Structure spaces of rings and Banach algebras (Q1079155)

\textit{I. Kaplansky} [Duke Math. J. 16, 399-418 (1949; Zbl 0033.187)] proved that a \(B^*\)-algebra has a structure space (set of primitive ideals with its hull-kernel topology) consisting of isolated points iff it is the topological sum of its minimal closed ideals. This was the starting point of the author's investigation of the set \({\mathcal J}(B)\) of isolated points of the structure space \({\mathcal P}(B)\) for \(B^*\)-algebras B, basing on an extensive study of these notions in semi-simple topological rings A. As a main tool the concept of a purely primitive ring (i.e. a ring for which (0) is the only primitive ideal) is employed. It is shown that \({\mathcal J}(B)\) consists of the annihilators of the minimal closed ideals and \({\mathcal J}(B)\) is dense in \({\mathcal P}(B)\) iff every non-zero ideal contains a minimal closed ideal. Characterizations and necessary conditions for A to have a discrete structure space are obtained and finally further answers to the question when \({\mathcal J}(A)\) is dense in \({\mathcal P}(A)\) are given.