every Tuesday and Thursday at 2 pm Room 210
|December 2, 2014
Notion of the standardness in the theory of AF-algebras, and iterations
of Kantorovich transport metric
The notion of the standardness came from the theory of filtration
(=decreasing sequences) of sigma-fields in 70-th. In the theory of
AF-algebras we also have the natural filtration in the space of paths
of Bratelli diagrams ---"tail-filtration" (the sets of n-th
sigma-filed consist with the classes of paths up to first n levels
of the path.) So by definition the standard $AF C^*$ -algebra is algebra
with standard tail filtration.
The criteria of standardness uses so called iteration of Kantorovich
metric/ which leads to the notion of intrinsic metric on the paths.
More exactly, the criteria reduces to the uniform compactness of the
levels of graph. The following theorem shows the role of the standardness
The set of the indecomposable traces of standard AF-algebra is a
compactification under the intrinsic semimetric n the space of paths.
|November 18, 2014
Inner quasidiagonality and AF Embeddings
We isolate the obstruction to AF embeddings of nuclear UCT QD algebras
via the path we have previously discussed and propose a solution to
|November 12, 2014
Abelian amenable operator algebras are isomorphic to C*-algebra
I will present the proof of Marcoux and Popov that amenable abelian
operator algebras are isomorphic to a C*-algebras.
|October 16, 2014
Correspondences for Smale spaces
We discuss joint work with Brady Killough and Michael Whittaker.
This work centers around the functorial properties of the homology
for Smale spaces introduced by Ian Putnam. In the case of a shift
of finite type this homology theory is Krieger's dimension group;
this case will be discussed in detail.
The fundamental object of study are correspondences between Smale
spaces; the precise definition will be given in the talk. However,
the idea is to encode both types of functorial properties of Smale
spaces (with respect to Putnam's homology theory) into a single object.
No knowledge of Smale spaces or Putnam's homology is required for
|October 14, 2014
Relative constructions in geometric K-homology
The Baum-Douglas model realizes K-homology using geometric cycles
(i.e., via data from manifold theory). The isomorphism between this
realization of K-homology and analytic definition of Kasparov naturally
leads to a proof of the Atiyah-Singer index theorem.
We discuss the process of proving other index theorems using the
framework of "relative geometry K-homology". Example of
such theorems include the Freed-Melrose index theorem and theorems
from R/Z-valued index theory.
|September 30, 2014
Characterization of spectral flow in a type II factor
I will start with an introduction to spectral flow. In B(H) (the
set of bounded operators on a Hilbert space), the spectral flow counts
the net number of eigenvalues which change sign as one travels along
a path of self-adjoint Fredholm operators. It is possible to generalize
the concept of spectal flow to a semifinite von Neumann algebra, as
we can use a trace on the algebra to measure the amount of spectrum
which changes sign. I will give a characterization of spectral flow
in a type II factor, and a sketch of the proof.
September 29, 2014
On what I learnt in Glasgow about the UCT (Part 2)
After the reviewing on the basic definitions related to TAF algebras,
I will give a proof of the fact that, in the TAF simple case, homotopic
*-homomorphisms are necessarily approximately unitarily equivalent.
If there is a time I will sketch an idea of how to prove that KK-equivalent
*-homs are approximately unitarily equivalent, passing through the
mod n KK-theory.
|September 25, 2014
|September 23, 2014
Finite nuclear dimension from Z-stability in the unique trace case
We report on a proof of the main result in a paper of Sato, White,
and Winter which establishes finite nuclear dimension from Z-stability
for the usual Toms-Winter C*-algebras with a unique tracial state.
A key idea here, which also appeared in an earlier paper of Matui
and Sato, is the two-colored approximately inner flip. The end result
is the estimate dr(A) < 4 -obtained by using UHF-stable approximations
twice, with each approximately inner flip contributing two colors.
September 22, 2014
On what I learnt in Glasgow about the UCT (Part 1)
I will present an introduction on the UCT and KK-theory and a collection
of results related to the UCT in the TAF case. This seminar will follow
the lectures of Marius Dadarlat and Christian Voigt in a recent Masterclass
August 21, 2014
Nuclear dimension and Z-stability
We will sketch the original construction of the Jiang-Su algebra
and discuss some of its properties. Apart from serving as the first
(and only) example of its kind, understanding its role in K- theoretic
classification of simple separable nuclear of C*-algebras has been
of great interest in recent years. A number of results indicate that
the largest class of C*-algebras for which Elliott's classification
conjecture can hold is for those which are also Z- stable. Various
disparate regularity properties, each of which is helpful for classification,
arise as a consequence of being Z-stable. A well-known conjecture
characterizing manifestations of Z-stability in a certain class of
C*-algebras is the Toms-Winter conjecture. We discuss one of the properties,
finite nuclear dimension, and progress on the conjecture.
|August 21, 2014
The necessity of quasidiagonality and unique tracial states
On Matui and Sato's paper, titled "Decomposition rank of UHF-
absorbing C*-algebras'", proves that if A is a unital separable
simple C*-algebra with a unique tracial state, then if A is nuclear,
quasidiagonal and has strict comparison or projections, then the decomposition
rank of A is at most 3. As a corollary, a partial affirmation of the
Toms and Winters conjecture can be proven. In this talk, we focus
on how quasidiagonality and having a unique tracial state are necessary
conditions for this theorem.
August 19, 2014
| James Lutley, University of Toronto
AF algebras of higher rank graphs
We will introduce sufficient and necessary conditions to ensure
that the algebras associated with a higher rank graph are AF. These
algebras are the Toeplitz extension, which has a representation on finite
paths, and the main algebra, represented on infinite paths, which correspond
to ultrafilters of the finite path space. We will discuss what obstructions
exist, and whether or not the quotient can be AF when the Toeplitz extension