Abstract
Let A and B be uniformly closed function algebras on locally compact Hausdorff spaces with Choquet boundaries Ch A and ChB, respectively. We prove that if T: A → B is a surjective real-linear isometry, then there exist a continuous function κ: ChB → {z ∈ ℂ: |z| = 1}, a (possibly empty) closed and open subset K of ChB and a homeomorphism φ: ChB → ChA such that T(f) = κ(f ∘φ) on K and \(T\left( f \right) = \kappa \overline {fo\phi }\) on ChB \ K for all f ∈ A. Such a representation holds for surjective real-linear isometries between (not necessarily uniformly closed) function algebras.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Araujo J., Font J.J., On Šilov boundaries for subspaces of continuous functions, Topology Appl., 1997, 77(2), 79–85
Burckel R.B., Characterizations of C(X) among its Subalgebras, Lecture Notes in Pure and Appl. Math., 6, Marcel Dekker, New York, 1972
Ellis A.J., Real characterizations of function algebras amongst function spaces, Bull. Lond. Math. Soc., 1990, 22(4), 381–385
Fleming R.J., Jamison J.E., Isometries on Banach Spaces: Function Spaces, Chapman Hall/CRC Monogr. Surv. Pure Appl. Math., 129, Chapman & Hall/CRC, Boca Raton, 2003
Fleming R.J., Jamison J.E., Isometries on Banach Spaces. Vol. 2: Vector-Valued Function Spaces, Chapman Hall/CRC Monogr. Surv. Pure Appl. Math., 138, Chapman & Hall/CRC, Boca Raton, 2008
Hatori O., Hirasawa G., Miura T., Additively spectral-radius preserving surjections between unital semisimple commutative Banach algebras, Cent. Eur. J. Math., 2010, 8(3), 597–601
Hatori O., Lambert S., Luttman A., Miura T., Tonev T., Yates R., Spectral preservers in commutative Banach algebras, In: Function Spaces in Modern Analysis, Contemp. Math., 547, American Mathematical Society, Providence (in press)
de Leeuw K., Rudin W., Wermer J., The isometries of some function spaces, Proc. Amer. Math. Soc., 1960, 11(5), 694–698
Mazur S., Ulam S., Sur les transformations isométriques d’espaces vectoriels normés, C. R. Acad. Sci. Paris, 1932, 194, 946–948
Nagasawa M., Isomorphisms between commutative Banach algebras with an application to rings of analytic functions, Kōdai Math. Sem. Rep., 1959, 11(4), 182–188
Rao N.V., Roy A.K., Multiplicatively spectrum-preserving maps of function algebras. II, Proc. Edinb. Math. Soc., 2005, 48(1), 219–229
Rao N.V., Tonev T.V., Toneva E.T., Uniform algebra isomorphisms and peripheral spectra, In: Topological Algebras and Applications, Contemp. Math., 427, American Mathematical Society, Providence, 2007, 401–416
Tonev T., Toneva E., Composition operators between subsets of function algebras, In: Function Spaces in Modern Analysis, Contemp. Math., 547, American Mathematical Society, Providence (in press)
Tonev T., Yates R., Norm-linear and norm-additive operators between uniform algebras, J. Math. Anal. Appl., 2009, 357(1), 45–53
Väisälä J., A proof of the Mazur-Ulam theorem, Amer. Math. Monthly, 2003, 110(7), 633–635
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Miura, T. Real-linear isometries between function algebras. centr.eur.j.math. 9, 778–788 (2011). https://doi.org/10.2478/s11533-011-0044-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.2478/s11533-011-0044-9