SCIENTIFIC PROGRAMS AND ACTIVITIES

July 25, 2014

Operator Algebras Seminars
July 2009 - June 2010

For more information about this program please contact George Elliott

PAST SEMINARS

June 29, 2010
Adam Sierakowski
Purely infinite C*-algebras arising from crossed products (part I)
This will be the first of a series (maybe two or three) of talks reporting new progress made in the study of purely infinite C*-algebras arising from crossed products by M. Roerdam and A. Sierakowski. We study conditions that will ensure that a crossed product of a C*-algebra by a discrete exact group is purely infinite (simple or non-simple). We are particularly interested in the case of a discrete non-amenable exact group acting on a commutative C*-algebra, where our sufficient conditions for example can be phrased in terms of paradoxicality of subsets of the spectrum of the abelian C*-algebra. As an application of our results we show that every discrete countable non-amenable exact group admits a free amenable minimal action on the Cantor set such that the corresponding crossed product C*-algebra is a Kirchberg algebra in the UCT class.

June 15, 2010
Working Seminar

June 10, 2010
Working Seminar

June 8, 2010
Working Seminar

June 3, 2010
Aaron Tikuisis
The Cuntz semigroup of continuous functions into a strongly self-absorbing C*-algebra
The Cuntz semigroup continues to establish itself as a powerful instrument in determining the structure of C*-algebras. Its usefulness shall increase as we better understand how to compute it. As it stands, we do not know what the Cuntz semigroup looks like for very many C*-algebras, including most abelian C*-algebras. In this talk, I shall describe some Cuntz semigroup computations I have made for commutative C*-algebras tensored with a UHF algebra or tensored with the Jiang-Su algebra (these are all the known examples of finite strongly self-absorbing C*-algebras). These computations apply whenever the commutative tensor factor is separable and has finite-dimensional spectrum, although one would hope that these restrictions may be removed.

June 1, 2010
Working Seminar

May 28, 2010
Working Seminar

May 20- Working Seminar

May 18- Working Seminar

May 13, 2010
Working Seminar

May 11, 2010
Franek Szafraniec, Jagiellonian University, Krakow
Naimark extensions for indeterminacy in the moment problem
Based on a simple and natural example a family of solutions to a Hamburger moment problem is described. It is completely different than that of N-extremal ones. Both are discrete however for present one the polynomials are of infinite codimension while for the N-extremal measures the polynomials are dense in their L2 spaces.

May 6, 2010
Aaron Tikuisis
The Cuntz semigroup of C(X,Z), where Z is the Jiang-Su algebra
The Jiang-Su algebra Z is considered an extremely important object in the classification of C*-algebras, as it is believed that simple nuclear C*-algebras may be classified up to Z-stability. In this talk I will discuss the realization of the Jiang-Su algebra as an inductive limit of dimension drop algebras, and demonstrate how this realization allows for a computation of the Cuntz semigroup of C(X,Z) when X is a separable compact Hausdorff space with finite covering dimension.

May 4, 2010
Working Seminar

April 29, 2010
Working Seminar

April 27, 2010
Leonel Robert
Classification by the functor Cu~

April 22, 2010
Working Seminar
Stewart Library

April 20, 2010
Aaron Tikuisis
Ordered vector spaces with Riesz interpolation
Riesz interpolation is a property of ordered groups allowing the solution to certain computations. It is slightly weaker than the property of being lattice ordered, but sufficiently weaker to permit many interesting examples. However, it is much more difficult to verify that an ordered group has Riesz interpolation than that to verify that it is lattice ordered. In this talk, I will describe joint work by myself and Greg Maloney towards on the problem of describing the finite dimensional ordered vector spaces with Riesz interpolation. A canonical form is found for such an ordered space, which is determined by the data consisting of the ideals and the spaces of positive homomorphisms on each ideal.

April 15, 2010
Working Seminar

April 8, 2010
Working Seminar

April 6, 2010
Marco Gualtieri
Gerbes and generalized Kahler geometry
I will explain what gerbes are (in a simple case), how they enter in generalized geometry, and in particular what happens on a generalized Kahler manifold. Generalized Kahler manifolds differ from usual Kahler manifolds in that they have two, rather than one, complex structure compatible with the Riemannian metric. These two complex structures needn't be isomorphic as complex manifolds, although they coexist on a smooth real manifold. I will explain how the two complex structures are related to each other as well as to a certain natural gerbe.

