Noncommutative end theory
The corona algebra \(M({\mathfrak A})/{\mathfrak A}\) contains essential information on the global structure of \({\mathfrak A}\), as demonstrated for instance by Busby theory. It is an interesting and surprisingly difficult task to determine the ideal structure of \(M({\mathfrak A})/{\mathfrak A}\) by means of the internal structure of \({\mathfrak A}\). Toward this end, we generalize Freudenthal's classical theory of ends of topological spaces to a large class of \(C^*\)-algebras. However, mirroring requirements necessary already in the commutative case, we must restrict attention to \(C^*\)-algebras \({\mathfrak A}\) which are \(\sigma\)-unital and have connected and locally connected spectra. Furthermore, we must study separately a certain pathological behavior which occurs in neither commutative nor stable \(C^*\)-algebras. We introduce a notion of sequences determining ends in such a \(C^*\)-algebra \({\mathfrak A}\) and pass to a set of equivalence classes of such sequences, the ends of \({\mathfrak A}\). We show that ends are in a natural 1-1 correspondence with the set of components of \(M({\mathfrak A})/{\mathfrak A}\), hence giving a complete description of the complemented ideals of such corona algebras. As an application we show that corona algebras of primitive \(\sigma\)-unital \(C^*\)-algebras are prime. Furthermore, we employ the methods developed to show that, for a large class of \(C^*\)-algebras, the end theory of a tensor product of two nonunital \(C^*\)-algebras is always trivial.
scientific article; zbMATH DE number 1060823