


Operator Algebras / Algèbres d'opérateurs Org: Marcelo Laca (Victoria) and/et Ian Putnam (Victoria)
 MARTIN ARGERAMI, University of Regina
Young's inequality for traceclass operators

If a and b are traceclass operators, and if u is a partial
isometry, then
u ab^{*} u^{*}_{1} £  
1
p

a^{p} + 
1
q

b^{q}_{1}, 

where ·_{1} denotes the norm in the trace class. In joint work
with D. Farenick we characterised the cases of equality in this Young
inequality. As an application we are able to characterize the cases of
equality in the operator version of Young's Inequality
u ab^{*} u £ 
1
p

a^{p}+ 
1
q

b^{q}, 

which was proved by Ando (1995) for matrices and by Erlijman, Farenick
and Zeng for compact operators (2002).
 BERNDT BRENKEN, Department of Mathematics, University of Calgary
A C^{*}algebra for a winding on the torus

We consider the Hilbert bimodule associated with a closed relation on
the torus obtained by an infinite winding of an interval about the
torus. The algebra of compact operators is identified and the
associated CuntzPimsner C^{*}algebra formed.
 JULIAN BUCK, University of Northern British Columbia,
Prince George, British Columbia V2N 4Z9
ConnesChern characters of hexic and cubic modules

Let A_{q} denote the rotation C^{*}algebra generated by
unitaries U and V satisfying VU = e^{2 pi q} UV where
q is a fixed real number. Let r and k denote the
canonical order 6 and 3 transforms on A_{q} respectively, and
let 6_{q} = A_{q} \rtimes_{r} Z_{6} and
3_{q} = A_{q} \rtimes_{k} Z_{3} denote the
associated C^{*}crossed products by corresponding cyclic groups. In
this talk we discuss how to obtain injections Z^{10}\hookrightarrow K_{0}(6_{q}) and Z^{8} \hookrightarrowK_{0}(3_{q}) by computing the ConnesChern characters of the
projections and modules which form linearly independent classes in the
K_{0} groups. Along the way, we obtain the unbounded traces on the
canonical dense smooth *subalgebras, which for irrational q
give Connes' cyclic cohomology groups of order zero HC^{0}(6_{q}) @ C^{9}, HC^{0}(3_{q}) @ C^{7}.
 MICHAEL BURNS, Department of Mathematics, University of Victoria, Victoria,
British Columbia V8W 3P4
Extending Jones' planar algebras to broader classes of
subfactors

Jones' planar algebra formalism provides the most elegant and powerful
description of the standard invariant of a finite index extremal II_{1}
subfactor, allowing the use of diagramatic techniques to prove results
in the theory of subfactors. We will first review the theory of planar
algebras and subfactors and then discuss extensions of the planar
algebra results to general finite index subfactors and also to infinite
index II_{1} subfactors.
 KEN DAVIDSON, University of Waterloo, Waterloo, Ontario N2L 3G1
A Kaplansky Theorem for free semigroup algebras

A free semigroup algebra is the unital weak operator topology closed
algebra generated by n isometries with pairwise orthogonal ranges.
We show that the unit ball of the norm closed algebra is weakly dense
in the whole ball if and only if the weak* closure agrees with the
weak operator closure. This fails only when the weak closure is a von
Neumann algebra but the weak* closure is notand no examples of
this phenomenon are known to exist.
 ANDREW DEAN, Lakehead University, Thunder Bay, Ontario P7B 5E1
Classification of C^{*}dynamical systems

We shall discuss the problem of classifying various kinds of
C^{*}dynamical systems up to equivariant isomorphisms using
invariants.
 GEORGE ELLIOTT, University of Toronto
Amenable C^{*}algebras: The gathering storm

An attempt is made to evaluate the significance of recent results on
the classification of amenable C^{*}algebras, which suggest that
new invariants may be needed.
 JULIANA ERLIJMAN, University of Regina
On braid subfactors and generalisations

I will discuss some of the tools used for computation of invariants of
braid subfactors. I will also mention a generalisation to subfactors
from tensor categories. Work in progress with H. Wenzl.
 REMUS FLORICEL, University of Ottawa, Department of Mathematics and Statistics,
Ottawa, Ontario
E_{0}semigroups in standard form

