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.

中文翻译：

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无分析盘的最大理想空间中格里森零件的拓扑结构

摘要 我们在各个方向上加强了 Garnett 定理，即每个 $\unicode[STIX]{x1D70E}$-紧凑的、完全规则的空间 $X$ 作为一些统一代数的 Gleason 部分出现。特别是，我们表明始终可以选择一致代数，使其最大理想空间不包含解析盘。我们表明，当空间 $X$ 是可度量的时，可以选择一致代数，以便其最大理想空间也是可度量的。我们还表明，对于欧几里德空间的每个局部紧致子空间 $X$，在某些 $\mathbb{C}^{N}$ 中存在紧致集 $K$，使得 $\widehat{K}\backslash K$包含与 $X$ 同胚的 Gleason 部分，而 $\widehat{K}$ 不包含解析盘。