April 5, 2010 at 4:10PM
BA6183, Bahen Center, 40 St. George St
Barry Rowe, University of Toronto
Basis map operators arising from regular semigroup representations
For a given orthonormal basis of a Hilbert space, a map from this basis (as a set) to itself can be extended as a (possibly unbounded) operator on the Hilbert space. We call such an operator a basis map operator, and classify these operators that arise as multiplications in the regular representation of a countable semigroup. We will review some of these results for various semigroup types (eg: finite, bounded, commutative, etc.), and discuss in greater detail the case of basis map operators arising from connected, but non-commutative semigroups. Finally we will end with an application of the above results to show that finitely generated, commutative, bounded semigroups must be also be uniformly bounded.

April 1, 2010
Working Seminar

March 30, 2010
Working Seminar

March 25, 2010
Working Seminar

March 23, 2010
Working Seminar

March 16, 2010
Snigdhayan Mahanta
TFT and topological K-theory
After a brief introduction to topological field theories (TFTs), I shall explain how the topological K-theory of C^*-algebras can be relevant to their study. In particular, I will discuss some features that a possible extension of TFTs to "noncommutative manifolds" might have.

MATH/PHYSICS LEARNING SEMINAR -- Special Relativity
Tuesday, 16 March 2010, 3:00PM
BA6183, Bahen Center, 40 St. George St
Daniel Rowe, University of TorontoI will use the general framework of representation theory to explain relativity. Then I will present the classical motivation and experimental results behind special relativity, which then forces one to admit that the symmetry group of our affine spacetime is the Poincare group. Time permitting, I may go into more representation theory to explain spinors, clifford algebras, etc.

March 11, 2010
Working Seminar

March 9, 2010
Working Seminar

March 4, 2010 at 2pm in BA6183
Nigel Higson
A C*-algebraist looks at [Q,R]=0
I'll give a brief introduction to K-homology theory, the generalized homology theory that is dual to Atiyah-Hirzebruch K-theory, as it is viewed from C*-algebra theory and from index theory. As I hope to illustrate, the two perspectives are in many respects complementary. Then I'll turn to the [Q,R]=0 problem. It is easy to sketch a simple connection between the C*-algebraic version of K-homology and the analytic proof of [Q,R]=0. But the geometric side of the story remains quite mysterious.

March 4, 2010 at 3pm in Fields Room 210
Working Seminar

March 2, 2010 **3:15pm**
Leonel Robert
On his most recent work

February 23, 2010
Working Seminar

February 25, 2010
Uffe Haagerup
On Thompson's group F: http://www.utoronto.ca/reg/list_full.pl?20100125-1410.24814

February 16, 2010
Working Seminar

February 11, 2010
Leonel Robert
On his most recent work

February 9, 2010
Leonel Robert
Properties of Z-absorbing C*-algebras
Rordam has shown that a simple unital C*-algebra that absorbs the Jiang-Su algebra has stable rank 1. I will prove a similar result for Z-absorbing C*-algebras that either don't have simple subquotients or are simple and nonunital. However, I will not show that these algebras have stable rank 1, but rather that they are contained in the closure of the invertibles of their unitization (this is slightly weaker but suffices for many applications).

February 4, 2010
Leonel Robert
Properties of Z-absorbing C*-algebras
Rordam has shown that a simple unital C*-algebra that absorbs the Jiang-Su algebra has stable rank 1. I will prove a similar result for Z-absorbing C*-algebras that either don't have simple subquotients or are simple and nonunital. However, I will not show that these algebras have stable rank 1, but rather that they are contained in the closure of the invertibles of their unitization (this is slightly weaker but suffices for many applications).

***SPECIAL TALK***
Monday, February 1, 2010 at 4:10 p.m. in Room BA6183
Abhijnan Rej
Introduction to KK-theory (and its categorical properties)
After taking care of some notions related to higher K-theory of C*-algebras, I introduce KK-theory. I also describe the categorical underpinnings of Kasparov's construction and show how KK-theory specializes to both K-theory and K-homology. I conclude with some basics of the theory of extensions and crossed products (needed for later!), following Blackadar's exposition.

http://sites.google.com/site/abhijnanrej/stringsncg

February 2, 2010
Aaron Tikuisis
The Cuntz semigroup of an AF algebra tensor a commutative algebra
There is a natural map from Cu(C_0(X) \otimes A) to Lsc(X,Cu(A)), the semigroup of lower semicontinuous maps from X to Cu(A) (using the right notion of lower semicontinuity). We will investigate the Cuntz semigroup of an AF algebra tensor a commutative algebra whose spectrum has dimension at most 3, building on previous results of Leonel Robert and myself. Examples may be computed, such as when A is a UHF algebra.

