


Operator Algebras
Org: Marcelo Laca and John Phillips (Victoria) [PDF]
 BEATRIZ ABADIE, Universidad de la Republica, Montevideo, Uruguay
Morita equivalence for Quantum Heisenberg Manifolds
[PDF] 
Quantum Heisenberg manifolds were constructed by M. Rieffel as a
quantization deformation of certain homogeneous spaces H/N_{c}, H
being the Heisenberg group. We describe Morita equivalence within
this family by adapting to this setting some of the techniques
employed in the analogous discussion for noncommutative tori and
Heisenberg C^{*}algebras. One of the main tools employed is the
generalization of a result of P. Green and M. Rieffel about Morita
equivalence of transformation group C^{*}algebras corresponding to
two commuting free and proper actions on a topological space; this
result is generalized to the context of crossed products by Hilbert
C^{*}bimodules.
 BERNDT BRENKEN, University of Calgary
The isolated ideal of a correspondence associated with a
topological quiver
[PDF] 
For a general C^{*}correspondence over a C^{*}algebra A we
identify a hereditary saturated ideal of isolated points of A. The
CuntzPimsner C^{*}algebra of the quotient correspondence is both a
relative CuntzPimsner algebra and a quotient of the CuntzPimsner
algebra of the original correspondence. For the C^{*}correspondence
arising from a topological quiver one obtains a correspondence
associated with the restriction of the quiver to the nonisolated
vertices. The restricted topological quiver satisfies conditions (L)
or (K) if and only if the original quiver does.
 KEN DAVIDSON, University of Waterloo
Nest representations of directed graph algebras
[PDF] 
To each directed graph, there are two natural nonselfadjoint operator
algebras. We investigate nest representations of these algebras,
i.e., representations with range that is dense in an algebra of
upper triangular matrices or a nest algebra. In particular, we show
that the finite dimensional nest representations separate points and
we characterize when there is a faithful nest representation.
This is joint work with Elias Katsoulis.
 GEORGE ELLIOTT, University of Toronto, Toronto, Ontario M5S 2E4
On the classification of nonsimple inductive limits of
matrix algebras over the intervalthe theorem of Robert
[PDF] 
An outline is given of the statement and proof of the theorem of
Leonel Robert the brief statement of which is as follows: The
inductive limits of matrix algebras over the interval are classified
by certain elementary Ktheoretical invariants (including traces).
(At the date of writing, the statement is restricted to the case that
the set of closed twosided ideals of the inductive limit
C^{*}algebra is totally ordered.)
 HEATH EMERSON, University of Victoria
A Gysin sequence in noncommutative geometry
[PDF] 
For a class of group actions on compact spaces, we construct an
analogue of the Gysin sequence in classical topology (which computes
the map on cohomology induced by the projection from the sphere bundle
of a compact manifold, to the manifold). The noncommutative sequence
computes the map on Ktheory induced by the inclusion of the reduced
C^{*}algebra of G into the crossproduct C(X) ×G.
Applications are given to a problem in index theory: namely, we give a
KKtheoretic proof of the theorem of Luck and Rosenberg identifying
the class of the de Rham operator in equivariant KK, with the class
of a certain combinatorial Euler class.
 THIERRY GIORDANO, University of Ottawa, Ottawa
Topological orbit equivalence of free, minimal actions of
Z^{2} on the Cantor set
[PDF] 
In 1959, H. Dye introduced the notion of orbit equivalence and proved
that any two ergodic finite measure preserving transformations on a
Lebesgue space are orbit equivalent. He also conjectured that an
arbitrary action of a discrete amenable group is orbit equivalent to a
Zaction. This conjecture was proved by Ornstein and Weiss and its
most general case by Connes, Feldman and Weiss by establishing that an
amenable nonsingular countable equivalence relation R can be
generated by a single transformation, or equivalently is hyperfinite,
i.e., R is up to a null set, a countable increasing union of
finite equivalence relations.
In the Borel case, Weiss proved that actions of Z^{n} are (orbit
equivalent to) hyperfinite Borel equivalence relations, whose
classification was obtained by Dougherty, Jackson and Kechris.
In 1995, Giordano, Putnam and Skau proved that minimal Zactions on
the Cantor set were orbit equivalent to approximately finite (AF)
relations and their classification was given. Since then some special
classes of minimal free actions of Z^{2} on the Cantor set were shown
to be affable (i.e., orbit equivalent to AFrelations).
In this talk I will indicate 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, free Z^{2}action on the Cantor set is affable.
 DANIEL GONCALVES, University of Victoria
An Introduction to New C^{*}algebras from Tilings
[PDF] 
Given a tiling that satisfies some standard conditions, we introduce
an equivalence relation on R^{d} by saying that two points, x and
y, are equivalent if the patch defined by y on the tiling matches
the patch defined by x translated by yx. We then consider the
C^{*}algebra associated to this equivalence relation and describe
some of its ideals. If time permits we will show that this is a
recursive subhomogeneous C^{*}algebra.
In the case of a substitution tiling we use the inflation map to get
an inductive limit of C^{*}algebras, which is simple. We finish with
a few Ktheory computations for some examples.
