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The corona algebra
M(A)∕mathfrakA 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.
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