January 28, 2010
2:10 p.m. - Aaron Tikuis
Ordered Vector Spaces with Interpolation
I will add some remarks to Greg Maloney's recent talk about the problem of classifying finite-dimensional ordered vector spaces with Riesz interpolation.
3:15 p.m. - Arthur Huang
Irrational Rotation Algebras and Their Canonical Extensions
We will briefly review what we did last time on the extended rotation algebras and show that each of these algebras has a unique tracial state, as is the case before the rotation algebras were extended. Time permitted, we will give a criterion for the faithfullness of the canonical tracial state.

January 26, 2010
Working Seminar

January 21, 2010
Barry Rowe
On his most recent work

January 19, 2010
2:10 p.m. - Working Seminar
3:15 p.m. - Leonel Robert, On his most recent work

***SPECIAL TALK***
Monday, January 18, 2010 at 4:10 p.m. in Room BA6183
Abhijnan Rej
K-theory basics: the Atiyah-Hirzebruch spectral sequence and K-homology
I introduce topological K-theory and focus on the construction and meaning of the Atiyah-Hirzebruch spectral sequence in K-theory. I also define K_0 and K_1 for an arbitrary C^*-algebra. I end with relating geometric K-homology to K-theory, and time permitting, discuss the Atiyah-Bott-Shapiro construction.

January 14, 2010
Greg Maloney
Finite-dimensional ordered real vector spaces with the Riesz interpolation property
Ordered real vector spaces play an important role in the representation theory of the K_0 groups of AF C*-algebras. I will explain some recent progress that has been made towards finding an explicit description of those finite-dimensional ordered real vector spaces that have the Riesz interpolation property. This is joint work with Aaron Tikuisis.

January 12, 2010
Nemanja Kosovalic
Simple Construction of an Outer Automorphism of the Calkin Algebra Using CH

***SPECIAL TALK***
Monday, January 11, 2010 at 4:10 p.m. in Room BA6183
Abhijnan Rej
D-branes and K-theory: an overview
In this overview talk, I introduce the notion of D-branes in string theory in a mathematically suitable language. I discuss the notion of classification of charges of D-branes in this language using K-theory. In the presence of a nonvanishing B-field, we have to extend this program by replacing "ordinary" K-theory by twisted K-theory which, in turn, can be defined using K-theory of C*-algebras (following J. Rosenberg). No physics knowledge is required to understand this talk. In fact, the goal would be to axiomatically define all relevant physics objects.

(This is Lecture 0 (self-contained) of a series; see http://sites.google.com/site/abhijnanrej/stringsncg for further details.)

January 7, 2010
Aaron Tikuisis
Exploring AT systems
I undertake a preliminary analysis of an inductive system of circle algebras. Using a result of Choi and Elliott along with various manipulations, we reduce the system to one that can be expressed with a Bratteli diagram, with the following additional data attached to each edge:
- a continuous map from the circle to itself, and
- a winding number.
Here, "reducing the system" means replacing it by one whose inductive limit is the same.

January 5, 2010
Arthur Huang
Irrational Rotation Algebras and Their Canonical Extensions

This will be the first of a serise (maybe four or five) of talks reporting new progress made in the study of irrational rotation algebras by Prof. Elliott and Niu. The studies of irrational rotation algebras were firsted picked up by Marc Rieffel in 1981; till now the fundamental theorem (as I see it) is that irrational rotation algebras are AT, i.e. countable direct limits of "circle algebras" (some new term). However, a nicer guess is that they are actually AF (a well-known term). Recently, Prof. Elliott and Niu could see that there are actually lots of irrational points associated with which the rotation algebras are AF, from an existence proof by means of adjoining spectral projections and constructing a field of C*-algebras attached on the circle. In this talk, we will give the set up of the whole story and look at the tracial state on the extended rotation algebras.

December 17, 2009
Barry Rowe
The Von Neumann Algebra of a Semigroup
The Von Neumann group algebras form a very important object in factor theory (specifically, the ones that come from free groups), but rarely is the more general case of a semigroup considered. We will go over some problems that arise in constructing such an algebra from a semigroup, and one way of constructing them if S is locally diagonalizable.

