Structure of the gleason part of the algebra R(E)

1967 ◽  
Vol 1 (1) ◽  
pp. 84-86 ◽  
Author(s):  
M. S. Mel'nikov
Keyword(s):  
Gleason Part

Homeomorphic Analytic Maps into the Maximal Ideal Space of H∞

1999 ◽  
Vol 51 (1) ◽  
pp. 147-163 ◽  
Author(s):  
Daniel Suárez
Keyword(s):  
Maximal Ideal ◽  
Maximal Ideal Space ◽  
Ideal Space ◽  
Gleason Part ◽  
Interpolating Sequences ◽  
Closed Subalgebra ◽  
Analytic Maps ◽  
The Ideal

AbstractLet m be a point of the maximal ideal space of H∞ with nontrivial Gleason part P(m). If Lm : D → P(m) is the Hoffman map, we show that H∞ ° Lm is a closed subalgebra of H∞. We characterize the points m for which Lm is a homeomorphism in terms of interpolating sequences, and we show that in this case H∞ ° Lm coincides with H∞. Also, if Im is the ideal of functions in H∞ that identically vanish on P(m), we estimate the distance of any f ϵ H∞ to Im.

Topology of Gleason Parts in Maximal Ideal Spaces with no Analytic Discs

2019 ◽  
pp. 1-18
Author(s):  
Alexander J. Izzo ◽  
Dimitris Papathanasiou
Keyword(s):  
Maximal Ideal ◽  
Uniform Algebra ◽  
Maximal Ideal Space ◽  
Ideal Space ◽  
Regular Space ◽  
Compact Set ◽  
Gleason Part ◽  
Completely Regular ◽  
Analytic Discs ◽  
Compact Subspace

Abstract We strengthen, in various directions, the theorem of Garnett that every $\unicode[STIX]{x1D70E}$ -compact, completely regular space $X$ occurs as a Gleason part for some uniform algebra. In particular, we show that the uniform algebra can always be chosen so that its maximal ideal space contains no analytic discs. We show that when the space $X$ is metrizable, the uniform algebra can be chosen so that its maximal ideal space is metrizable as well. We also show that for every locally compact subspace $X$ of a Euclidean space, there is a compact set $K$ in some $\mathbb{C}^{N}$ so that $\widehat{K}\backslash K$ contains a Gleason part homeomorphic to  $X$ , and $\widehat{K}$ contains no analytic discs.

A Characterization of a Sparse Blaschke Product

1989 ◽  
Vol 32 (4) ◽  
pp. 385-390 ◽  
Author(s):  
Carroll Guillory
Keyword(s):  
Blaschke Product ◽  
Sufficient Condition ◽  
Gleason Part ◽  
Interpolating Blaschke Product ◽  
Support Sets

AbstractWe give a characterization of a sparse Blaschke product b in terms of the separation of support sets of its zeros in M(H∞ + C) and the structure of the nonanalytic points. We use this characterization to give a sufficient condition on an interpolating Blaschke product q to have the following property: there exists a non trivial Gleason part P on which q is nonzero and less than one.

A Topological Characterization of Gleason Parts of Real Function Algebras

1983 ◽  
Vol 26 (1) ◽  
pp. 44-49
Author(s):  
S. H. Kulkarni ◽  
B. V. Limaye
Keyword(s):  
Topological Space ◽  
Real Function ◽  
Complex Function ◽  
Function Algebra ◽  
Topological Characterization ◽  
Gleason Part ◽  
Completely Regular ◽  
Compact Topological Space ◽  
Function Algebras ◽  
Real Function Algebra

AbstractIt is well-known that a topological space is a Gleason part of some complex function algebra if and only if it is completely regular and σ-compact. In the present paper, a Gleason part of a real function algebra is characterized as a completely regular σ-compact topological space which admits an involutoric homeomorphism.

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