The Cuntz semigroup is a powerful invariant for C*-algebras introduced by Joachim Cuntz at the end of the seventies. It is build out of the so-called positive elements of a C*-algebra in a manner akin to the construction of the projection semigroup, out of which the Grothendieck group stems. However, its structure is far more complex, as it becomes an abelian semigroup with a non-algebraic order. Its order is the key feature that was used by A. Toms in 2008 to distinguish two C*-algebras that were otherwise the same under the light of K-Theory, traces, and many other continuous, homotopy stable invariants. This gave a huge impetus to the ongoing classification programme of simple, separable, nuclear C*-algebra launched by G. A. Elliott. In fact, suitably interpreted, the Cuntz semigroup is a classifying invariant equivalent to the so-called Elliott invariant.
Over time, it has proved itself to be an essential object in our understanding of structural properties of C*-algebras. Indeed, many aspects in the theory can be codified in algebraic properties of this semigroup. For example, the fact that its order can be read off from suitable functionals is one of the ingredients of the so-called Toms-Winter conjecture, that predicts that three formally disparate conditions are in fact equivalent.

The Cuntz semigroup lies in a category whose structure has been analysed in a lot of detail over the last years. Constructions in this category have been reflected back to C*-algebras and the Cuntz semigroup factor has been shown to preserve many of these constructions. The techniques introduced have allowed us to gain new understanding of the class of C*-algebras of stable rank one, solving open problems within this class. More recently, it also shows to be an essential ingredient in describing crossed products by certain actions of groups. Such crossed products are important examples that in many cases might satisfy additional properties that make them amenable to classification. This is therefore a vibrant area of research that continues to expand and produce exciting new results.

ORGANISING COMMITTEE

Ramon Antoine | Universitat Autònoma de Barcelona – Centre de Recerca Matemàtica
Associate Professor in the Department of Mathematics at the Universitat Autònoma de Barcelona specializing in the structure and classification of rings, modules, and C*-algebras, with a particular focus on their interactions with dynamics, combinatorics, and topology.

Laurent Cantier | Universitat Autònoma de Barcelona – Institut of Mathematics of the Czech Academy of Sciences
Margarita Salas postdoctoral fellow affiliated with the Institut of Mathematics at the Czech Academy of Sciences and the Department of Mathematics at the Universitat Autònoma de Barcelona. He earned their Ph.D. in 2020 from the UAB under the supervision of Dr. Antoine and Dr. Perera. His research lies in the classification of C*-algebras within Operator Algebras. He is geared towards algebraic aspects of the theory such as the study of the Cuntz semigroup (and some of its variations) while being attracted to construct tools for classification via Category Theory.

Francesc Perera | Universitat Autònoma de Barcelona – Centre de Recerca Matemàtica
Associate Professor in the Department of Mathematics at the Universitat Autònoma de Barcelona and the current Managing Editor of the journal Publicacions Matemàtiques. His research interests are in Operator Algebras, Noncommutative Algebra, Semigroup Theory, and the interplay between these; more specifically, he is interested in the structure of nuclear C*-algebras and their classification, with particular emphasis on the study of invariants such as K-Theory and the Cuntz semigroup. Moreover he is involved in the connections of C*-algebras with Dynamical Systems and with more general algebraic structures, such as Steinberg algebras and Leavitt Path algebras.

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