Geodesic
Topology of Gleason Parts in Maximal Ideal Spaces with no Analytic Discs
Canadian journal of mathematics, Tome 73 (2021) no. 1, pp. 177-194

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DOI

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. 
Izzo, Alexander J.; Papathanasiou, Dimitris. Topology of Gleason Parts in Maximal Ideal Spaces with no Analytic Discs. Canadian journal of mathematics, Tome 73 (2021) no. 1, pp. 177-194. doi: 10.4153/S0008414X19000567
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     title = {Topology of {Gleason} {Parts} in {Maximal} {Ideal} {Spaces} with no {Analytic} {Discs}},
     journal = {Canadian journal of mathematics},
     pages = {177--194},
     year = {2021},
     volume = {73},
     number = {1},
     doi = {10.4153/S0008414X19000567},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008414X19000567/}
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