Pullback and pushout constructions in \(C^*\)-algebra theory
The paper is dedicated to the category of \(C^*\)-algebras with *-homomorphisms as morphisms. Many of the results require proper morphisms. ``Pullback'' and ``pushout'' are certain commutative diagrams of \(C^*\)-algebras, often used in the theory of extensions. The author undertakes a thorough study of such diagrams in relation to restricted direct sums and amalgamated free products of \(C^*\)-algebras. It is shown that pullback and pushout diagrams are stable under tensoring with a fixed algebra and under crossed products with a fixed group. From the abstract: ``The crowning result is that if three short exact sequences of \(C^*\)-algebras are given, with appropriate morphisms between the sequences allowing for pullback or pushout constructions at the levels of ideals, algebras and quotients, then the three new \(C^*\)-algebras will again form a short exact sequence under some mild extra conditions. As a generalization of a theorem of T. A. Loring, it is shown that each morphism between a pair of \(C^*\)-algebras, combined with its extension to the stabilized algebras, gives rise to a pushout diagram''.
pullback and pushout diagrams are stable under tensoring with a fixed algebra and under crossed products with a fixed group
scientific article; zbMATH DE number 1367849
scientific article