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Montel Algebras on the Plane

Published online by Cambridge University Press:  20 November 2018

W. E. Meyers
W. E. Meyers*
Affiliation:
The University of British Columbia, Vancouver, British Columbia
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The results of Rudin in [7] show that under certain conditions, the maximum modulus principle characterizes the algebra A (G) of functions analytic on an open subset G of the plane C (see below). In [2], Birtel obtained a characterization of A(C) in terms of the Liouville theorem; he proved that every singly generated F-algebra of continuous functions on C which contains no non-constant bounded functions is isomorphic to A(C) in the compact-open topology. In this paper we show that the Montel property of the topological algebra A (G) also characterizes it. In particular, any Montel algebra A of continuous complex-valued functions on G which contains the polynomials and has continuous homomorphism space M (A) homeomorphic to G is precisely A(G).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 1 , 01 February 1970 , pp. 116 - 122
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Arens, R., A topology for spaces of transformations, Ann. of Math. (2) 47 (1946), 480–495.Google Scholar
2. Birtel, F. T., Singly-generated Liouville F-algebras, Michigan Math. J. 11 (1964), 89–94.Google Scholar
3. Birtel, F. T. and Lindberg, J. A. Jr., A Liouville algebra of non-entire functions, Studia Math. 25 (1964/65), 27–31.Google Scholar
4. Gunning, R. C. and Rossi, H., Analytic functions of several complex variables (Prentice-Hall, Englewood Cliffs, N.J., 1965).Google Scholar
5. Michael, E. A., Locally multiplicatively-convex topological algebras, Mem. Amer. Math. Soc. No. 11 (1952), 79 pp.Google Scholar
6. Rickart, C. E., Holomorphic convexity in general function algebras, Can. J. Math. 20 (1968), 272–290.Google Scholar
7. Rudin, W., Analyticity, and the maximum modulus principle, Duke Math. J. 20 (1953), 449–457.Google Scholar
8. Whyburn, G. T., Topological analysis (Princeton Univ. Press, Princeton, N.J., 1964).Google Scholar