Research Programme

Academic Year 2010-2011

Dates: From January 10 to July 31, 2011                                            Link to the Weekly Seminar

Place: Centre de Recerca Matemàtica

Pere Ara, Universitat Autònoma de Barcelona

Francesc Perera, Universitat Autònoma de Barcelona

Nathanial P. Brown, Pennsylvania State University

Joachim Cuntz, Universität Münster

Marius Dardarlat, Purdue University

George Elliott, University of Toronto

Mikael Rørdam, University of Copenhagen

Andrew S. Toms, York University

Weekly Seminar

Workshop on Dynamics and C*-Algebras. From April 6 to 8, 2011

Conference on Structure and Classification of C*-Algebras. From June 6 to 10, 2011

Advanced Course on Dynamical Systems. From June 14 to 23, 2011

Keywords. Classification, amenable, C*-algebra, K-Theory, Cuntz semigroup, dynamical systems, Z-stability, strict comparison, purely infinite.

Overview. Classification is a central theme in mathematics. It has driven some of the most exciting developments of the 20th century, and is particularly prominent in the theory of operator algebras. The Elliott Program seeks classification of simple, separable and amenable C*-algebras by means of K-theoretical data. Following a remarkable success in the nineties, it has enjoyed a resurgence of late, owing to the discovery by M. Rørdam (and later, in a different direction, A. Toms) that K-theory alone, at least in its naive topological formulation, does not suffice for the classification of all simple separable amenable C*-algebras. Rather than being a detracting factor, this opens up two ways forward: restrict the class of C*-algebras considered, or enlarge the proposed invariant. Both of these courses have been pursued vigorously over the past three or four years, leading to several breakthroughs in Elliott’s program. An account of these goings-on can be found in the April 2008 issue of the Bulletin of the American Mathematical Society. We propose to gather the prime movers in Elliott’s program to tackle its most difficult remaining problems.

Background. The earliest classification results in the area of operator algebras are due to the founders of the theory, F. Murray and J. von Neumann. In the 1930s they proved that a von Neumann algebra—a weakly closed self-adjoint subalgebra of the bounded linear operators on Hilbert space—could be decomposed into atoms they called factors, and gave a broad classification of these factors into types. In the 1970s, A. Connes and U. Haagerup gave a complete classification of injective factors with separable predual, garnering a Fields Medal for Connes. Another Fields Medal was awarded to V. Jones for his work on subfactors, which led to powerful new tools in the classification theory of knots. Over the past two decades, much effort has been concentrated on the topic of this Research Program: G. A. Elliott’s program to classify separable amenable C*-algebras via K-theoretic invariants.

In 1976 Elliott, building on work of J. Glimm and O. Bratteli, gave a classification of locally finitedimensional C*-algebras in terms of their ordered K-theory. His discovery led him to conjecture, around 1989, that separable amenable C*-algebras would be classified by K-theoretic invariants. He gave great impetus to the conjecture by proving, with D. Evans, that the C*-algebras associated to an irrational rotation on the circle were so classified. His results were seized upon by several talented mathematicians who continue to work on Elliott’s program today. Their work has been recognised by several ICM talks and countless publications in elite journals. Highlights of the classification program include the Kirchberg-Phillips classification of purely infinite simple C*-algebras via graded topological K-theory, the Elliott-Gong-Li classification of simple approximately homogeneous C*-algebras of bounded dimension, and Lin’s work leading up to the classification of real rank zero C*-dynamical systems.

Objectives. Much effort has recently gone into the following question: “Can one characterise the largest class of simple separable amenable C*-algebras for which Elliott’s original conjecture holds?” Two properties stand out: Z-stability and strict comparison of positive elements.

A C*-algebra A is Z-stable if it absorbs the Jiang-Su algebra Z tensorially. The algebra Z was discovered by Jiang and Su in 1991 as an infinite dimensional C*-algebra that has the same K-Theory and traces as the complex numbers.

The interest in the property of Z-stability stems from the fact that taking a tensor product with Z is inert at the level of K-theory. Thus, Elliott’s original conjecture predicts that all simple separable amenable C*-algebras are Z-stable. The first examples of non-Z-stable C*-algebras were given by Villadsen, and they are the basis for the aforementioned counterexamples of Rørdam and Toms. The study of Z and its close cousins, the strongly self-absorbing C*-algebras, is currently a hot topic. Winter has proved that the C*-algebras associated to minimal uniquely ergodic diffeomorphisms satisfy the original Elliott conjecture whenever they are Z-stable, giving urgency to the difficult but plausible task of proving that such C*- algebras are always Z-stable. This is but one problem regarding Z-stability that we will attack during our program.

The second property of interest—strict comparison of positive elements—is related to an object called the Cuntz semigroup. This semigroup can be associated to any C*-algebra and consists of equivalence classes of positive operators. Its recent popularity is due to its extreme sensitivity as an invariant. It is able to distinguish simple separable amenable C*-algebras which have the same K-theory and tracial state space, and will form an essential part of any enlarged invariant for separable amenable C*-algebras. (Recent work of Ciuperca and Elliott has already shown it to be a complete invariant for a large class of non-simple C*-algebras, the so-called AI algebras.) The property of strict comparison says, roughly, that the natural partial order on the Cuntz semigroup is determined by states. This property often—conjecturally, always—implies that the Cuntz semigroup can be recovered functorially from K-theory and traces; the semigroup is otherwise very difficult to describe. This “nice” description of the Cuntz semigroup goes some way to explaining the success of Elliott’s program so far: his proposed invariant already contains the ultra-sensitive Cuntz semigroup in a codified (and surprising!) way.

We aim to improve our understanding of the Cuntz semigroup and the property of strict comparison during our program. Some specific projects include the following:

(i) When does a C*-algebra have strict comparison of positive elements? We have positive answers in the case of unital simple Z-stable algebras, and for the C*-algebras of minimal diffeomorphisms, but we have no information yet in the case of C*-algebras associated to discrete and symbolic dynamical systems.

(ii) To what extent is the Cuntz semigroup a classifying invariant for non-simple C*-algebras? The Ciuperca-Elliott classification shows that the Cuntz semigroup is a complete invariant for AI algebras, but we believe that these results can be extended dramatically. Specific cases to consider include crossed products of C(X) by a non-minimal homeomorphism, and the C*-algebras of higher rank graphs.

Perspectives. Our program seeks to solve important problems related to K-theory, the Cuntz semigroup, and Z-stability, with the ultimate goal of proving classification theorems for C*-algebras previously out of reach for Elliott’s program. Many of these problems have satisfying solutions when the C*-algebras in question are approximated locally by subalgebras whose finite-dimensional representations are of bounded dimension, but are poorly understood for the C*-algebras of discrete and dynamical systems. By bringing together top researchers in the theory of C*-algebras, we hope to create the right environment for a breakthrough in Elliott’s program of lasting importance.

Please, send your inquiries to Neus Portet at nportet@crm.cat 

Last updated on 04/01/2011