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Topologies on the ideal space of a Banach algebra and spectral synthesis. (English) Zbl 0883.46032
Summary: Let the space $$\text{Id}(A)$$ of closed two-sided ideals of a Banach algebra $$A$$ carry the weak topology. We consider the following property called normality (of the family of finite subsets of $$A)$$: if the net $$(I_i)_i$$ in $$\text{Id}(A)$$ converges weakly to $$I$$, then for all $$a\in A\backslash I$$ we have $$\liminf_i|a+I_i|>0$$ (e.g. $$C^*$$-algebras, $$L^1(G)$$ with compact $$G,\ldots)$$. For a commutative Banach algebra normality is implied by spectral synthesis of all closed subsets of the Gelfand space $$\Delta(A)$$, the converse does not always hold, but it does under the following additional geometrical assumption:
$$\inf \{|\varphi_1-\varphi_2|;\varphi_1,\varphi_2 \in \Delta(A), \varphi_1\neq \varphi_2\}>0$$.

##### MSC:
 46J20 Ideals, maximal ideals, boundaries
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##### References:
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