The Cuntz semigroup

This workshop, sponsored by AIM and the NSF, will explore the Cuntz semigroup - an invariant of C*-algebras inspired by K-theory and recently shown to be important for classification.

In the 1970s Joachim Cuntz introduced this new invariant, based on a C$^*$-analogue of the comparison theory that was central to work of Murray and von Neumann. It initially received some attention, but soon fell out of favor with researchers. However, in the last decade interest has been renewed. Indeed, following Toms's use of this invariant to distinguish otherwise indistinguishable algebras, a number of researchers began studying the Cuntz semigroup and a flurry of papers soon followed.

The main goal of this workshop is to clarify the role of the Cuntz semigroup in the classification program, including a discussion of related (and relevant!) problems. More precisely, the following questions, among others, will be addressed:

1. How much information can be squeezed out of the Cuntz semigroup? For example, Elliott, Coward and Ivanescu have shown the Cuntz semigroup to be isomorphic to the semigroup of Hilbert modules; can this be used to prove new classification theorems? Does the Cuntz semigroup classify the "singular" cases (i.e. where the Elliott invariant is known not to be complete)?
2. When is the Cuntz semigroup functorially equivalent to other invariants, such as the Elliott invariant or Thomsen's invariant? For example, can we put the recent work of Ciuperca-Elliott and Elliott-Robert-Santiago into a common framework?
3. Conjecturally, strict comparison (a statement about the order structure of the Cuntz semigroup) is related to other important properties such as Z-stability or the classical notions of topological or mean dimension. Are these conjectures true?

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email workshops@aimath.org

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