




Operator Algebras / Algèbres des opérateurs (Michael Lamoureaux and Ian Putnam, Organizers)
 BERNDT BRENKEN, University of Calgary, Calgary, Alberta T2N 1N4
Representations of graph C^{*}algebras and endomorphisms
of type I algebras with discrete center

CuntzKrieger algebras associated with infinite, possibly infinite
valued matrices with any number of zero entries were introduced
previously, and found to correspond exactly to C^{*}algebras of
directed graphs with any number of edges, sources, sinks, and isolated
vertices. Here we show that the correspondence between representations
and endomorphisms established earlier involving the original
CuntzKrieger algebras extends to this setting, so to one between
representations of CuntzKrieger algebras for infinite matrices and
endomorphisms of a direct sum of type I factor von Neumann algebras.
This further demonstrates that this is a natural approach to defining
CuntzKrieger algebras in the more general setting.
 KEN DAVIDSON, Department of Pure Mathematics, University of Waterloo,
Ontario N2L 3G1
Primitive limit algebras and C^{*}envelopes

We study irreducible representations of regular limit subalgebras of
AFalgebras. The main result is twofold: every closed prime ideal
of a limit of direct sums of nest algebras (NSAF) is primitive, and
every prime regular limit algebra is primitive. A key step is that the
quotient of a NSAF algebra by any closed ideal has an AF
C^{*}envelope, and this algebra is exhibited as a quotient of a
concretely represented AF algebra. When the ideal is prime, the
C^{*}envelope is primitive. The GNS construction is used to
produce algebraically irreducible representations for quotients of NSAF
algebras by closed prime ideals. Thus the closed prime ideals of NSAF
algebras coincide with the primitive ideals. Moreover these
representations extend to *representations of the C^{*}envelope
of the quotient, so that a fortiori these algebras are also operator
primitive. The same holds true for arbitrary limit algebras and the
{0} ideal.
 GEORGE ELLIOTT, University of Toronto, Toronto, Ontario
On the classification of nonsimple C^{*}algebras

A brief survey is given of progress in the classification of nonsimple
amenable C^{*}algebras. (Many classification results, but by no
means all, are focussed on the simple case.) Some recent work in
progress is described.
 JULIANA ERLIJMAN, University of Regina, Regina, Saskatchewan S4S 0A2
Nsided braid type subfactors

We generalize the construction of twosided pairs of braid subfactors
of the hyperfinite II_{1} factor to nsided pairs. The nsided
inclusion contains a sequence of intermediate subfactors with the
property that the inclusion of any two consecutive subfactors is
conjugate to the twosided pair.
 DOUGLAS R. FARENICK, Department of Mathematics and Statistics, University of Regina
Regina, Saskatchewan S4S 0A2
An algebraic analogue of the FukamiyaKaplansky Lemma

The FukamiyaKaplansky Lemma is recast as: in any complex algebraic
algebra with positive involution, elements of the form a*a have
nonnegative spectrum. One application of this result is an algebraic
characterisation of finitedimensional C^{*}algebras (real or
complex) among all finitedimensional involutive algebras.
 DAVID KRIBS, University of Iowa, Iowa, USA
Completely positive maps in dilation and wavelet
decompositions

Certain representations of the Cuntz C^{*}algebra arise in wavelet
analysis and through the minimal isometric dilations of row
contractions acting on Hilbert space. Decomposition theories for
dealing
with such representations have recently been developed; however, the
computations required can become quite involved. An alternative method
will be presented for obtaining this information strictly in terms of a
completely positive map on finite dimensional space.
 DAN KUCEROVSKY, Department of Mathematics and Statistics, University of New
Brunswick, Fredericton, New Brunswick E3B 5A3
Idealrelated extensions

We explain the extension picture of KKtheory. The deeper
applications of the theory depend on the properties of socalled
absorbing extensions. We give a characterization of absorbing
extensions, and introduce an ``idealrelated'' version of the
absorption property.
 HUAXIN LIN, Department of Mathematics, University of Oregon, Eugene, Oregon 97403,
USA
Classification of simple amanable C^{*}algebras

We give a classification theorem for unital separable simple nuclear
C^{*}algebras with tracial topological rank zero which satisfy the
Universal Coefficient Theorem. We prove that if A and B are two
such C^{*}algebras and

æ è

K_{0}(A), K_{0}(A)_{+}, [1_{A}], K_{1}(A) 
ö ø

@ 
æ è

K_{0}(B), K_{0}(B)_{+}, [1_{B}], K_{1}(B) 
ö ø

, 

then A @ B.
 LAURENT MARCOUX, Department of Mathematics, University of Alberta, Edmonton,
Alberta T6G 2G1
Linear combinations of projections in certain C^{*}algebras

In this talk we shall show that if a C^{*}algebra A
admits a certain 3 ×3 matrix decomposition, then every
commutator in A can be expressed as a linear combination of
at most 210 (and often fewer) projections in A.
In certain C^{*}algebras, this is sufficient to allow us to express
every element as a linear combination of a (fixed) finite number of
projections.
 JAMES MINGO, Queen's University
qcircular and zcircular elements of a C^{*}algebra

(joint work with Alexandru Nica, University of Waterloo)
If 1 £ q £ 1 the qcommutation relations for a Hilbert space
K are
a(x) a(h)^{*} q a(h)^{*} a(x) = áx, hñ1 for x,h Î K 

