
THE
FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES 
JanuaryJune
2014
Thematic Program on Abstract Harmonic Analysis,
Banach and Operator Algebras
June
2014
Theme Period on
C*Algebras and Dynamical Systems
Organizer:
George Elliott



Week
of June 26: (video
of the talks)

Monday

Tuesday

Wednesday

Thursday

Friday


Room 230 
Room 230 
Room 230 
Room 230 
Stewart
Library 
11:00  12:00

David Kerr,
Texas A&M University
Dynamical systems and C*algebras

Thierry Giordano
University of Ottawa
Dynamical systems and C*algebras

David Kerr
Texas A&M University
Dynamical systems and C*algebras

Thierry Giordano,
University of Ottawa
Dynamical systems and C*algebras

David Kerr
Texas A&M University
Dynamical systems and C*algebras


Room 230 
Room 230 
Room 230 
Room 230 
Room 230 
2:00  3:30

Andrew Toms
Purdue University
Finite TomsWinter C*algebras

Andrew Toms
Purdue University
Finite TomsWinter C*algebras

Andrew Toms
Purdue University
Finite TomsWinter C*algebras

Andrew Toms
Purdue University
Finite TomsWinter C*algebras

*2:00  3:00*
Thierry Giordano
University of Ottawa
Dynamical systems and C*algebras

back
to top
Minicourse Week of June 913: (video
of the talks)

Monday

Tuesday

Wednesday

Thursday

Friday

11:00  12:00

Andrew Toms
Purdue University
Finite TomsWinter C*algebras

Andrew Toms
Purdue University
Finite TomsWinter C*algebras

Andrew Toms
Purdue University
Finite TomsWinter C*algebras

Andrew Toms
Purdue University
Finite TomsWinter C*algebras

N. C. Phillips
University of Oregon
The proof of the classification theorem for UCT Kirchberg
algebras

2:00  3:00

N. C. Phillips
University of Oregon
The proof of the classification theorem for UCT Kirchberg
algebras

N. C. Phillips
University of Oregon
The proof of the classification theorem for UCT Kirchberg
algebras

N. C. Phillips
University of Oregon
The proof of the classification theorem for UCT Kirchberg
algebras

N. C. Phillips
University of Oregon
The proof of the classification theorem for UCT Kirchberg
algebras

Eberhard Kirchberg
HumboldtUniversität zu Berlin
The proof of the classification theorem for UCT Kirchberg
algebras

