# The Cuntz semigroup

November 2 to November 6, 2009

at the

American Institute of Mathematics, Palo Alto, California

organized by

Nate Brown, George Elliott, and Andrew Toms

## Original Announcement

This workshop 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?

## Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Some basic facts about the Cuntz semigroup (link to latex file)