Resumen de Un nou invariant per la classificació de c*-àlgebres
Laurent Cantier
The main objective of this thesis is the study of a new invariant for C*-algebras, called the Cu1 semigroup. The classification of C*-algebras has gained a lot of ground during the last decades. For that matter, two main objects stand out: the original Elliott invariant, consisting of K-Theoretical data, together with traces, and a pairing between them, and the Cuntz semigroup. The former has produced a tremendous breakthrough in the classification, mostly for simple C*-algebras.
The latter was introduced at the end of the 70's, as an analogous version of the Murray von-Neumann monoid, considering the set of positive elements instead of the set of projections. However, it is only very recently that this semigroup has been used as an invariant and it has provided promising results. On the one hand, for a rather large class of simple C*-algebras, it has been shown that these invariants determine one another in a functorial way. On the other hand, the Cuntz semigroup alone has become a useful tool in the classification of non-simple C*-algebra. The main drawback is that one usually has to restrict to the trivial K1 group case, since the Cuntz semigroup does not capture the homotopy group of unitaries in a C*-algebra. The aim of the thesis is to define an augmented version of the Cuntz semigroup, incorporating the K1 information of the C*- algebras and its ideals.
In a first part, we define our new invariant and describe its first properties: the Cu1 semigroup satisfies the Cuntz axioms and is continuous as a functor from the category of (separable) C*-algebras with stable rank one to a certain category o Cuntz semigroups. Then on, we determine the ideal lattice structure for the abstract semigroups in the latter category and we link it to the ideal lattice of the C*-algebra. We also obtain some exactness results, such as the fact that the Cu1 semigroup preserves canonical short- exact sequences of ideals. Further, we functorially recover the Cuntz semigroup and the K1 group from the Cu1 semigroup. Finally, we build an example of two C*-algebras A and B that have isomorphic Cuntz semigroup and isomorphic K1 group, even though they are non-isomorphic, since the Cu1 semigroup distinguishes them.