In his work on classification of type IIE_{0}semigroups,
R. T. Powers has introduced a standard form for spatial
E_{0}semigroups that appears to correspond to the standard form of von
Neumann algebra theory. It was noticed that for some particular
examples of E_{0}semigroups in standard form (CAR/CCR flows,
Tsirelson's type II_{0}examples), the gauge groups act transitively
on the set of normalized units. A result of A. Alevras implies that
for these examples, conjugacy and cocycleconjugacy are equivalent
concepts. It was conjectured by A. Alevras and W. Arveson that the
equivalence between conjugacy and cocycleconjugacy holds for all
E_{0}semigroups in standard form.
The goal of my talk is to discuss my recent results on this problem.
 THIERRY GIORDANO, University of Ottawa, Ottawa, Ontario
Some developments on orbit equivalence

Both in the measurable and in the Borel case, hyperfinite actions are
well understood and classified up to orbit equivalence. In particular
any free Borel action of Z^{2} is (orbit equivalent to)
hyperfinite. In the case of (minimal) topological actions of
Z^{2} on the Cantor set, the same question is still open. In
this talk, I will present recent developments in the study of this
problem. These developments come from a work in progress with
I. Putnam (Victoria) and C. Skau (Trondheim).
 DAVID KERR, University of Muenster, Mathematisches Institut,
Einsteinstraße 62, 48149 Muenster, Germany
Noncommutative geodesic flows and dynamical growth

A substantial part of Riemannian geometry has been concerned with
geodesic dynamics. For example, significant relationships have been
established between the entropy of the geodesic flow and topological
and geometric invariants. In noncommutative geometry, however, this
dynamical aspect has been largely absent. We initiate a study of growth
in noncommutative geodesic flows in the context of Rieffel's work on
group C^{*}algebras as quantum metric spaces.
 DAN KUCEROVSKY, University of of New Brunswick at Fredericton
Locally full extensions

Absorbing extensions are C^{*}algebra extensions having the
property that their sum with a trivial extension is equivalent to the
given extension. It can be shown that an extension that is absorbing
in this sense must be full. We modify the absorption property so that
fullness is replaced, in a natural way, by a property that we call
local fullness.
 MICHAEL LAMOUREUX, University of Calgary, Calgary, Alberta
Representation of linear operators by Gabor multipliers

(Joint work with Peter C. Gibson, and Gary F. Margrave)
We consider a continuous version of Gabor multipliers: operators
consisting of a shorttime Fourier transform, followed by
multiplication by a distribution on phase space (called the Gabor
symbol), followed by an inverse shorttime Fourier transform, allowing
different localizing windows for the forward and inverse transforms.
For a given pair of forward and inverse windows, which linear operators
can be represented as a Gabor multiplier, and what is the relationship
between the (nonclassical) KohnNirenberg symbol of such an operator
and the corresponding Gabor symbol? These questions are answered
completely for a special class of "compatible" window pairs. In
addition, concrete examples are given of windows that, with respect to
the representation of linear operators, are more general than standard
Gaussian windows. The results in the paper help to justify techniques
developed for seismic imaging that use Gabor multipliers to represent
nonstationary filters and wavefield extrapolators.
 JAMIE MINGO, Queen's University
Orthogonal Polynomials and noncrossing annular diagrams

If X_{n} is a n ×n selfadjoint Gaussian random matrix then a
theorem of Johansson (1998) shows that the random variables {Tr(T_{k}(X_{n}))}_{k} converge, as n ® ¥,
to independent Gaussian random variables, where {T_{k}} are the
Chebyshev polynomials of the second kind (suitably centred).
CabanalDuvillard (2001) extended this result to the case of a pair of
independent selfadjoint Gaussian random matrices X_{n} and Y_{n}, in
that he showed that the random variables {Tr
(T_{j}(X_{n})),Tr(T_{k}(Y_{n})),Tr(S_{l,m}(X_{n},Y_{n}))}_{j,k,l,m} converge to independent Gaussian random
variables, where S_{m,n} is a family of Chebyshev polynomials of the
first kind in two noncommuting variables.
I shall give a diagrammatic interpretation of the result of
CabanalDuvillard motivated by a theorem of Andu Nica and me which
shows that the correlation of certain ensembles of random matrices is
given by noncrossing annular diagrams. Some extensions to the case of
Wishart ensembles will be given.
This is joint work with Tim Kusalik (Queen's) and Roland Speicher
(Queen's).
 ANDU NICA, University of Waterloo, Waterloo, Ontario
On and around the qcircular element

The circular element of Voiculescu is a remarkable generator for the
von Neumann algebra of the free group on two generators. The
qcircular element is a deformation of the circular one, which
appears in the context of the qcommutation relations of Bozejko and
Speicher. The talk will survey a few results and (mostly) problems
related to qcircular elements and to some of their siblings called
"zcircular elements".
 IGOR NIKOLAEV, University of Calgary
Moduli of Riemann surfaces and classification of simple dimension
groups of rank 2n

