Topologies of compact families on the ideal space of a Banach algebra

Let be a family of compact sets in a Banach algebra A such that is stable with respect to finite unions and contains all finite sets. Then the sets ${\displaystyle U\left(K\right):=\{I\in Id\left(A\right):I\cap K=\varnothing \}}$, K ∈ define a topology τ() on the space Id(A) of closed two-sided ideals of A. is called normal if ${\displaystyle I_i\to I}$ in (Id(A),τ()) and x ∈ A╲I imply ${\displaystyle lim\in f_i\parallel x+I_i\parallel >0}$. (1) If the family of finite subsets of A is normal then Id(A) is locally compact in the hull kernel topology and if moreover A is separable then 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 ⊂ A such that for all closed two-sided ideals I we have ${\displaystyle \overline{I\cap S}=I}$. Examples are separable C*-algebras, the convolution algebras ${\displaystyle L^p\left(G\right)}$ where 1 ≤ p < ∞ and G is a metrizable compact group, and others; but not all separable Banach algebras share this property.