back to top
June 1620,
2014 Workshop (abstracts)
Video of the talks
Organizers: Zhuang Niu and Luis Santiago
C*Algebras
and Dynamical Systems Abstracts
Joan Bosa
The category Cu. Which maps are the correct ones? *homomorphisms
or cpc order zero maps? (slides)
In this talk we focus on the fact
that the map induced by a cpc order zero $\varphi : A \to
B$ in the category Cu does not preserve the compactly containment
relation. In particular, these kinds of maps are not in
the category Cu, so that, in general, they may not be used
in the classification of C*algebras via the Cuntz Semigroup.
Nevertheless, there is a subclass of these maps which preserves
the relation, and so they can be used in the above mentioned
classification. Our main result characterizes these maps
via the positive element induced by the description of cpc
order zero maps shown in [1]}.
References
[1] Winter, W. and Zacharias, J.,Completely positive maps
of order zero,
Munster J. Math., 2, 2009, 311324.
Julian Buck
Large Subalgebras of C*Algebras
We survey work in progress by Dawn Archey, N. Christopher
Phillips, and myself on various formulations of large
subalgebras of C*algebras. Such definitions provide abstract
formulations of the properties observed in the approximating
subalgebras used to study transformation group C*algebras.
Applications to the structure of crossed products will
be presented.
Eusebio Gardella
The continuous Rokhlin property and permanence of
the Universal Coefficient Theorem
We define a continuous analog of the Rokhlin
property for circle actions, asking for a continuous path
of unitaries instead of a sequence. Besides being classifiable,
these actions enjoy a number of nice properties that do
not hold in general for Rokhlin actions. This talk will
focus on the connections between the continuous Rokhlin
property and Etheory, with the goal of showing that if
$\alpha\colon \mathbb{T}\to \mbox{Aut}(A)$ is an action
with the continuous Rokhlin property on a nuclear C*algebra
$A$, then $A$ satisfies the UCT if and only if the fixed
point algebra satisfies the UCT, if and only if the crossed
product satisfies the UCT.
Guihua Gong
TBA
Ilan Hirshberg
TBA
Benjamin Johannesen
The core of a certain oriented transformation groupoid
algebra
In a recent work by Thomas Schmidt and Klaus
Thomsen on $C^\ast$algebras arising from circle maps, they
introduced orientation preserving groupoids as an intermediate
step. It was shown, under some assumptions on the circle
maps, that the oriented transformation groupoid algebras
introduced there are classifiable by Ktheory due to the
KirchbergPhillips classification theorem. In the same
spirit, the cores of the oriented transformation groupoid
algebras, i.e., the fixed point algebras of a gauge action
on the oriented transformation groupoid algebras, are classifiable
by ordered Ktheory using a classification result by Andrew
Toms.
back
to top
Eberhard Kirchberg
Filling families and strong pure infiniteness of some
endomorphism crossed products
We give first a short overview on the 
possibly different  notions of pure infiniteness and describe
then a method to prove with help of the study of a ``filling
family'' ``strong'' pure infiniteness, or of its permanence
under the operations like e.g. tensor products or endomorphism
crossed products. Always with some additional assumptions
like e.g.\ exactness and generalized Rokhlin type conditions.
Related are Sections 3 and 6 of the joint work with A.Sierakowski:
``Strong pure infiniteness of crossed products''
Huaxin Lin
Crossed products from minimal dynamical systems on connected
odd dimensional spaces (slides)
Let : S^{2n+1} > S^{2n+1} be a minimal
homeomorphism (n a natural number ). We show that the crossed
product C(S^{2n+1})x Z has rational tracial rank at most
one.
More generally, let M be a connected compact metric space
with finite covering dimension and with H^1(M, Z)=0. Suppose
that K_i(C(M))=Z\oplus G_i for some finite abelian group
$G_i,$ $i=0,1.$ Let M > M be a minimal homeomorphism.
We also show that A=C(M) x Z has rational tracial rank at
most one and is classifiable. In particular, this applies
to the minimal dynamical systems on odd dimensional real
projective spaces.
Terry Loring
Homtopy classification of freeparticle, gapped,
finite systems
We consider the problem of three matrices,
two unitaries $U$ and $V$ that commute with each other and
the third matrix being Hermitian, with bounds on $\H\\leq
L,\H^{1}\\leq L$, so that $H$ almost commutes with $U$
and $V$. When can we deform such a system continuously,
keeping all the exact and almost keeping the approximate
relations, so that at the end of the path we have three
commuting matrices? The answer, not surprisingly, has to
do with $K$theory. The problem is inspired by finite models
of topological insulators and superconductors. To address
all the flavors of topological insulators and superconductorswe
need to consider one or two real structures on $\mathbf{M}_{n}(\mathbb{C})$and
consider invariants in both $KU$ and $KO$.
This talk includes many joint results with Adam S\o rensen.
Martino Lupini
Conjugacy and cocycle conjugacy of automorphisms of O2
are not Borel
I will present the result, obtained in joint
work with Eusebio Gardella, that the relations of conjugacy
and cocycle conjugacy of automorphisms of the Cuntz algebra
O2 are not Borel. I will focus on the motivations and implications
of such result, and I will provide the main ideas of the
proof. No previous knowledge of Borel complexity theory
will be assumed.
James Lutley
TBA
Cornel Pasnicu
Permanence properties for crossed products and fixed point algebras
of finite groups (slides)
For an action of a finite group on a C*algebra, we present
some conditions under which properties of the C*algebra
pass to the crossed product or the fixed point algebra.
We mostly consider the ideal property, the projection property,
topological dimension zero, and pure infiniteness. In many
