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

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.

ON ALGEBRAIC EQUATION WITH COEFFICIENTS FROM THE $ \beta $-UNIFORM ALGEBRA $ C_{\beta} (\Omega) $

In this work the question of algebraic closeness of $ \beta $-uniform algebra $ A (\Omega) $ defined on locally compact space $ \Omega $ is investigated.

A Note on Some Uniform Algebra Generated by Smooth Functions in the Plane

We determine, via classroom proofs, the maximal ideal space, the Bass stable rank as well as the topological and dense stable rank of the uniform closure of all complex-valued functions continuously differentiable on neighborhoods of a compact planar set and holomorphic in the interior of . In this spirit, we also give elementary approaches to the calculation of these stable ranks for some classical function algebras on .

Szegö's Theorem and Uniform Algebras

We study Szegö's theorem for a uniform algebra. In particular, we do it for the disc algebra or the bidisc algebra.