Course on Modern Theory of Cuntz Semigroups

Period: Tuesdays and Thursdays from February 13th to March 7th, 2024
Time Schedule: 11h to 13h
February: days 13, 15, 20, 22, 27 and 29
March: days 5 and 7

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.

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.

It is desirable that students have a background on aspects of functional analysis and ring theory. It is also desirable that students know what a C*-algebra is, as well as the construction of the Grothendieck group.

• Basic Theory of C*-algebras
  • Spectral radius
  • Finite dimensional algebras
  • Commutative algebras and the Gelfand representation
  • Hilbert spaces, the Gelfand-Naimark-Segal representation
• Projections, positive elements, and unitaries. \(\textbf{K}\)-Theory 
  • Comparison of projections. The Grothendieck group \(K_0\)
  • Comparison of positive elements
  • Traces, dimension functions
  • Unitaries. The group \(K_1\)
• The Elliott programme
  • The Elliott invariant and success of the programme
  • Counterexamples to the original conjecture
  • The Toms-Winter conjecture
• The Cuntz semigroup
  • Classical definition
  • Cuntz’s Theorem on existence of quasitraces
  • Non-continuity of the functor W(-), stable definition
• The Category Cu
  • Definition of the category and relation with C*-algebras
    • Compact containment
    • Axioms (O1)-(O4)
    • The category Cu and the Coward-Elliott-Ivanescu Theorem
    • Axiom (O5)
    • Limits in the category
    • The Completion functor: Cu(-) as a completion of W(-)
  • Examples and applications
    • Some examples
    • Brown-Perera-Toms’ Theorem for \(\mathcal Z\)-stable algebras
    • The relationship of the Cuntz semigroup with the Elliott invariant
    • Almost unperforation and almost divisibility: the relationship to the Toms-Winter conjecture
  • The Cuntz semigroup for C-algebras of stable rank one.
    • Additional properties: axioms (O6) and (O6+), inf-semilattice structure.
    • The Blackadar-Handelman conjectures
    • Glimm Halving
    • Realization of Ranks
  • Further categorical structure
    • The  \(\tau\)-construction: bivariant Cuntz semigroups
    • Closed monoidal structure
    • Products, coproducts, and ultraproducts. Preservation of such structures for C*-algebras.

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