# Classification of amenable C*-algebras (10w5092)

Arriving in Banff, Alberta Sunday, September 19 and departing Friday September 24, 2010

## Organizers

(Purdue University)

Soren Eilers (University of Copenhagen)

(University of Toronto)

(University of Copenhagen)

Andrew Toms (Purdue University)

## Objectives

Specific projects

The first project which we expect to further at our meeting relates to crossed product C*-algebras. Given a C*-algebra $A$, a (typically discrete countable) group $G$, and a homomorphism $alpha: G to mathrm{Aut}(A)$, we may form the crossed product $A times_alpha G$, a universal C*-algebra containing $A$ in which the action of $G$ is implemented by unitary operators. If both $A$ and $G$ are amenable, then so is $A times_alpha G$, but it is very difficult to understand the structure of $A times_alpha G$ in general. For instance, Connes observed some 20 years ago that if $A = mathrm{C}(mathrm{S}^{2n+1})$ and $alpha: mathbb{Z} to mathrm{Aut}(mathrm{C}(mathrm{S}^{2n+1}))$ is given by $z mapsto gamma^z$ for some automorphism $gamma$ induced by a minimal diffeomorphism, then the crossed products $mathrm{C}(mathrm{S}^{2n+1}) times_alpha mathbb{Z}$ all have the same $mathrm{K}$-theory. This invited the question of whether the algebras were in fact isomorphic, a question answered only recently by Winter using results of H. Lin, Q. Lin, and N. C. Phillips. Winter's breakthrough has led to huge progress on the structure of crossed products. We aim to confirm Elliott's classification for all crossed products $mathrm{C}(X) times_alpha mathbb{Z}$, where $alpha$ is given by a minimal homeomorphism of mean dimension zero (this last condition is known to be necessary for $mathrm{K}$-theoretic classification). We also expect to make progress on the structure of more general crossed products by finding a replacement in countable discrete amenable groups for a technique of Berg used in crossed products by $mathbb{Z}$. New results due to Orfanos give hope for success in this direction.

A second project concerns the classification of C*-algebras which are either nonsimple, or not classifiable by $mathrm{K}$-theory and traces (the latter invariant was successful in classifying simple C*-algebras throughout the 1990s and early 2000s, but was shown by M. Rordam in 2002 to be insufficient among all simple amenable C*-algebras). Examples of both types of algebras abound. They may arise from symbolic dynamics, topological dynamics of nonzero mean dimension, extensions, various inductive limit constructions, and continuous fields of C*-algebras. Recent work on Universal Coefficients Theorems for the Kasparov-Kirchberg bivariant $mathrm{K}$-theory of C*-algebras with finite ideal lattices by Bonkat, Meyer, Nest, Restorff, and Ro rdam, has led to strong classification results for extensions of C*-algebras which were previously known only in the simple case. A goal of the workshop will be to discover how general an ideal lattice may be while still admitting a Universal Coefficients Theorem (it is known that limits exist), and to apply these results to classification problems in nonsimple C*-algebras. (These applications depend on Kirchberg's powerful lifting results for ideal-related $mathrm{KK}$-elements.) Another goal will be to understand the role of the Cuntz semigroup in the classification theory of C*-algebras. This semigroup is related to the $mathrm{K}_0$-group, but instead of recording the structure of finitely generated projective modules over a C*-algebra, it records the structure of its countably generated modules. It has been surprising to discover how sensitive this invariant is. For instance, it suffices for the classification of all inductive limits of C*-algebras of the form $F otimes mathrm{C}([0,1])$, with $F$ finite-dimensional, giving a broad generalisation of Elliott's seminal classification in the unital simple case c. 1990. Two goals of the workshop related to the Cuntz semigroup will be to prove the first classification theorem for simple C*-algebras which are not classifiable by $mathrm{K}$-theory and traces, and to understand the interaction of this semigroup with the algebraic $mathrm{K}_1$-group of a C*-algebra (this interaction is known to exist, and to be relevant to the question of isomorphism of C*-algebras). Finally, we will seek to extend recent results on the structure of continuous fields of C*-algebras. These fields---bundles of C*-algebras, if you like---have been shown to be trivial whenever their base spaces are finite-dimensional and their fibres are all isomorphic to "very well behaved" C*-algebras.

Our third project is one of unification. There is a well developed theory of C*-algebras associated to graphs and their higher rank analogues. These algebras are always separable and amenable, and are known to be classifiable by $mathrm{K}$-theory in the case of simple algebras associated to graphs. There is strong evidence that the C*-algebras of higher rank graphs are also so classifiable, but the principals working on the general structure of these algebras are oddly disjoint from the principals working on Elliott's conjecture. We will ensure that both camps attend our workshop, in order to finally attack the classification problem for the C*-algebras of higher rank graphs. We feel that recent advances in the theory of covering dimension for C*-algebras will allow us to make significant progress.

Finally, we have a project which seeks to understand the foundations of classification-by-$mathrm{K}$-theory. As mentioned above, $mathrm{K}$-theory and traces do not suffice for the classification of all simple separable amenable C*-algebras. But for what subclass $mathcal{C}$ 'do' they suffice? Toms and Winter have conjectured that there are three equivalent characterisations of $mathcal{C}$: one operator algebraic, one homological, and one topological. The three conditions may be viewed as the minimal instances of three 'a priori' disparate notions of dimension for amenable C*-algebras. Some implications are known to hold, and we seek to establish the remainder. The equivalence of the operator algebraic and homological conditions will provide a broad generalisation of Kirchberg's celebrated characterisation of simple nuclear purely infinite C*-algebras.

