


Operator Algebras / Algèbres des opérateurs (Org: Thierry Giordano and/et David Handelman)
 BERNDT BRENKEN, University of Calgary, Calgary, Alberta T2N 1N4
C^{*}algebras associated with topological relations
[PDF] 
With a closed relation on a locally compact Hausdorff space X arising
from a continuous positive map of X to the space of Radon measures on
X we associate a C^{*}algebra, namely the CuntzPimsner algebra
of a particular Hilbert bimodule constructed from the relation. The
relations may have branch points and there are no local homeomorphism
requirements. This family of C^{*}algebras contains the
C^{*}algebras of directed graphs, and the crossed product
C^{*}algebras of topological dynamical systems. Some examples are
considered.
 MAN DUEN CHOI, Department of Mathematics, University of Toronto, Toronto,
Ontario M5S 3G3
The ultimate norm estimate for complex matrices
[PDF] 
It is often a complicated matter to estimate the C^{*}norm (the
usual Hilbertspace operatornorm) of a complex matrix. Nevertheless,
an ultimate answer (without hard computation) can be sought for the
best bound of the norm of T=A + iB where A and B are
(noncommuting) hermitian operators with known eigenvalues. Moreover,
the main result can be extended to cover the case of the sum of two
normal matrices.
(This is a joint work with ChiKwong Li.)
 KEN DAVIDSON, Department of Pure Mathematics, University of Waterloo, Waterloo,
Ontario N2l 3G1
Spans and sums of unitary and similarity orbits of a single operator
[PDF] 
If T is a bounded operator on a separable Hilbert space H
which is not of the form scalar plus compact, then every bounded linear
operator on H can be written as a linear combination of 14
or fewer operators unitarily equivalent to T, as a linear combination
of 6 or fewer operators similar to T, and as a sum of 8 or fewer
operators similar to T. When T is not polynomially compact, the
set of all sums of 2 operators similar to T is dense in
B(H), while if T is polynomially compact, but
not of the form scalar plus compact, then the set of sums of 3
operators similar to T is dense in B(H).
 GEORGE ELLIOTT, University of Toronto, Toronto, Ontario M5S 3G3
Recent progress in the classification of amenable
C^{*}algebras
[PDF] 
A summary is given of recent progress in the classification of
separable amenable C^{*}algebras.
 REMUS C. FLORICEL, University of Ottawa, Ottawa, Ontario
E_{0}semigroups of von Neumann algebras
[PDF] 
The study of E_{0}semigroups of a type I_{¥} factor was
initated by R. Powers and W. Arveson, and many interesting
classification results were obtained in the last years. Our purpose,
in this talk, is to investigate the structure of E_{0}semigroups that
act on arbitrary von Neumann algebras. We show that such a semigroup
can be canonically decomposed as the direct sum of an inner
E_{0}semigroup and a properly outer E_{0}semigroup. This
decomposition is stable under conjugacy and cocycle conjugacy. We
also show that the class of inner E_{0}semigroups can be completely
characterized in terms of product systems.
 NIGEL HIGSON, Penn State University, University Park, Pennsylvania 16802, USA
The residue cocycle of Connes and Moscovici
[PDF] 
The index theorem of Connes and Moscovici provides a formula for Chern
character in cyclic cohomology involving residues of zeta functions
associated to elliptic operators. I shall give a streamlined account
of the `residue cocycle' discovered by Connes and Moscovici and the
associated Fredholm index formula.
 MASOUD KHALKALI, University of Western Ontario, London, Ontario
Invariant cyclic homology
[PDF] 
I will present joint work with B. Rangipour on invariant cyclic
homology. This theory extends cyclic homology of Hopf algebras defined
by Connes and Moscovici and its dual theory defined by present authors.
I will also give several computations and conjectures regarding this
new theory.
 CLAUS KOESTLER, Queen's University
L^{p}martingales on qwhite noises
[PDF] 
A qwhite noise is the von Neumann algebra generated by qBrownian
motion on qFock space. In the case 1 < q < 1 we characterize bounded
L^{p}martingales (1 < p < ¥) w.r.t a canonical filtration as
noncommutative Hardy spaces. This result generalizes work of Pisier
and Xu on ItôClifford martingales which correspond to the case
q=1.
 DAN KUCEROVSKY, Department of Mathematics and Statistics,
University of New Brunswick, Fredericton, New Brunswick E3B
5A3
Locally absorbing extensions
[PDF] 
Absorbing extensions are C^{*}algebra extensions having the property
that their sum with a trivial extension are unitarily equivalent to the
given extension. It can be shown that an extension that is absorbing in
this sense must necessarily be full. In this talk, we attempt to
modify the absorption property so that fullness is replaced, in a
natural way, by a weaker condition we call local fullness.
 MARCELO LACA, University of Victoria, Victoria, British Columbia
KMS states of Pimsner algebras
[PDF] 
I will discuss the structure of the equilibrium state space of
quasifree dynamics on the C^{*}algebras associated by M. Pimsner
to a Hilbert bimodule. (Current joint work with S. Neshveyev.)
 JAMIE MINGO, Queen's University
Two Point functions for families of random matrices
and noncrossing annular partitions
[PDF] 
(joint work with A. Nica)
Let (W, P) be a probability space and X be selfadjoint
Gaussian random matrix, i.e. X: W®M_{n}(C)_{s.a.} is a matrix valued random variable with
independent and normally distributed entires. If we write X = (f_{i,j})_{i,j = 1}^{n} and X is normalized so that
E(f_{i,j}) = 0 for all i and j, and
E(Â(f_{i,j})^{2}) = E(Á(f_{i,j})^{2}) = 1/(2n) for i ¹ j and
E(f_{i,i}^{2}) = 1/n, then E. Wigner showed that
E 
æ è