of our results, additional conditions are necessary on the
group, the algebra, or the action. Sometimes the action
must be strongly pointwise outer, and in a few results it
must have the Rokhlin property. When the group is finite
abelian, we prove that crossed products and fixed point
algebras preserve topological dimension zero with no condition
on the action. We give an example to show that the ideal
property and the projection property do not pass to fixed
point algebras (even for the two element group). The construction
also gives an example of a C*algebra which does not have
the ideal property but such that the algebra of 2 by 2 matrices
over it does have the ideal property; in fact, this matrix
algebra has the projection property. This is joint work
with N. Christopher Phillips, and it will be published in
the Transactions of the A.M.S.
Ulrich Pennig
Topological invariants of C(X)algebras
I will report on joint work with Marius Dadarlat. We
showed that the DixmierDouady theory of continuous fields
of C*algebras with compact operators as fibers extends
to a more general theory of fields with fibers stabilized
strongly selfabsorbing C*algebras. The classification
of the corresponding locally trivial fields involves a
generalized cohomology theory obtained from the unit spectrum
of topological KTheory, which is computable via the AtiyahHirzebruch
spectral sequence. An important feature is the appearance
of characteristic classes in higher dimensions. We found
a necessary and sufficient Ktheoretical condition for
local triviality of these continuous fields over spaces
of finite covering dimension. If time permits I will also
explain how the torsion elements in the classifying generalized
cohomology group arise from locally trivial fields with
fibers isomorphic to matrix algebras over the strongly
selfabsorbing algebra.
Francesc Perera
Structural aspects of the category Cu
The Cuntz semigroup $W(A)$ of a C$^*$algebra
$A$ is an important ingredient, both in the structure theory
of C$^*$algebras, and also in the current format of the
Classification Programme. It is defined analogously to the
Murrayvon Neumann semigroup $V(A)$ by using equivalence
classes of positive elements. The lack of continuity of
$W(A)$, considered as a functor from the category of C$^*$algebras
to the category of abelian semigroups, led to the introduction
(by Coward, Elliott and Ivanescu) of a new category Cu of
(completed) Cuntz semigroups. They showed that the Cuntz
semigroup of the stabilized C$^*$algebra is an object in
Cu and that this assignment extends to a sequentially continuous
functor.
We introduce a category W of (precompleted)
Cuntz semigroups such that the original definition of Cuntz
semigroups defines a continuous functor from local C$^*$algebras
to W. There is a completion functor from W to Cu such that
the functor Cu is naturally isomorphic to the completion
of the functor W. Using this, we show that the functor Cu
is continuous.
We also indicate how the category Cu should be recasted,
by adding additional axioms. If time allows, we will discuss
the construction of tensor products in the category Cu.
(This is joint work with Ramon Antoine and
Hannes Thiel.)
back to top
Henning Petzka
Infinite multiplier projections and dichotomy of
simple C*algebras
We study infiniteness of multiplier
projections of a stabilized C*algebra and the connection
to dichotomy of C*algebra A in the sense of A being
either stably finite or purely infinite. We discuss
when all infinite multiplier projections are equivalent
to the multiplier unit, and we reduce the dichotomy
problem for real rank zero algebras to a property on
multiplier projections, which could possibly hold for
a general separable C*algebra.
Yasuhiko Sato
Thomas Schmidt
$C^*$algebras from noninjective circle maps
Consider a continuous surjection of the circle which is
piecewise monotone, but not locally injective. To this,
we associate a locally compact étale groupoid 
the so called \emph{amended transformation groupoid} 
and study the relationship between the dynamical system,
the groupoid, and the associated groupoid $C^*$algebra.
Under modest assumptions on the dynamics, we apply the work
of Katsura on $C^*$correspondences to develop an algorithm
that reduces calculating the $K$theory of the $C^'$algebra
to elementary linear algebra. This is joint work with Klaus
Thomsen.
Adam Sierakowski
Purely infinite $C^*$algebras associated to etale groupoids
(slides)
Let $G$ be a Hausdorff, etale groupoid that is minimal
and topologically principal. We show that $C^*_r(G)$ is
purely infinite simple if and only if all the nonzero
positive elements of $C_0(G^0)$ are infinite in $C_r^*(G)$.
If $G$ is a Hausdorff, ample groupoid, then we show that
$C^*_r(G)$ is purely infinite simple if and only if every
nonzero projection in $C_0(G^0)$ is infinite in $C^*_r(G)$.
We then show how this result applies to $k$graph $C^*$algebras.
Finally, we investigate strongly purely infinite groupoid
$C^*$algebras. This is joint work with Jonathan Brown
and Lisa Orloff Clark.
Adam Sørensen
Nuclear dimension of UCTKirchberg algebras
When Winter and Zacharias introduced nuclear dimension
they showed that the Cuntz algebras have nuclear dimension
1. Recently, Tomforde, Ruiz and Sims adapted the techniques
developed by Winter and Zacharias to show that all purely
infinite graph algebras with finitely many ideals have
nuclear dimension 1. We will explain these techniques
and how they extend to 2graph algebras. This will lets
us show that certain tensor products of UCTKirchberg
algebras have nuclear dimension 1. Combined with results
of Enders and a direct limit argument we get that all
UCTKirchberg algebras have nuclear dimension 1.
This is joint work with Ruiz and Sims.
back to top
Nicolai Stammeier
On C*algebras of right LCM semigroups
Xin Li's construction of C*algebras for arbitrary leftcancellative
semigroups S has raised interest in semigroup C*algebras
