The Cuntz semigroup is an isomorphism invariant for C*-algebras consisting of a semigroup with a compatible (though not algebraic) ordering. Its construction is similar to that of the Murray-von Neumann semigroup (from which the ordered K_0-group arises by the Grothendieck construction), but using positive elements in place of projections. Both rich in structure and sensitive to subtleties of the C*-algebra, the Cuntz semigroup promises to be a useful tool in the classification program for nuclear C*-algebras. It has already delivered on this promise, particularly in the study of regularity properties and the classification of nonsimple C*-algebras. The first part of this thesis introduces the Cuntz semigroup, highlights structural properties, and outlines some applications. The main result of this thesis, however, contributes to the understanding of what the Cuntz semigroup looks like for particular examples of (nonsimple) C*-algebras. We consider separable C*-algebras given as the tensor product of a commutative C*-algebra C_0(X) with a simple, approximately subhomogeneous algebra A, under the regularity hypothesis that A is Z-stable. (The Z-stability hypothesis is needed even to describe of the Cuntz semigroup of A.) For these algebras, the Cuntz semigroup is described in terms of the Cuntz semigroup of A and the Murray-von Neumann semigroups of C(K,A) for compact subsets K of X. This result is a marginal improvement over one proven by the author in [Tikuisis, A. "The Cuntz semigroup of continuous functions into certain simple C*-algebras." Internat. J. Math., to appear] (there, A is assumed to be unital), although improvements have been made to the techniques used. The second part of this thesis provides the basic theory of approximately subhomogeneous algebras, including the important computational concept of recursive subhomogeneous algebras. Theory to handle nonunital approximately subhomogeneous algebras is novel here. In the third part of this thesis lies the main result. The Cuntz semigroup computation is achieved by defining a Cuntz-equivalence invariant I(.) on the positive elements of the C*-algebra, picking out certain data from a positive element which obviously contribute to determining its Cuntz class. The proof of the main result has two parts: showing that the invariant I(.) is (order-)complete, and describing its range.

The Cuntz semigroup is an isomorphism invariant for C*-algebras consisting of a semigroup with a compatible (though not algebraic) ordering. Its construction is similar to that of the Murray-von Neumann semigroup (from which the ordered K_0-group arises by the Grothendieck construction), but using positive elements in place of projections. Both rich in structure and sensitive to subtleties of the C*-algebra, the Cuntz semigroup promises to be a useful tool in the classification program for nuclear C*-algebras. It has already delivered on this promise, particularly in the study of regularity properties and the classification of nonsimple C*-algebras. The first part of this thesis introduces the Cuntz semigroup, highlights structural properties, and outlines some applications. The main result of this thesis, however, contributes to the understanding of what the Cuntz semigroup looks like for particular examples of (nonsimple) C*-algebras. We consider separable C*-algebras given as the tensor product of a commutative C*-algebra C_0(X) with a simple, approximately subhomogeneous algebra A, under the regularity hypothesis that A is Z-stable. (The Z-stability hypothesis is needed even to describe of the Cuntz semigroup of A.) For these algebras, the Cuntz semigroup is described in terms of the Cuntz semigroup of A and the Murray-von Neumann semigroups of C(K,A) for compact subsets K of X. This result is a marginal improvement over one proven by the author in [Tikuisis, A. "The Cuntz semigroup of continuous functions into certain simple C*-algebras." Internat. J. Math., to appear] (there, A is assumed to be unital), although improvements have been made to the techniques used. The second part of this thesis provides the basic theory of approximately subhomogeneous algebras, including the important computational concept of recursive subhomogeneous algebras. Theory to handle nonunital approximately subhomogeneous algebras is novel here. In the third part of this thesis lies the main result. The Cuntz semigroup computation is achieved by defining a Cuntz-equivalence invariant I(.) on the positive elements of the C*-algebra, picking out certain data from a positive element which obviously contribute to determining its Cuntz class. The proof of the main result has two parts: showing that the invariant I(.) is (order-)complete, and describing its range.