December 15, 2009
Working Seminar

*** POSTPONED***
December 10, 2009
Abhijnan Rej
What is K-theory, really?
I sketch some classical facts about the K-functor, with emphasis on algebraic K-theory and the K-theory of C*-algebras. Time permitting, I'll present my own (extremely limited!) understanding of KK-theory. This is a "talk 0" in a planned series of talks leading up to cyclic homology and Chern-Connes theory.

December 8, 2009
Working Seminar

December 3, 2009
Aaron Tikuisis
Homotopies of Hilbert C*-modules

Inspired by homotopy theory of projections (ie. finitely generated projective Hilbert modules), I have been investigating homotopies of Hilbert modules. Two good notions of homotopy present themselves. The first is a map from [0,1] to submodules of a fixed module, such that the map is continuous with respect to Hausdorff distance between unit balls. The second is a Hilbert (C[0,1] \otimes A)-module such that each Cuntz functional is constant over the fibres (ie. at each t in [0,1]). The second of these notions is strictly weaker than the first. A result of Larry Brown shows that the first of these notions implies homotopy (when the ambient Hilbert module is l^2(A)). Using this result, I show that in the commutative case, the second notion implies Cuntz equivalence.

December 1, 2009
Working Seminar

November 24, 2009
Working Seminar

November 17, 2009
Working Seminar

November 12, 2009
Greg Maloney
Dimension group structures on finite-dimensional real vector spaces
Real dimension groups play an important role in the representation theory of the K_0 groups of AF C*-algebras. To improve my understanding of the free, finite rank dimension groups, I am investigating the possible dimension group structures on finite dimensional real vector spaces.

November 10, 2009
Henning Petzka
Hochschild Cohomology of von Neumann algebras continued
The theory of bounded Hochschild cohomology for von Neumann algebras was initiated by Johnson, Kadison and Ringrose in 1972. They showed that H^n(M,M) = 0 for all n ? 1 when M is an injective von Neumann algebra and they conjectured that this should be true for all von Neumann algebras. We will look at the tools they used in their early papers. We will verify vanishing cohomology for type I von Neumann algebras, and afterwards discuss the approach to the general case of injective von Neumann Algebras.

November 5, 2009
Henning Petzka
Hochschild Cohomology of von Neumann algebras
The theory of bounded Hochschild cohomology for von Neumann algebras was initiated by Johnson, Kadison and Ringrose in 1972. They showed that H^n(M,M) = 0 for all n ? 1 when M is an injective von Neumann algebra and they conjectured that this should be true for all von Neumann algebras. We will look at the tools they used in their early papers. We will verify vanishing cohomology for type I von Neumann algebras, and afterwards discuss the approach to the general case of injective von Neumann Algebras.

November 3, 2009
Nicola Watson
The Wold Decomposition
I will present a simple proof of Von Neumann's Wold Decomposition of isometries on a Hilbert space, and then discuss how to generalize this to the decomposition of pairs of doubly commuting. I will talk about the smaller, interesting results leading to the proofs and some possible generalizations and applications.

October 29, 2009
Abhijnan Rej
What are motives? And why should we care? - Part II
This talk will be a pedagogical lecture on certain aspects of the theory of motives that have recently found generalizations in the context of operator algebras and noncommutative geometry, in the theory of "endomotives" of Connes-Consani-Marcolli's 'Weil proof and the geometry of adele class space' paper.

October 27, 2009
Abhijnan Rej
What are motives? And why should we care?
This talk will be a pedagogical lecture on certain aspects of the theory of motives that have recently found generalizations in the context of operator algebras and noncommutative geometry, in the theory of "endomotives" of Connes-Consani-Marcolli's 'Weil proof and the geometry of adele class space' paper.

October 22, 2009
Working Seminar

October 20, 2009
Working Seminar

October 16, 2009
Working Seminar

October 13, 2009
***CANCELLED***

October 8, 2009
Adam Sierakowski
Crossed product C*-algebras

October 6, 2009
Adam Sierakowski
Crossed product C*-algebras

October 1, 2009
Barry Rowe
The Isomorphism Problem for Semigroups
An important question in representation theory is whether or not isomorphic representations of groups imply that the underlying groups themselves must be isomorphic. In general, this need not be true (in fact, the problem is undecidable), though for some conditions on the group it is the case. We will look at what happens when one considers a semigroups instead, and give some results.

