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

Let the space of closed two-sided ideals of a Banach algebra carry the weak topology. We consider the following property called normality (of the family of finite subsets of : if the net in converges weakly to , then for all we have (e.g. -algebras, with compact . For a commutative Banach algebra normality is implied by spectral synthesis of all closed subsets of the Gelfand space , the converse does not always hold, but it does under the following additional geometrical assumption: .