Upcoming Seminars:
every Tuesday and Thursday at 2:00 pm Room 210 

TBA 
Past Seminars 
May 21 
James Lutley
Nuclear Dimension and the Toeplitz Algebra
After reviewing classical Toeplitz matrices, we will briefly review
the CPC approximation used on the CuntzToeplitz algebras by Winter
and Zacharias to study the Cuntz algebras. We will then show in full
detail how these maps operate on the Toeplitz algebra itself. This
calculation fails to determine the nuclear dimension of the Toeplitz
algebra itself but we will show how these techniques can be extended
to put this within reach.

May 14 
Dave Penneys
Computing principal graphs part 2
Last week we looked at the Jones tower, the relative commutants,
and the principal graph for some subfactors associated to finite groups.
This week, we'll continue our analysis of the relative commutants
and the principal. We will show how minimal projections in the relative
commutants correspond to bimodules, and we'll discuss how we view
the principal graph as the fusion graph associated to these bimodules,
where fusion refers to Connes fusion of bimodules. We'll then compute
a particular example of importance in the classification of subfactors
at index 3+\sqrt{5}.

May 9 
Dave Penneys
Computing principal graphs
Subfactors are classified by their standard invariants, and standard
invariants are classified by their principal graphs. I will give the
appropriate definitions, and then I will compute some principal graphs.
In particular, we will look at examples coming from groups and examples
coming from compositions of subfactors.

May 7 
David Kerr
Turbulence in automorphism groups of C*algebras 
Apr. 30 
Nicola Watson
Connes's Classification of Injective Factors 
Apr. 11 
Greg Maloney 
Apr. 9 
Danny Hay 
Apr. 4 
Nicola Watson

Mar. 28 
Luis Santiago
The $Cu^\sim$semigroup of a C*algebra 
Mar. 26 
Dave Penneys
GJS C*algebras
GuionnetJonesShlyakhtenko (GJS) gave a diagrammatic proof of a
result of Popa which reconstructs a subfactor from a subfactor planar
algebra. In the process, certain canonical graded *algebras with
traces appear. In the GJS papers, they show that the von Neumann algebras
generated by the graded algebras are interpolated free group factors.
In ongoing joint work with Hartglass, we look at the C * algebras
generated by the graded algebras. We are interested in a connection
between subfactors and noncommutative geometry, and the first step
in this process is to compute the Ktheory of these C * algebras.
I will talk about the current state of our work.

Mar. 21 
Makoto Yamashita
Deformation of algebras from group 2cocycles
Algebras with graded by a discrete can be deformed using 2cocycles
on the base group. We give a Ktheoretic isomorphism of such deformations,
generalizing the previously known cases of the thetadeformations
and the reduced twisted group algebras. When we perturb the deformation
parameter, the monodromy of the GaussManin connection can be identified
with the action of the group cohomology.

Jan. 17 
Zhiqiang Li
Certain group actions on C*algebras
We will discuss group actions of certain groups, mainly, discrete
groups, for example, \mathbb{Z}^d, and finite groups, then look at
several classifiable classes of such group actions, and finally we
will give a classification of inductive limit actions of
cyclic groups with prime orders on approximate finite dimensional
C* algebras.

Jan. 15

Ask Anything Seminar

Jan 10

George Elliott

Dec 20

Nadish de Silva

Dec 18

James Lutley

Dec 13

Dave Penneys

Dec 11

George Elliott

Dec 6

Greg Maloney
A constructive approach to ultrasimplicial groups
I will review the result of Riedel that says that every simple finitely
generated dimension group with a unique state is ultrasimplicial.
The proof involves explicitly constructing a sequence of positive
integer matrices using a multidimensional continued fraction algorithm.
This approach is similar to that used by Elliott
and by Effros and Shen in their earlier results.

Dec 4

Danny Hay
Computing the decomposition rank of Zstable AH algebras
We will take a look at a recent paper of Tikuisis and Winter, in
which it is shown that the decomposition rank of Zstable AH algebras
is at most 2. The result is important not only because establishing
finite decomposition rank is significant for the classification program,
but also because the computation is direct—
previous results of this type generally factor through classification
theorems, and so shed no light on why finite dimensionality occurs.

Nov 27

Zhiqiang Li (U of Toronto; Fields)
Finite group action on C*algebra
I am going to talk about some result of M. Izumi on finite group action
on C*algebras. Mainy, there is a cohomology obstruction for C*algebra
having finite group action with Rokhlin property.

Nov 29

Mike Hartglass (Berkley)
Rigid $C^{*}$ tensor categories of bimodules over interpolated
free group factors
The notion of a fantastic (or factor) planar algebra will be presented
and some examples will be given. I will then show how such an object
can be used to diagrammatically describe a rigid, countably generated
$C^{*}$ tensor category $\mathcal{C}$. Following in the steps of Guionnet,
Jones, and Shlyakhtenko, I will present a diagrammatic construction
of a $II_{1}$ factor $M$ and a category of bimodules over $M$ which
is equivalent to $\mathcal{C}$. Finally, I will show that the factor
$M$ is an interpolated free group factor and can always be made to
be isomorphic to $L(\mathbb{F}_{\infty})$. Therefore we will deduce
that every rigid, countably generated $C^{*}$ tensor category is equivalent
to a category of bimodules over $L(\mathbb{F}_{\infty})$.
This is joint work with Arnaud Brothier and David Penneys.

