Topologies on the ideal space of a Banach algebra and spectral synthesis

Abstract:

Let the space $\operatorname {Id}(A)$ 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 $\operatorname {Id}(A)$ 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: $\inf \{\|\varphi _1-\varphi _2\|;\varphi _1,\varphi _2 \in \Delta (A), \varphi _1\neq \varphi _2\}>0$.