, 981–993. MR 1400575, DOI 10.1006/jmaa.1996.0296
J.-P. Kahane and W. Żelazko, A characterization of maximal ideals in commutative Banach algebras, Studia Math. 29 (1968), 339–343. MR 226408, DOI 10.4064/sm-29-3-339-343
Irving Kaplansky, Lattices of continuous functions, Bull. Amer. Math. Soc. 53 (1947), 617–623. MR 20715, DOI 10.1090/S0002-9904-1947-08856-X
Irving Kaplansky, Lattices of continuous functions. II, Amer. J. Math. 70 (1948), 626–634. MR 26240, DOI 10.2307/2372202
Miguel Lacruz and José G. Llavona, Composition operators between algebras of uniformly continuous functions, Arch. Math. (Basel) 69 (1997), no. 1, 52–56. MR 1452159, DOI 10.1007/s000130050092
Denny H. Leung and Wee-Kee Tang, Banach-Stone theorems for maps preserving common zeros, Positivity 14 (2010), no. 1, 17–42. MR 2596461, DOI 10.1007/s11117-008-0002-3
Maurice D. Weir, Hewitt-Nachbin spaces, North-Holland Mathematics Studies, No. 17, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR 0514909