Invitees

There are more than 100 faculty, postdoctoral fellows, and senior graduate students working partly or wholly on the classification theory of amenable C*-algebras. They span the globe: Australia, Brazil, Britain, Canada, China, Denmark, France, Germany, Japan, Korea, Mexico, Norway, Scotland, Spain, and the United States. This endeavour is moreover a Canadian creation, seeded by George Elliott at the end of the 1980s.

We plan to invite roughly 25-30 senior scholars. These will include (in addition to the organisers) Claire Anantharaman-Delaroche (Orleans), Etienne Blanchard (Jussieu), Berndt Brenken (Calgary), Nate Brown (Penn State), Joachim Cuntz (Munster), Siegfried Echterhoff (Munster), Thierry Giordano (Ottawa), Guihua Gong (Puerto Rico), Astrid an Huef (Newcastle), Masaki Izumi (Kyoto), David Kerr (Texas A&M), Eberhard Kirchberg (Berlin), Nadia Larsen (Oslo), Liangqing Li (Puerto Rico), Huaxin Lin (Oregon), Ralf Meyer (G"ottingen), Ryszard Nest (Copenhagen), Francesc Perera (Barcelona), N. Christopher Phillips (Oregon), Ian Putnam (Victoria), Iain Raeburn (Wollongong), Wilhelm Winter (Nottingham), and Simon Wassermann (Glasgow). This list includes four ICM speakers (two among our organisers), present or erstwhile editors of
Crelle's Journal, Journal of Functional Analysis, Canadian Journal of Mathematics, Documenta Mathematica, Transactions of the American Mathematical Society, and Journal of Noncommutative Geometry, winners of the Andre-Aisenstadt Prize, the CRM-Fields-PIMS prize, the Leibniz Prize, the Medal of the College de France, and the holders of two Tier I Canada Research Chairs.

Among junior faculty and postdoctoral fellows we will include Dawn Archey (Ber Sheva), Alin Ciuperca (Fields Institute), Ilan Hirshberg (Ber Sheva),
Takeshi Katsura (Keio), Zhuang Niu (Oregon), Gunnar Restorff (Torshavn), Benjamin Itza-Ortiz (Hidalgo), Stefanos Orfanos (Cincinnati), Leonel Robert (York), Efren Ruiz (Hilo), Luis Santiago (Fields Institute), and Stuart White (Glasgow). We also plan to have selected senior graduate students in attendance: Sara Arklint (Copenhagen), Tao Lei (York), Randi Rohde (Odense), Adam Sierakowski (Copenhagen), and Aaron Tikuisis (Toronto), Selke-Kristin Zschauer (Munster), and others who distinguish themselves between now and the time of our workshop. (Already we have listed more than 42 participants, but one has to overbook.)

We would like to point out that in addition to US and Canadian researchers, we will have Mexican participants (Itza-Ortiz and his students). We hope that this will help to maintain CONACYT's funding commitment to BIRS. Also, the majority of the invitees above were contacted in August 2008 at the Oberwolfach conference on C*-algebras, and are committed to attending our BIRS workshop.

Timing and timeliness

2010 will be a busy year for researchers in the classification theory of C*-algebras worldwide. The Centre de Recerca Mathematica (CRM) in Barcelona
will host a three month program on operator algebras, their classification, and their connections to dynamics from January to March, and will host a ten day Advanced Course on the classification theory of operator algebras in June. There is also an application afoot for a PIMS Summer School in Noncommutative Geometry at Victoria in July, which will have a 10 hour course on the classification theory of C*-algebras. (And just this year, incidentally,
the Universidad Internacional Menendez Pelayo, Santander, Spain, hosted a Summer School on the Cuntz semigroup---see the second of the specific projects
above.) These activities evidence the high profile of our workshop's topic, and make 2010 a particularly good time for BIRS to remind the operator
algebras community of Canada's primacy in this area. Let us note that the CRM program can manage at most 4-6 researchers at one time, and will therefore be unable to achieve the goals that we have set out in section 2. for reasons of scale. The other 2010 activities have an educational theme. Our research-oriented BIRS workshop will both complement these activities and be primed for success by them.

We believe that a meeting in the August-December range will be best for us, as it will allow graduate students and postdoctoral fellows to make the most of
2010's other operator algebras activities, particularly the CRM Advanced Course and the PIMS Summer School. They can then bring their ideas, enthusiasm, and
good grounding to our workshop. Another time period which would suit us is April through mid-May. All other times of the year will conflict with the already planned activities listed above, although we could perhaps live with a week in the January-March range.

Summary

The classification theory of C*-algebras has Canadian origins and global reach. It has seen several breakthroughs in the past five years, leading to high profile publications and support from mathematical research institutes the world over (CRM Barcelona, PIMS, Oberwolfach, Universidad Internacional Menendez Pelayo). We believe that 2010 is the ideal time for a 5-day BIRS workshop on this topic.