Topologies on the space of ideals of a Banach algebra

Some topologies on the space Id(A) of two-sided and closed ideals of a Banach algebra are introduced and investigated. One of the topologies, namely ${\displaystyle {\tau }_{\infty }}$, coincides with the so-called strong topology if A is a C*-algebra. We prove that for a separable Banach algebra ${\displaystyle {\tau }_{\infty }}$ coincides with a weaker topology when restricted to the space Min-Primal(A) of minimal closed primal ideals and that Min-Primal(A) is a Polish space if ${\displaystyle {\tau }_{\infty }}$ is Hausdorff; this generalizes results from [1] and [5]. All subspaces of Id(A) with the relative hull kernel topology turn out to be separable Lindelöf spaces if A is separable, which improves results from [14].