Ideals generated by projections and inductive limit \(C^*\)-algebras
The author introduces two classes of inductive limit \(C^*\)-algebras: \(GAH\)-algebras (generalized \(AH\)-algebras) and its subclass -- strong \(GAH\)-algebras. A necessary and sufficient condition is given for an ideal of a \(GAH\)-algebra to be generated by projections. It is proved that if \[ 0 \to I \to A \to B \to 0 \] is an exact sequence of \(C^*\)-algebras such that \(A\) is a \(GAH\)-algebra then \(A\) has the ideal property (i.e. any ideal is generated by projections) if and only if \(I\) and \(B\) have the ideal property. The lattice of ideals generated by projections of a strong \(GAH\)-algebra and also the partially ordered set of the stably cofinite ideals generated by projections of a strong \(GAH\)-algebra \(A\) are described under an additional assumption on \(A\).
scientific article; zbMATH DE number 1744065