Nov 22

Paul McKenney (Carnegie Mellon)
Approximate *homomorphisms
Abstract: I will discuss various notions of "approximate
homomorphism", and show some averaging techniques that have been
used
to produce an actual homomorphism near a given approximate homomorphism.

Nov 20

Brent Brenken (Univeristy of Calgary)
Universal C*algebras of *semigroups and the C*algebra of a
partial isometry
Certain universal C*algebras for *semigroups will be introduced.
Some basic examples, and ones that occur in describing the C*algebra
of a partial isometry, will be discussed. The latter is a Cuntz Pimsner
C*algebra associated with a C*correspondence, and can be viewed
as a form of crossed product C*algebra for an action by a completely
positive map. The C*algebras involved occur as universal C*algebras
associated with contractive *representations, and complete order
*representations, of certain *semigroups.

Nov 13

Nicola Watson (U of Toronto)
Noncommutative covering dimension
There have been many fruitful attempts to define noncommutative versions
of the covering dimension of a topological space, ranging from the
stable and real ranks to the decomposition rank. In 2010, Winter and
Zacharias defined the nuclear dimension of a C*algebra, which has
turned out to be a major development in the study of nuclear C*algebras.
In this talk, we introduce nuclear dimension, discuss the differences
between it and other dimension theories, and focus on why
nuclear dimension is so important.
(This is a practice for a talk I'm giving at Penn State, so it will
be more formal than usual.)

Nov 8

Danny Hay

Nov 6

Greg Maloney
Connes' fusion
I'll give a basic introduction to Connes' fusion for bimodules over
finite von Neumann algebras.

Nov 6,8,13,15

Working seminars

Octr 30 and Nov 1

"Wiki Week"

Oct 25

Dave Penneys (U of Toronto)
Infinite index subfactors and the GICAR algebra
We will show how the GICAR algebra is the analog of the TemperleyLieb
algebra for infinite index subfactors. As a corollary, we will see
that the centralizer algebra M_0'\cap M_{2n} is nonabelian for all
n\geq 2.

Oct 18

Greg Maloney
Ultrasimplicial groups
An ordered abelian group is called a dimension group if it is the
inductive limit of a sequence of direct sums of copies of Z. Dimension
groups are of interest in the study of operator algebras because they
are the K0groups of AF C*algebras.
If, in addition, a dimension group admits such an inductive limit
representation in which the maps are injective, then it is called
an ultrasimplicial group. The question then arises: exactly which
dimension groups are ultrasimplicial?
There have been positive and negative results on this subject. Elliott
showed that every totally ordered (countable) group is ultrasimplicial,
and Riedel showed that a free simple dimension group of finite rank
with a unique state is ultrasimplicial. Much later, Marra showed that
every lattice ordered abelian group is ultrasimplicial. On the other
hand, Elliott produced an example of a simple dimension group that
is not ultrasimplicial, and later Riedel produced a collection of
simple free dimension groups that are not ultrasimplicial. I will
discuss the history of this subject and go through some calculations
in detail.

Oct 16

Martino Lupini (York)
The complexity of the relation of unitary equivalence for automorphisms
of separable unital C*algebras
A classical result of Glimm from 1961 asserts that the irreducible
representations of a given separable C*algebra A are classifiable
by real numbers up to unitary equivalence if and only if A is type
I. In 2008, KerrLiPichot and, independently, Farah proved that when
A is not type I, then the irreducible representations are not even
classifiable by countable structures. I will show that a similar dichotomy
holds for classification of automorphisms up to unitary equivalence.
Namely, the automorphisms of a given separable unital C* algebra
A are classifiable by real numbers if and only if A has continuous
trace, and not even classifiable by countable structures otherwise.

Oct 11

Xin Li (University of Muenster)
Semigroup C*algebras
The goal of the talk is to give an overview of recent results about
semigroup C*algebras. We discuss amenability, both in the semigroup
and C*algebraic context, and explain how to compute Ktheory for
semigroup C*algebras.

Oct 4

Zhi Qiang Li (U of Toronto; Fields)
Finite group action on C*algebra
I am going to talk about some result of M. Izumi on finite group
action on C*algerbras. Mainy, there is a cohomology obstruction for
C*algebra having finite group action with Rokhlin property.

Sept. 18

Aaron Tikuisis
Regularity for stably projectionless C*algebras
There has been significant success recently in proving that unital
simple C*algebras are Zstable, under other regularity hypotheses.
With certain new techniques (particularly concerning traces and algebraic
simplicity), many of these results can be generalized to the nonunital
setting. In particular, it can be shown that the following C*algebras
are Zstable: (i) (nonunital) ASH algebras with slow dimension growth
(TToms); (ii) (nonunital) C*algebras with finite nuclear dimension
(T); and (iii) (nonunital) C*algebras with strict comparison and
finitely many extreme traces (Nawata). I will discuss the proofs of
these results, with emphasis on the innovations required for the nonunital
setting.
