Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-23T00:30:08.909Z Has data issue: false hasContentIssue false
  • English
  • Français

Automatic continuity of certain isomorphisms between regular Banach function algebras

Published online by Cambridge University Press:  18 May 2009

Juan J. Font
Juan J. Font
Affiliation:
Departamento De MatemÁticas, Universidad Jaume 1, Campus Penyeta, E-12071, Castellón, Spain E-mail address: font@mat.uji.es
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let A and B be regular semisimple commutative Banach algebras; that is to say, regular Banach function algebras. A linear map T denned from A into B is said to be separating or disjointness preserving if f.g = 0 implies Tf.Tg = 0, for all f, g ∈ A In this paper we prove that if A satisfies Ditkin's condition then a separating bijection is automatically continuous and its inverse is separating. If also B satisfies Ditkin's condition, then it induces a homeomorphism between the structure spaces of A and B.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 39 , Issue 3 , September 1997 , pp. 333 - 343
Copyright
Copyright © Glasgow Mathematical Journal Trust 1997

References

REFERENCES

1.Abramovich, Y., Multiplicative representation of disjointness preserving operators. Indag. Math. 45 (1983), 265–279.CrossRefGoogle Scholar
2.Araujo, J., Beckenstein, E. and Narici, L., Biseparating maps and homeomorphic realcompactifications, J. Math. Anal. Appl. 192 (1995), 258–265.CrossRefGoogle Scholar
3.Araujo, J. and Font, J. J., Linear isometries between subspaces of continuous functions, Trans. Amer. Math. Soc. 349 (1997), 413–428.CrossRefGoogle Scholar
4.Araujo, J. and Font, J. J., On Shilov boundaries for subspaces of continuous functions. Topology and Appl. 77 (1997), 79–85.CrossRefGoogle Scholar
5.Arendt, W., Spectral properties of Lamperti operators. Indiana Univ. Math. J. 32 (1983), 199–215.CrossRefGoogle Scholar
6.Arendt, W. and Hart, D. R., The spectrum of quasi-invertible disjointness preserving operators, J. Functional Analysis, 68 (1986), 149–167.CrossRefGoogle Scholar
7.Banach, S., Théorie des opérations linéaires (Chelsea Publishing Company, New York, 1932).Google Scholar
8.Beckenstein, E., Narici, L. and Todd, R., Automatic continuity of linear maps on spaces of continuous functions, Manuscripla Math. 62 (1988), 257–275.CrossRefGoogle Scholar
9.Dales, H. G., Banach algebras and automatic continuity (Oxford University Press), to appear.CrossRefGoogle Scholar
10.Font, J. J. and Hernández, S., Separating maps between locally compact spaces. Arch. Math. (Basel) 63 (1994), 158–165.CrossRefGoogle Scholar
11.Font, J. J. and Hernández, S., Automatic continuity and representation of certain linear isomorphisms between group algebras. Indag. Math. 6 (4) (1995), 397–409.CrossRefGoogle Scholar
12.Hernández, S., Beckenstein, E. and Narici, L., Banach-Stone theorems and separating maps, Manuscripta Math. 86 (1995), 409–416.CrossRefGoogle Scholar
13.Hewitt, E. and Ross, K. A., Abstract harmonic analysis II (Springer-Verlag, 1963).Google Scholar
14.Huijsmans, C. and de Pagter, B., Inverlible disjointness preserving operators, Proc. Edinburgh Math. Soc. 37 (1993), 125–132.CrossRefGoogle Scholar
15.Johnson, B. E., The uniqueness of the (complete) norm topology, Bull. Amer. Math. Soc. 73 (1967), 537–539.CrossRefGoogle Scholar
16.Jarosz, K., Automatic continuity of separating linear isomorphisms, Canad. Math. Bull. 33 (2) (1990), 139–144.CrossRefGoogle Scholar
17.Koldunov, A., Hammerstein operators preserving disjointness, Proc. Amer. Math. Soc. 123 (1995), 1083–1095.CrossRefGoogle Scholar
18.Lamperti, J., On the isometries of certain function spaces, Pacific J. Math. 8 (1958), 459–466.CrossRefGoogle Scholar
19.Nagasawa, M., Isomorphisms between commutative Banach algebras with an application to rings of analytic functions, Kodai Math. Sem. Rep. 11 (1959), 182–188.CrossRefGoogle Scholar
20.Reiter, H., Classical harmonic analvsis and locally compact groups (Oxford University Press, 1968).Google Scholar
21.Yap, L. Y. H., Every Segal algebra satisfies Ditkin's condition, Studia Math. 40 (1971), 235–237.CrossRefGoogle Scholar