 CRISTIAN IVANESCU, University of Northern British Columbia
Continuous trace C^{*}algebras with spectrum [0,1],
inductive limits of continuous trace C^{*}algebras and their
classification
[PDF] 
A classification result will be presented for the class of simple
separable C^{*}algebras which are inductive limits of continuous
trace C^{*}algebras with spectrum [0,1] or finitely many copies of
such a space. An important step that will be discussed is the pulling
back of the invariant from the inductive limit stage to the finite
stage. As will be explained, a new technique is necessary, namely the
use of what we call "a gap" interpreted in a suitable sense.
Joint work with George A. Elliott.
 DAVID KERR, Texas A&M University, College Station, TX, USA 778433368
Dynamics and rigidity in simple C^{*}algebras
[PDF] 
The achievements of the Elliott classification program have
established a striking picture of cohomological rigidity within a
large class of simple nuclear C^{*}algebras. I will discuss some
consequences of this rigidity for the dynamics of typical
automorphisms.
 CLAUS KOESTLER, Carleton University
Central Limit Laws on Jones Towers
[PDF] 
The reinterpretation of commuting squares as a noncommutative version
of stochastic independence gives rise to the notion of Bernoulli
shifts on Jones towers. We investigate central limit theorems on such
towers and show that such laws exist for shifts with a homogeneous
product representation. In particular, we report experimental results
indicating that Wigner's semicircle law is among them.
This is joint work with Rolf Gohm.
 HANGFENG LI, SUNY at Buffalo, Buffalo, New York
A Topology for Ergodic Actions of Compact Quantum Groups
[PDF] 
We show that there is a natural topology on the set of all isomorphism
classes of ergodic actions of any fixed compact quantum group.
 HUAXIN LIN, University of Oregon
The Rokhlin property for automorphisms on simple C*algebras
[PDF] 
We study a general Kishimoto's problem for automorphisms on simple
C^{*}algebras with tracial rank zero. The original problem of
Kishimoto is the following: Let A be a unital simple
ATalgebra with real rank zero and a is an
approximately inner automorphism. Is A\rtimes_{a}Z again a
simple ATalgebra? Let A be a unital separable simple
C^{*}algebra, with tracial rank zero and let a be an
automorphism. Under the assumption that a has certain Rokhlin
property, A\rtimes_{a}Z has tracial rank zero. We also show
that if the induced map a_{*0} on K_{0}(A) fixes a "dense"
subgroup of K_{0}(A) then the tracial Rokhlin property implies a
stronger Rokhlin property. Consequently, the induced crossed product
C^{*}algebras have tracial rank zero. By applying the classification
theorem, this answers affirmatively Kishimoto's original question.
 JAMIE MINGO, Queen's University
Second Order Freeness and Compound Wishart Matrices
[PDF] 
In 1986, D. Voiculescu introduced the Rtransform of a random
variable. The coefficients of this power series, now called free
cumulants, were shown in 1994 by R. Speicher to have a combinatorial
interpretation using noncrossing partitions. In 2004, A. Nica and I
showed that the covariance, when suitably magnified, of the traces of
some standard families of random matrices has a limiting value that
can be interpreted in terms of noncrossing annular partitions.
Indeed the order of elements in a block of a partition is now
significant and the relevant concept is a noncrossing permutation.
These noncrossing annular permutations give rise to second order
cumulants which in turn form the basis of second order freeness. We
shall give a number of natural examples of families of matrices that
exhibit second order freeness.
This work was done in collaboration with B. Collins, A. Nica,
P. Sniady, and R. Speicher.
 IAN PUTNAM, University of Victoria
Relative Ktheory for C^{*}algebras and applications
[PDF] 
I will give a description for a relative Ktheory determined by an
inclusion of a pair of C^{*}algebras. I will describe a result which
shows two such pairs have isomorphic relative Kgroups. Finally, I
will present some examples arising from problems in orbit equivalence
of Cantor minimal systems and from projection method tilings.
 GUNNAR RESTORFF, University of Copenhagen, Denmark
Classification of CuntzKrieger algebras
[PDF] 
Mikael Rørdam and Danrun Huang initiated the classification of
CuntzKrieger algebras in the midnineties, and, recently, the
classification up to stable isomorphism of CuntzKrieger algebras
(satisfying the condition (II) of Cuntz) was completed. This work
raised other new interesting questions.
In my talk I will talk about the classification of CuntzKrieger
algebras, and state the classification theorem. Also, I am planning
to comment on some of the questions raised by this classification
result.
 ANA SAVU, University of Northern British Columbia
Closed and exact functions in interacting particle systems
[PDF] 
Statistical physics has developed a whole variety of interacting
particle systems to explain the transition from microscopic to
macroscopic scale. One of the main difficulties in finding the
scaling limit of a nongradient particle system is to make rigorous
sense of the fluctuationdissipation equations. As has been shown by
Varadhan, the fluctuationdissipation equation is equivalent to a
certain direct sum decomposition of a Hilbert space. For a general
vector field we exhibit two Hilbert spaces, namely the space of so
called "closed functions" and the space of "exact functions" and
we calculate the codimension of the space of "exact functions"
inside the space of "closed functions". In particular we provide a
new approach based on Fourier analysis for the two known cases, the
Glauber field and the secondorder GinzburgLandau field, and we
provide a solution for a new case, the fourthorder GinzburgLandau
field.
The author is grateful to George A. Elliott for many discussions and
ideas that led to the completion of this work.

