Some characterizations of trivial parts for

Authors:
Thomas J. Abram and Max L. Weiss

Journal:
Proc. Amer. Math. Soc. **99** (1987), 455-461

MSC:
Primary 46J15; Secondary 30H05, 46J20

DOI:
https://doi.org/10.1090/S0002-9939-1987-0875380-7

MathSciNet review:
875380

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Abstract: The unit disc in the complex plane is made into a locally compact topological group. This group acts as a transformation group on the maximal ideal space of the Banach algebra of bounded analytic functions on the disc. Among other characterizations the trivial parts are shown to be the minimal closed invariant sets of this transformation group. A point in the maximal ideal space is a trivial part if and only if it is the limit of a maximal invariant filter. An example shows that the correspondence between such points and filters is not one-to-one.

**[1]**T. J. Abram,*Parts in the maximal ideal space of**--A harmonic analysis approach*, Doctoral Dissertation, University of California, Santa Barbara, 1983, pp. 1-94.**[2]**L. Carleson,*Interpolations by bounded analytic functions and the corona problem*, Ann. of Math. (2)**76**(1962), 542-559. MR**0141789 (25:5186)****[3]**R. Ellis,*Lectures on topological dynamics*, Benjamin, New York, 1969, pp. 1-211. MR**0267561 (42:2463)****[4]**K. Hoffman,*Banach spaces of analytic functions*, Prentice-Hall, Englewood Cliffs, N. J., 1962, pp. 1-217. MR**0133008 (24:A2844)****[5]**-,*Bounded analytic functions and Gleason parts*, Ann. of Math. (2)**81**(1967), 74-111. MR**0215102 (35:5945)**

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DOI:
https://doi.org/10.1090/S0002-9939-1987-0875380-7

Keywords:
Bounded analytic functions,
parts,
transformation group

Article copyright:
© Copyright 1987
American Mathematical Society