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Monday, May 30, 2011, 4:10 p.m., in BA 6183, 40 St. George St. PhD Candidate: Aaron Tikuisis PhD Advisor: George A. Elliott PhD Thesis Title: The Cuntz Semigroup of C(X,A) (http://www.math.toronto.edu/aptikuis/thesis.pdf) PhD Thesis Abstract: The Cuntz semigroup is an isomorphism invariant for C*-algebras comprised of a semigroup together with a compatible (though not algebraic) ordering. It is constructed akin to the Murray-von Neumann semigroup (from which the ordered K_0-group arises by the Groethendieck construction), but using positive elements in place of projections and with a pre-order which generalizes Murray-von Neumann equivalence on projections. Both rich in structure and sensitive to subtleties of the C*-algebra (especially those relating to ideals), the Cuntz semigroup promises to be a useful tool in the classification program for nuclear C*-algebras. It has already delivered on this promise, contributing particularly in the study of regularity properties and in the classification of nonsimple C*-algebras. The first part of this thesis introduces the Cuntz semigroup, highlights structural properties, and delves into an exposition of 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 seperable 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 get a clear picture 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 A. 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 the techniques in the proof have been added to and improved upon. 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 appears absent in the literature, and is therefore 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 a positive element which obviously contribute to determining its Cuntz class. The proof of the min result is divided into two parts: showing that the invariant I(.) is (order-) complete, and describing its range.

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