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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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