A non-commutative generalization of Gelfand-Naimark theorem
Let A be a \(C^*\)-algebra, its structure space, i.e. the set of primitive ideals with the hull-kernel topology, denoted by \(A^ j\), and the subspace of minimal primitive ideals denoted by \(A^ b\). \textit{W. M. Ching} [Pac. J. Math. 67, 131-153 (1976; Zbl 0339.46040)] defined a \(C^*\)-algebra to be bounded (cf. bounded chain condition) if every primitive ideal contains a unique minimal primitive ideal. He defined a \(C^*\)-algebra to be *-bounded if a more technical condition, involving \(A^ j\) and \(A^ b\), holds, which ensures that \(A^ b\) is locally compact. He called A normal if it is bounded and *-bounded. He called A standard if it is normal and \(A^ b\) is Hausdorff, and proved that a standard \(C^*\)-algebra is isomorphic to the \(C^*\)-algebra defined by a continuous field of primitive \(C^*\)-algebras over \(A^ b\). The author proves the converse, that any \(C^*\)-algebra defined by a continuous field of primitive \(C^*\)-algebras is standard. His Theorem 2.2 characterizing continuous fields of simple \(C^*\)-algebras is not correct as stated. In answer to a query by Ching(loc. cit.) as to whether a normal \(C^*\)-algebra is necessarily GCR, the author illustrates with Bratelli diagrams that a certain AF-algebra is GCR but not bounded.
set of primitive ideals with the hull-kernel topology
continuous field of primitive \(C^*\)-algebras
scientific article; zbMATH DE number 4192429