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Topologies on the space of ideals of a Banach algebra. (English) Zbl 0836.46038
Summary: Some topologies on the space $$\text{Id} (A)$$ of two-sided and closed ideals of a Banach algebra are introduced and investigated. One of the topologies, namely $$\tau_\infty$$, coincides with the so-called strong topology if $$A$$ is a $$C^*$$-algebra. We prove that for a separable Banach algebra $$\tau_\infty$$ coincides with a weaker topology when restricted to the space $$\text{Min-Primal} (A)$$ of minimal closed primal ideals and that $$\text{Min-Primal} (A)$$ is a Polish space if $$\tau_\infty$$ is Hausdorff; this generalizes results from R. J. Archbold [J. Lond. Math. Soc., II. Ser. 35, No. 3, 524-542 (1987; Zbl 0613.46048)] and the author [Mich. Math. J. 40, No. 3, 477-492 (1993; Zbl 0814.46042)]. All subspaces of $$\text{Id} (A)$$ with the relative hull kernel topology turn out to be separable Lindelöf spaces if $$A$$ is separable, which improves results from D. W. B. Sommerset [Math. Proc. Camb. Philos. Soc. 115, No. 1, 39-52 (1994; Zbl 0818.46054)].

##### MSC:
 46H10 Ideals and subalgebras 46J20 Ideals, maximal ideals, boundaries
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