SCIENTIFIC PROGRAMS AND ACTIVITIES

James Lutley
Nuclear Dimension and the Toeplitz Algebra

After reviewing classical Toeplitz matrices, we will briefly review the CPC approximation used on the Cuntz-Toeplitz 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.

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}.

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.

Nicola Watson

Dave Penneys
GJS C*-algebras

Guionnet-Jones-Shlyakhtenko (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 non-commutative geometry, and the first step in this process is to compute the K-theory of these C * -algebras. I will talk about the current state of our work.

Makoto Yamashita
Deformation of algebras from group 2-cocycles

Algebras with graded by a discrete can be deformed using 2-cocycles on the base group. We give a K-theoretic isomorphism of such deformations, generalizing the previously known cases of the theta-deformations and the reduced twisted group algebras. When we perturb the deformation parameter, the monodromy of the Gauss-Manin connection can be identified with the action of the group cohomology.

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 Z-stable 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 Z-stable 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

Oct 25

We will show how the GICAR algebra is the analog of the Temperley-Lieb 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

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 K0-groups 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

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, Kerr-Li-Pichot 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

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 K-theory 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 Z-stable, 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 Z-stable: (i) (nonunital) ASH algebras with slow dimension growth (T-Toms); (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.