tr(X^{2p}) 
ö ø

= c_{p} + O(1/n^{2}) 

where tr is the normalized trace and c_{p} is the pth Catalan
number.
We shall show that
E 
æ è

tr(X^{p})tr(X^{q}) 
ö ø

E 
æ è

tr(X^{p}) 
ö ø

E 
æ è

tr(X^{q}) 
ö ø

= 
a_{p, q} n^{2}

+ O(1/n^{4}) 

where a_{p,q} is the number of noncrossing annular partitions
of an annulus with p vertices on the inner circle and q vertices on
the outer circle.
 MATTHIAS NEUFANG, Carleton University, Ottawa, Toronto K1S 5B6
Amplification of completely bounded operators and applications
[PDF] 
It is a characteristic feature of completely bounded operators on
B(H) to admit an amplification to the level of B(H Ä_{2} K),
where H and K are Hilbert spaces. Using Wittstock's HahnBanach
principle and Tomiyama's slice map theorem, one deduces that, more
generally, any completely bounded map on M can be amplified to a map
on the von Neumann tensor product M [`(Ä)] N, whenever M and
N are either von Neumann algebras or dual operator spaces with at
least one of them sharing the w^{*} operator approximation property.
Our aim is to show that there is a simple and explicit formula of an
amplification of completely bounded operators for all such pairs
(M,N), thus providing a constructive approach to the
amplification problem. The key idea is to combine two fundamental
concepts in the theory of operator algebras, one being classical, the
other one fairly modern: Tomiyama's slice maps on the one hand, and the
description of the predual of M [`(Ä)] N given by
EffrosRuan in terms of the projective operator space tensor product,
on the other hand.
We will further discuss the question of uniqueness of such an amplification,
but mainly focus on various applications of our construction, such as:
 a generalization of the socalled GeKadison Lemma;
 the amplification of completely bounded module homomorphisms;
 an algebraic characterization of normality for completely
bounded
maps in terms of a commutation relation for the associated amplification.
In particular, the latter result leads us to a new concept in operator
algebra theory which may be viewed as a tensor product version of Arens
regularity.
 VLADIMIR PESTOV, University of Ottawa, Ottawa, Ontario K1N 6N5
The fixed point on compacta property of some topological
groups related to operator algebras and ergodic theory
[PDF] 
A topological group G has the fixed point on compacta (f.p.c.)
property if there is a fixed point in each compact space upon which G
acts continuously. This is a very strong version of amenability, which
is why such groups are also called extremely amenable. The property is
closely linked to Ramsey theory and to geometry of highdimensional
structures. Among a number of known groups with the f.p.c. property,
many are linked to operator algebras and ergodic theory, and we will
dwell on some of the recent developments in this direction.
 JOHN PHILLIPS, University of Victoria, Victoria, British Columbia V8W 3P4
Centrevalued index of Toeplitz operators with noncommuting
symbols
[PDF] 
We begin with a unital C^{*}algebra A and a unital C^{*}subalgebra,
Z of the centre of A. We assume that we have a faithful, unital
Ztrace t and a continuous action a:R®Aut(A) leaving t and hence Z invariant. We let
d be the infinitesimal generator of a on A.
We have in this setting a largest (in the sense of
quasicontainment) *representation of A on a Hilbert space which
carries a faithful, unital u.w.continuous Z^{u.w.}trace
[`(t)]: A^{u.w.} ® Z^{u.w.} extending t. We assume
that A is concretely represented on this Hilbert space. We denote by
A and Z respectively, the ultraweak closures
of A and Z. One shows that there is an u.w.continuous action
[`(a)]:R ®Aut(A) extending a
and leaving [`(t)] and Z invariant.
At this point we construct a representation, Ind = [(p)\tilde]×l of A\rtimesR on a certain selfdual
HilbertZ module H_{A} constructed from a
certain ``ZHilbert Algebra,'' A. We let
M=Ind(A\rtimesR)^{¢¢} which contains
Z in its centre and has a faithful, normal semifinite
Ztrace [^(t)]. This construction is half the
battle. We let H denote the image of the Hilbert Transform in
M and let P = [1/2](H+1) in M. We then
consider the semifinite von Neumann algebra,
with the faithful, normal, semifinite Ztrace obtained by
restricting [^(t)]. For a Î A we define the Toeplitz
operator
We prove the following theorem.
Theorem 1
Let A be a unital C^{*}algebra and let Z Í Z(A) be a unital
C^{*}subalgebra of the centre of A. Let t: A ® Z be a
faithful, unital Ztrace which is invariant under a continuous action
a of R. Then for any a Î A^{1}Çdom(d),
the Toeplitz operator T_{a} is Fredholm relative to the trace
[^(t)] on N=P(Ind(A\rtimesR)^{¢¢})P,and

^ t

ind(T_{a})= 
1 2pi

t 
æ è

d(a)a^{1} 
ö ø

. 

 IAN PUTNAM, University of Victoria, Victoria, British Columbia
Recent results on topological orbit equivalence
[PDF] 
I will describe recent work with T. Giordano (Ottawa) and C. Skau
(Trondheim) on topological orbit equivalence for actions of higher rank
free abelian groups on the Cantor set.
 ROLAND SPEICHER, Queen's University, Kingston, Ontario K7L 3N6
Free probability
[PDF] 
I will present some recent results from free probability theory.

