# C*-Algebras Associated to Discrete and Dynamical Systems (08w5034)

Arriving in Banff, Alberta Sunday, January 27 and departing Friday February 1, 2008

## Organizers

Soren Eilers (University of Copenhagen)

(University of Toronto)

Alex Kumjian (University of Nevada, Reno)

Iain Raeburn (University of Otago, New Zealand)

Andrew Toms (Purdue University)

## Objectives

Evidently, the goal of this workshop is to connect researchers who are
working on the various types of C$^*$-algebras associated to discrete and
dynamical structures. The branches of this subject have, despite their common
roots in Bratteli diagrams and Cuntz-Krieger algebras, diverged both
mathematically and geographically. But they have much to offer each other
in terms of techniques and philosophy, if only the principals can be brought
under one roof. A BIRS workshop will enable an exchange of ideas whose scale
and range cannot be matched by small collaborations. Here is a sampling of the questions
which could be considered at such a workshop, but would be diffcult to consider
otherwise:
begin{itemize}
item When can a given C$^*$-algebra be modeled as the C$^*$-algebra of more than
one type of discrete or dynamical structure?
item Can one translate well known properties of one type of structure into properties
of another through their C$^*$-algebras? For instance, can flow equivalence be viewed as
a property of graphs? Or can it at least suggest the investigation
of graph properties not previously considered, and which might entail important
properties of the graph C$^*$-algebra?
item At what level of generality can we model classifiable C$^*$-algebras using
discrete and dynamical structures?
item Can tools for computing $mathrm{K}$-theory in one branch of the subject
be applied in others?
item Is it possible to construct, in the spirit of Ro rdam and Toms,
a C$^*$-algebra from a discrete or dynamical system
which is not amenable to classification via $mathrm{K}$-theory?
item What kind of noncommutative geometry can arise in such algebras? Can they give
essentially new examples of spectral triples?
item Can discrete models for certain classes of C$^*$-algebras be used to establish
semiprojectivity for such, generalising the work of Spielberg?
end{itemize}

The workshop will also have the advantage of bringing together researchers
who concentrate on one type of C$^*$-algebra, be it graph, $k$-graph,
dynamical system, shift space, etc., allowing rapid progress on outstanding
projects in each branch of the subject. These branches typically consist of 5-10
researchers, and so one could expect several of them to attend in their
entirety. Some of the projects here include:
begin{itemize}
item Can C$^*$-algebra techniques be used to obtain good descriptions
of flow equivalence classes of general shift spaces?
item Can simple AH algebras be modeled by topological $k$-graphs in
complete generality?
item What do the C$^*$-algebras of multiply generated dynamical systems
look like?
end{itemize}

A major goal of the proposed workshop is to involve young
mathematicians. Graph algebras and the
C$^*$-algebras of discrete structures have proved especially well-suited to
the task of exposing young mathematicians to problems of current interest in
operator algebras;
the cost of admission'' to operator algebras defined using discrete
structures is really quite low, with easy-to-understand definitions
and a wealth of examples. Our list of proposed participants will show
that our subject is very youthful indeed.

We expect the requested timing of the
workshop to be fortuitous for several post-doctoral
The meeting will also be timely from other points of view. Two recent conferences indicate that
there is much current international interest in our subject area, but these meetings were
too narrowly focused to achieve our aims. The CBMS conference at Iowa in 2004,
for example, was centred on a series of introductory lectures, and its
goal was primarily educational; a meeting on graph algebras and ring theory
at Malaga 2006 was only on the fringes of C$^*$-algebra theory.
Our workshop will improve dramatically upon these conferences,
and sustain a high level of research activity for a subject very much in vogue.
On another note, there will be a Thematic Program on Operator Algebras
at the Fields Institute from July to December, 2007. A BIRS workshop
in the spring, combined with the planned noncommutative geometry
activity at Fields and the annual Canadian Operator Symposium, would
encourage international researchers participating in the Fields program
to stay on into early 2008, with the obvious collaborative benefits.
Finally, we aim to be easy on your scheduling: we would not only be
amenable to an off-season'' meeting, but actually prefer it.

This meeting will help anchor future research in the
theory and application of C$^*$-algebras associated to discrete
and dynamical structures. It will bring young mathematicians
into contact with an active frontier in mathematics, and give