## Departmental PhD Thesis Exam – Aaron Tikuisis

Monday, May 30, 2011, 4:10 p.m., in BA 6183, 40 St. George St.

PhD Candidate:  Aaron Tikuisis

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.