September 29, 2009
Adam Sierakowski
Crossed product C*-algebras

September 24, 2009
Leonel Robert
On his current work (Fields of Hilbert spaces)

September 17, 2009
Leonel Robert
On his current work

September 15, 2009
Benoit Jacob
The structure of selfadjoint matrix fields over compact 3-manifolds
Selfadjoint matrix fields are well-understood as long as the base space has dimension at most 2, as a small perturbation then ensures that eigenvalues have multiplicity 1 everywhere. However, there isn't such a nice result when the base space has dimension more than 2, and a classical counterexample exists already as a 2x2 matrix over the Euclidean 3-ball,
namely:
z x+iy
x-iy -z
In this talk, we'll see that, as far as 3-manifolds are concerned, this simple counterexample is actually the whole story: we'll prove that over compact 3-manifolds, selfadjoint matrix fields may be perturbed so that outside of a finitely many points, the eigenvalues have multiplicity 1, and at these finitely many points, we just have two eigenvalue curves meeting, and the corresponding 2x2 matrix is similar to the one in the aforementioned counterexample

September 8, 2009
Branimir Cacic
Almost-Commutative Spectral Triples
In this talk, I shall propose a generalisation of the conventional notion of almost-commutative spectral triple (i.e. the product of the canonical spectral triple of a compact spin manifold with some finite spectral triple) and shall discuss some constructions and examples for this class of spectral triples. If time permits, I shall discuss possible relations to B. Mesland's recent work on unbounded KK-theory.

September 3, 2009
Aaron T
The Automatic Embedding of a Hilbert C_0(X)-module into a Sufficiently Larger One
Leonel Robert and I have been studying the structure of Hilbert modules over commutative C*-algebras. I will discuss our general approach and one of our main results: that if M,N are Hilbert C_0(X)-modules for which the dimension of M|_x exceeds that of N|_x by at least (dim X - 1)/2 then M embeds into N. In fact, our result generalizes to Hilbert modules over RSH algebras.

August 27, 2009
Barry Rowe

August 25, 2009
*** LOCATION CHANGE TO STEWART LIBRARY***

August 20, 2009
Kevin Teh
Introduction to Harmonic Analysis on the Infinite Symmetric Group
&
Nemanja Kosovalic: TBA
Steve Lowdon:
TBA
Brian Skinner:
TBA

August 18, 2009
Ilijas Farah, York University
Graphs and CCR algebras
Abstract: I introduce yet another way to associate a C*-algebra to a graph. While in the separable case we more-or-less always end up with the CAR algebra, uncountable graphs provide new examples of C*-algebras. For example, I will construct a simple nuclear C*-algebra that has irreducible representations on both separable and nonseparable Hilbert spaces. This gives a simple nuclear C*-algebra with a nonhomogeneous state space and answers a question of Kishimoto-Ozawa-Sakai.
&
Arthur Huang
TBA

August 13, 2009
Terry Loring
Weakly projective and weakly semiprojective C*-algebras

August 11, 2009
Thierry Giordano
Orbit Equivalence for Cantor Minimal Z^d Systems
Abstract: In this talk, I will present the main steps of the proof of the general result obtained in a joint effort with H. Matui, I.Putnam and C. Skau and whose statement is the following:
Theorem: Any minimal action of a finitely generated abelian group on the Cantor set is affable (i.e., orbit equivalent to an AF-relation).

August 4, 2009
Greg Maloney
On his most recent work

July 30, 2009
Leonel Robert
On his most recent work

July 28, 2009
Barry Rowe
For this talk I will continue with the topic of the left regular representation L(S) of a semigroup S. In particular, we will concentrate on what conditions on S are required in order to make the algebra L(S) reflexive (that is, L(S)=Alg(Lat(L(S))), the algebra of all operators that leave the invariant subspaces of L(S) invariant). There will also be review of results covered previously.

July 23, 2009
Zhuang Niu
On his most recent work

July 16, 2009
Greg Maloney
Dimension groups and C*-algebras associated to multidimensional continued fractions
Thirty years ago, Effros and Shen classified the simple free dimension groups with rank two. Every such group is parametrized by an irrational number, and can be constructed as an inductive limit using that number's continued fraction expansion.
There is a natural generalization of continued fractions to higher dimensions, and this invites the following question: What dimension groups correspond to multidimensional continued fractions? We describe this class of groups and show how some properties of a continued fraction are reflected in the structure of its dimension group.

July 9, 2009
Andrew Toms
Ranks of operators in simple C*-algebras
We explore the question of which rank functions may occur for positive operators in a simple unital stably finite C*-algebra. We give a complete answer for separable algebras with strict comparison and finite dimensional extreme tracial boundary, and for ASH algebras with slow dimension growth. Some applications will be discussed, time permitting.

 

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