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Topologies of compact families on the ideal space of a Banach algebra. (English) Zbl 0854.46045
Summary: Let $${\mathcal K}$$ be a family of compact sets in a Banach algebra $$A$$ such that $${\mathcal K}$$ is stable with respect to finite unions and contains all finite sets. Then the sets $$U(K):= \{I\in \text{Id} (A): I\cap K= \emptyset\}$$, $$K\in {\mathcal K}$$, define a topology $$\tau ({\mathcal K})$$ on the space $$\text{Id} (A)$$ of closed two-sided ideals of $$A$$. $${\mathcal K}$$ is called normal if $$I_i\to I$$ in $$(\text{Id} (A), \tau ({\mathcal K}))$$ and $$x\in A\setminus I$$ imply $$\liminf_i |x+ I_i|>0$$.
(1) If the family of finite subsets of $$A$$ is normal then $$\text{Id} (A)$$ is locally compact in the hull kernel topology and if moreover $$A$$ is separable then $$\text{Id} (A)$$ is second countable.
(2) If the family of countable compact sets is normal and $$A$$ is separable then there is a countable subset $$S\subset A$$ such that for all closed two-sided ideals $$I$$ we have $$\overline {I\cap S} =I$$.
Examples are separable $$C^*$$-algebras, the convolution algebras $$L^p (G)$$ where $$1\leq p< \infty$$ and $$G$$ is a metrizable compact group, and others; but not all separable Banach algebras share this property.

##### MSC:
 46H10 Ideals and subalgebras
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