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

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... 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.