Noncommutative End Theory

The corona algebra M(A)/A contains essential information on the global structure of A, as demonstrated for instance by Busby theory. It is an interesting and surprisingly difficult task to determine the ideal structure of M(A)/A by means of the internal structure of 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 A which are σ-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 A and pass to a set of equivalence classes of such sequences, the ends of A. We show that ends are in a natural 1–1 correspondence with the set of components of M(A)/A, hence giving a complete description of the complemented ideals of such corona algebras. As an application we show that corona algebras of primitive σ-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.