Operator algebras and dynamical systems from number theory (13w5152)

Arriving in Banff, Alberta Sunday, November 24 and departing Friday November 29, 2013


(Australian National University)

Marcelo Laca (University of Victoria)


In the last few years, there have been several exciting and promising developments on the interplay between
number theory and noncommutative geometry/operator algebras/dynamical systems. The program has moved
forward in several key directions, there are now several junior researchers who have been trained specifically
on the subject and, moreover some established researchers from either area have become involved and made
contributions to the subject. The aims of the workshop are
1) to bring together a fairly diverse group of researchers with expertise in the different strands of the field in
order to spark new ideas and directions.
2) to promote an efficient exchange of the recent results so that they can be interpreted from the various
perspectives, and
3) to provide an opportunity to researchers entering the field (either early career or established ones who
are reaching out to this area)
to share their results and keep abreast of recent developments.

There are common threads that need to be developed explicitly and thoroughly; a 5-day workshop at BIRS
will be the perfect forum and should lead to rapid progress.
There are some key developments which we see as both exciting and promising because they open new directions
and also because they may yield answers to some long-standing problems in number theory.
Results in which the guidance and the techniques of
noncommutative geometry/operator algebras lead to, or directly yield, conjectures or proofs of statements
formulated entirely within number theory are central to the effort.
We are beginning to see evidence that these results are at hand.
For example this has been achieved in recent work of Cornelissen-Marcolli and also
of Cuntz-Laca-Deninger and we would like to accelerate this process by enhancing the communication
between the key players.

Among the current developments we count:

1) The C*-algebras associated by Cuntz and Li to algebraic number fields.
These are similar in spirit to those constructed by Bost-Connes and Connes-Marcolli, but appear to be
more tractable in terms of K-theory. We expect that the application of the associated invariants to number
theoretic questions will be fruitful.

2) The C*-algebras and dynamical systems studied by Carey, Rennie, Phillips and Putnam. These involve
similar crossed products to the above ones, and, in fact, yield Cuntz's $C*$-algebras $O_n$ and $Q_n$
as particular cases. They have interesting applications involving KMS states and modular index theory
which have the potential to bear fruit in a number theoretic context.

3) The study of the equilibrium states and symmetries of the Toeplitz algebra of the affine semigroup of
the natural numbers, by Laca and Raeburn. They highlight the role of the Toeplitz extensions and their
realisation in terms of product systems of Hilbert bimodules as the natural setting for a very rich
KMS state structure.

4) Recent results of Cornelissen and Marcolli that use Bost-Connes systems as an invariant for number
fields to prove, among other things, the isomorphism of number fields that have isomorphic maximal
abelian Galois groups with compatibly isomorphic semigroups of integral ideals.

5) New results of Laca, Neshveyev, and Trifkovic and of Yalkinoglu on induction of Bost-Connes systems,
with applications to functoriality, the connection with Hecke algebras, and the elusive arithmetic subalgebras
that make a Bost-Connes system relevant for explicit class field theory

6) Recent work of Cuntz, Deninger, and Laca on C*-algebras from rings of algebraic integers.
These are Toeplitz-type extensions of the algebras of Cuntz and Li which generalise the Toeplitz
algebra of Laca and Raeburn. New results on the asymptotics of partial zeta functions arising in
this work were inspired by the operator algebraic paradigm of uniqueness of equilibrium at high temperature.

7) Recent work of Connes and Consani on the L-functions for fields of characteristic $p$ using generalised
Bost-Connes systems.

We expect to have a highly interactive, workshop focused on the above topics. Open questions that might
be investigated in the workshop include:

(a) To explore further the relationship of Toeplitz algebras such as those from (3) and (6) to the algebras
of Carey, Rennie, Phillips and Putnam, in particular, to decide whether the
latter can be obtained as quotients of (subalgebras or other variation of) the former, along the same lines
in which the C*-algebras associated to rings of integers by Cuntz and Li in (1) above are quotients of
the Toeplitz type algebras from (6).

(b) While some of the results of Cornelissen and Marcolli do not have Operator Algebras or Noncommutative Geometry
in their statement, these methods play a key role in the motivation and in the proof. We see this as a key feature because
it is the realisation of a long standing hope for the application of new analytic methods from Operator Algebras, namely
to shed light on questions that have not been answered by traditional methods.

(c) The research in (5) (6) and (7) is still very recent, however by 2013 its implications will begin to have impact. Results
proved there elucidate the relation between various constructions of C*-algebraic dynamical systems from number fields
(Hecke systems and Bost-Connes type systems)
and produce new systems with still unexplored behavior.
This is important because although the analysis of equilibrium has been completed on the side of Bost-Connes type systems,
the Hecke systems come with natural presentations derived from the multiplication table of double cosets, and this has the
potential to shed light on the Galois symmetries and thus on the connection with Hilbert's 12th problem of explicit class field
theory. One focus point of this aspect would be the results of Yalkinoglu on arithmetic models of Bost Connes systems.

(d) To investigate further the functoriality of Bost-Connes systems and the relation to other C*dynamical systems from number
theory. In particular, we would like to understand better the functoriality observed by Cuntz, Deninger and Laca for their Toeplitz-type
construction in contrast to the lack of functoriality of the Cuntz-Li construction. This is a key issue that will have further applications.
At the level of Bost-Connes systems the initial difficulties arising from wrong way maps have been resolved and the appropriate
morphisms turn out to be equivariant C*-correspondences (via right-Hilbert bimodules), in particular, they are not homomorphisms
of C*-algebras as one may have expected, which highlights the role of noncommutative geometry.

(e) The results of Connes and Consani advance the whole program through showing that the
Bost-Connes technology generalises to study classical L-functions.

Consequently the time seems ripe for a workshop in which the key people involved in this research
and some experts in related areas as well as their students and postdoctoral fellows get
together to disseminate their results, exchange ideas, connect the different strands that are being developed
and spark new results and collaborations.
The participants we are planning to invite include several women and junior mathematicians; we also plan to include a few
graduate students who would be writing their thesis at the time the workshop is held.