over the last years. Right LCM semigroups constitute a
large class of leftcancellative semigroups. For instance,
it encompasses semigroups associated to selfsimilar actions
and suitable semidirect products of groups by semigroups.
In this talk I will indicate how the right LCM property
simplifies the study of the full semigroup C*algebra
C*(S). This leads to a uniqueness theorem for C*(S) based
on its diagonal subalgebra (in the spirit of a result
by LacaRaeburn for quasilattice ordered groups from
1996). As a byproduct, we obtain a criterion to ensure
that C*(S) is purely infinite simple. I will discuss several
examples arising as semidirect products of groups by semigroups.
This is joint work with Nathan Brownlowe and Nadia S.
Larsen.
Karen Strung
TBA
Wojciech Szymanski
Product systems and dynamics
I will present some recent work on product systems of
Hilbert bimodules and their corresponding C*algebras.
The focus will be on dynamical properties, including topological
aperiodicity and (time permitting) KMS states. The talk
will contain some results obtained in collaboration with
Jeong Hee Hong, Bartosz K. Kwasniewski and Nadia S. Larsen.
Gabor Szabo
Rokhlin dimension for certain residually finite groups
(slides)
In 2012, Ilan Hirshberg, Wilhelm Winter and Joachim Zacharias
introduced the concept of Rokhlin dimension for actions
of finite groups and the integers. Shortly thereafter,
this was adapted to actions of Z^m. The main motivation
for introducing this concept was that actions with finite
Rokhlin dimension preserve the property of having finite
nuclear dimension, when passing to the crossed product
C*algebra. Since then, this has been successfully used
to verify finite nuclear dimension for a variety of nontrivial
examples of C*algebras, in particular transformation
group C*algebras. We extend the notion of Rokhlin dimension
to cocycle actions of countable, residually finite groups.
If the group in question has a box space of finite asymptotic
dimension, then one gets an analogous permance property
concerning finite nuclear dimension. We examine the case
of topological actions and indicate that Rokhlin dimension
is closely related to amenability dimension in the sense
of Erik Guentner, Rufus Willett, and Guoliang Yu. Moreover,
it turns out that the recent result concerning the Rokhlin
dimension of free Z^mactions on finite dimensional spaces
generalizes to actions of finitely generated nilpotent
groups. (joint work with Jianchao Wu and Joachim Zacharias)
Hannes Thiel
Structure of simple Cuntzsemirings
We introduce the concepts of Cuntzsemirings
and their modules. Natural examples are given by Cuntz semigroups
of C*algebras that are strongly selfabsorbing and of C*algebras
that tensorially absorb such a C*algebra.
We characterize the modules over the Cuntzsemiring
of the JiangSu algebra as those Cuntzsemigroups that are
almost unperforated and almost divisible.
Then, we study simple Cuntzsemirings. Under
mild assumptions, they are automatically almost unperforated
and almost divisible. We also classify all solid Cuntzsemirings.
A semiring is called solid if the multiplication map induces
an isomorphism of the tensorsquare of the semiring with
the semiring. One can think of solidity as an algebraic
analog of being strongly selfabsorbing.
(joint work with Ramon Antoine and Francesc Perera)
Alessandro Vignati
A complete theory whose saturated C*algebras are
characterized in terms of Boolean algebras
After a brief introduction, we will see how saturation
in a continuous model theory setting for an abelian real
rank zero C*algebra without minimal projections corresponds
to saturation of the associated Boolean algebra of projections,
in the classical model theoretical sense. Moreover we
show that the theory of this class of C*algebras is complete,
that is, that every two such C*algebras are elementary
equivalent.
This is a joint work with Christopher Eagle.
Qingyun Wang
Tracial Rokhlin property for amenable
groups (slides)
Tracial Rokhlin property for actions
on unital simple C*algebras has been proved to be very
useful in determine the structure of the crossed product.
But most of the results dealt with actions of finite groups
or group of integers only. In this talk, we will give a
definition of tracial Rokhlin property for actions of countable
discrete amenable groups. We shall see that most of the
previous results could be generalized to our case. Among
other things, we show that the crossed products of actions
with tracial Rokhlin property preserve the class of C* algebras
with real rank zero, stable rank one and strict comparison
for projections, and the crossed products of actions with
weak tracial Rokhlin property preserve the class of tracially
$\mathcal{Z}$stable C*algebra. We shall also give some
interesting examples if time permits.
Nicola Watson
Local lifting properties of C*algebras
Lifting and perturbation results have played an important,
but largely undervalued, role in the classification program
thus far. Motivated by these applications, and also the
difference between alternative characterisations of nuclear
dimension and decomposition rank for C*algebras with
real rank zero, we introduce various local lifting properties.
We show that these properties are fairly general (being
satisfied by many classes of 'nice' algebras) and are
useful. Our main application will be to show that simple,
separable, unital, finite, nonelementary C*algebras with
finite nuclear dimension, real rank zero and finitely
many extremal traces that are locally liftable (in a suitable
sense) are TAF, and so those that also satisfy the Universal
Coefficient Theorem are classifiable.
Wilhelm Winter
Dstability and nuclear dimension
I will report on recent results on finite nuclear dimension
of a C*algebra; in particular, I will outline how this
can be derived from Dstability (where D is strongly selfabsorbing)
in certain situations.
Back to top

