Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-26T06:59:08.310Z Has data issue: false hasContentIssue false
  • English
  • Français

Maximum modulus algebras and singularity sets

Published online by Cambridge University Press:  14 November 2011

John Wermer
John Wermer
Affiliation:
Department of Mathematics, Brown University, Providence, Rhode Island, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

A classical theorem of Hartogs gives conditions on the singularity set of an analytic function of several complex variables in order for such a set to be an analytic variety. A result of E. Bishop from 1963 gives an analogous condition of the maximal ideal space of a uniform algebra in order for this space to have analytic structure. We show that algebras of functions satisfying a maximum principle serve to explain both of these results.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 86 , Issue 3-4 , 1980 , pp. 327 - 331
Copyright
Copyright © Royal Society of Edinburgh 1980

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Aupetit, B.. Caractérisation spectrale des algèbres de Banach de dimension finie. J. Functional Analysis 26 (1977), 232–250.CrossRefGoogle Scholar
2Aupetit, B.. Propriétés spectrales des algèbres de Banach. Lecture Notes in Mathematics 735 (Berlin: Springer, 1979).Google Scholar
3Aupetit, B. and Wermer, J.. Capacity and uniform algebras. J. Functional Analysis 28 (1978), 386–400.CrossRefGoogle Scholar
4Bishop, E.. Holomorphic completions, analytic continuations, and the interpolation of semi-norms. Ann. of Math. 78 (1963), 468–500.CrossRefGoogle Scholar
5Hartogs, F.. Über die aus den singulären Stellen einer analytischen Funktion mehrerer Veränderlichen bestehenden Gebilde. Acta Math. 32 (1909), 57–79.CrossRefGoogle Scholar
6Nishino, T.. Sur les ensembles pseudoconcaves. J. Math. Kyoto Univ. 1 (1962), 225–245.Google Scholar
7Oka, K.. Note sur les familles de fonctions analytiques multiformes etc. J. Sci. Hiroshima Univ. Series A 4 (1934), 93–98.Google Scholar
8Rossi, H.. The local maximum modulus principle. Ann. of Math. 72 (1960), 1–11.CrossRefGoogle Scholar
9Rudin, W.. Analyticity and the maximum modulus principle. Duke Math. J. 20 (1953), 449–457.CrossRefGoogle Scholar
10Slodkowski, Z.. Characterisation of spectral multifunctions. Abstract 770–B 25. Notices Amer. Math. Soc. 26 (October 1979).Google Scholar
11Wermer, J.. Subharmonicity and hulls. Pacific J. Math. 58 (1975), 283–290.CrossRefGoogle Scholar
12Narasimhan, R.. Several complex variables (Univ. of Chicago Press, 1971).Google Scholar