Let S be Riemann surface of genus n ³ 1. Denote by SpecS the
length spectrum of S, i.e. the (countable) set of length of all
simple periodic geodesics on S. Selberg proved that SpecS is a
conformal invariant of S. Later on Wolpert showed that in
"generic" case SpecS is in fact a complete conformal invariant,
i.e. "module" of S.
In present talk we discuss how one can relate to SpecS a simple
dimension group of rank 2n. In this setting, classification of
dimension groups "translates" into the language of moduli of Riemann
surfaces. In the simplest case n=1, one gets a correspondence between
complex and noncommutative tori.
 VLADIMIR PESTOV, University of Ottawa, Department of Mathematics and Statistics,
Ottawa, Ontario K1N 6N5
Distortion property and the infinite unitary group

The distortion property, proved by Odell and Schlumprecht in 1994, has
been one of the most important and stunning discoveries ever made about
the geometry of Hilbert space. To date, there is no direct proof of the
distortion property for Hilbert space, rather it is deduced indirectly
using distortion in some special Banach spaces. Vitali Milman has
suggested that one way to understand the distortion is to try and
formulate its analogues for infinitedimensional groups, such as the
unitary group of the separable Hilbert space. In this talk, we will
report on some preliminary progress in this direction. It turns out
that the distortion property admits a very natural reformulation in the
language of topological transformation groups, and when applied to
concrete examples other than the unitary group, it allows to
incorporate some known results of infinite combinatorics into the same
scheme. Besides, there is an interesting open question which the
speaker is unable to answer (as of September 15): does every
topological group have the distortion property?
 JOHN PHILLIPS, University Of Victoria, Victoria, British Columbia V8W 3P4
From spectral flow to the local index formula

We generalise the local index formula of Connes and Moscovici to the
case of unbounded spectral triples (A,N,D) for a *subalgebra A of a general
semifinite von Neumann algebra, N. In this setting it gives
a formula for (type II) spectral flow along a path joining
D, an unbounded self adjoint BreuerFredholm operator,
affiliated to N, to uDu^{*} for a unitary u Î A.
We start from the spectral flow formula for finitely summable triples
developed by CareyPhillips and show how the the seemingly innocuous
normalising constant in that formula actually gives a new approach to
the ConnesMoscovici results. This is joint work with Alan Carey, Adam
Rennie and Fyodor Sukochev.
 N.C. PHILLIPS, Department of Mathematics, University of Oregon, Eugene,
Oregon 974031222, USA
Crossed products of irrational rotation algebras by
actions of finite groups

The crossed product of every irrational rotation algebra by the
noncommutative Fourier transform is AF. The crossed products by the
standard actions of Z/3Z and Z/6Z always have tracial rank zero, and we are close to proving
that these crossed products are also always AF. (Joint work with
Wolfgang Lück and Sam Walters.)
 BAHRAM RANGIPOUR, University of Victoria
Cyclic homology of Hopf algebras and duality in cyclic
category

We show that various cyclic and cocyclic modules attached to Hopf
algebras and Hopf modules are related to each other via Connes' duality
isomorphism for the cyclic category. This is a joint work with
M. Khalkhali.
 CHRISTIAN SKAU, Norwegian University of Science and Technology(NTNU), N7491
Trondheim, Norway
Free continuous actions of countable groups on the Cantor
set

We give a construction ,applicable to any countable group G, of a
continuous action of G on the Cantor set, which is free and minimal.
Furthermore, the construction in question yields an invariant
probability measure for the action, which is remarkable since the group
do not have to be amenable. We will apply this to prove some new
results.
 KEITH TAYLOR, Dalhousie University, Halifax, Nova Scotia B3H 4J1
From projections to tight frames

Let A be a nonsingular n by n real matrix and let d = detA. For w Î L^{2}(R^{n}), k Î Z and x Î R^{n}, let w_{k,x}(y) = d^{k/2}w(A^{k}yx), for all y Î R^{n}. We call w a tight frame generator (TFG) if
{w_{k,x}:k Î Z,x Î R^{n}} forms a normalized
tight frame in L^{2}(R^{n}). We think of TFGs as semidiscrete
wavelets. The existence of a TFG depends on the nature of A and
smooth TFGs can exist only if A, or its inverse, is expansive. In
this case, we show how smooth TFGs are intimately related to
projections in L^{1}(G), where G is the semidirect product group
formed by the action of the integers on R^{n} through A.