These interpolate between the canonical commutation (q = 1)
and the anticommutation (q = 1) relations.
Bo\.zejko and Speicher showed that the annihilation operators
{a(x)}_{x} can be realized as bounded operators on qFock
space. Elements of the form a(x) + a(x)^{*} are the q
analogues of semicircular elements and so for x^h
the qcircular elements

a(x)+a(x)^{*}+i 
æ è

a(h) + a(h)^{*} 
ö ø

Ö2



are the analogues of the circular elements introduced by Voiculescu.
qcircular elements can be characterized by combinatorial relations
which we show can be extended to the case where q is a complex number
z with z < 1. For these elements we show that there is an
asymptotic model of random matrices.
 ALEXANDRU NICA, University of Waterloo, Waterloo, Ontario N2L 3G1
Rcyclic families of matrices in free probability

We introduce the concept of Rcyclic family of matrices with entries
in a noncommutative probability space; the definition consists in
asking that only the `cyclic' noncrossing cumulants of the entries of
the matrices are allowed to be different from zero.
Let A_{1}, ¼, A_{s} be an Rcyclic family of d ×d
matrices over a noncommutative probability space ( A, j). We prove a convolutiontype formula for the explicit computation of
the joint distribution of A_{1}, ¼, A_{s} (considered in M_{d}(A), with the natural state), in terms of the joint
distribution (considered in the original space (A,j)) of the entries of the s matrices. We present several
applications of this formula.
Moreover, let A_{1},¼,A_{s} be as above, and let D Ì M_{d} (A) denote the algebra of scalar diagonal
matrices. We prove that the Rcyclicity of the family A_{1},¼,A_{s} is equivalent to a freeness requirement, when one works with
amalgamation over the subalgebra D.
 JOHN PHILLIPS, Victoria
Spectral Flow and the Dixmier trace

(joint work with A. Carey and F. Sukochev)
Suppose that (H,D) is an unbounded Fredholm Module for the
C^{*}algebra, A. That is, (1+D^{2})^{1} is compact and A is
represented on H in such a way that the commutators [D,a] are
bounded for a in a dense *subalgebra, A, of A. Let
{m_{n}} be the decreasing sequence of eigenvalues of the compact
operator (1+D^{2})^{1/2} (which is a smooth replacement for D^{1}
which may not exist). If

N å
n = 0

m_{n} = O(logN) 

then we say that (H,D) is L^{(1,¥)}summable. We observe that:
m_{N} £ 
1 N+1


N å
n = 0

m_{n} = O 
æ è


logN N+1


ö ø

. 

Hence the sequence {m_{n}} (and therefore the Fredholm Module (H,D))
is psummable for each p > 1.
Now for each unitary u Î A, the spectral flow along the linear path
joining D to uDu^{*} makes sense and is known to be the Fredholm index of
PuP on P(H) where P is the projection on the nonnegative eigenspace
of D. By previous work of Carey and Phillips, we have the following
analytic formula for this index for each p > 1:
sf(D,uDu^{*}) = ind(PuP) = 
1 M_{p}


ó õ

1
0

Tr 
æ è

u[D,u^{*}](1+D_{t}^{2})^{p/2} 
ö ø

dt, 

where D_{t} = (1t)D+tuDu^{*} and M_{p} = ò_{¥}^{¥}(1+x^{2})^{p/2} dx.
By taking the limit of this formula (which is by no means obvious) we are able
to deduce a formula of Connes' implicit in his book. That is,
sf(D,uDu^{*}) = ind(PuP) = 
1 2

T_{w} 
æ è

u[D,u^{*}](1+D^{2})^{1/2} 
ö ø

. 

Where, T_{w} is ``the'' Dixmier trace on the ideal
L^{(1,¥)}. In the case that Connes usually considers (
i.e., D^{1} exists as a bounded operator) our formula is easily
seen to imply:
sf(D,uDu^{*}) = ind(PuP) = 
1 2

T_{w}(u[D,u^{*}]D^{1}), 

which is the usual form of Connes' formula.
 N. CHRISTOPHER PHILLIPS, Department of Mathematics, University of Oregon, Eugene,
Oregon 974031222, USA
C^{*}algebras of minimal diffeomorphisms

This work is joint with Lin Qing.
Let M be a compact smooth manifold, and let h: M ® M be a
minimal diffeomorphism. Our main theorem is that the crossed product
C^{*}algebra C^{*}(Z,M,h) can be represented as a direct
limit, with no dimension growth, of what we call recursive
subhomogeneous C^{*}algebras. Many (but not all) properties of
simple direct limits, with no (or very slow) dimension growth, of
homogeneous C^{*}algebras, carry over to such direct limits. In
particular, the crossed product always has stable rank one. Moreover,
if h is uniquely ergodic, so that C^{*}(Z,M,h) has a unique
normalized trace t, then C^{*}(Z, M, h) has real rank
zero if and only if the range of t_{*} on
K_{0}(C^{*}(Z,M,h)) is dense in R, and, in
this case, C^{*} (Z,M,h) belongs to the class of simple
separable nuclear C^{*}algebras known to be determined up to
isomorphism by the Elliott invariant.
With suitable choices of M and h, we can show that the crossed
product C^{*}algebra preserves much less information about the
diffeomorphism h than is the case for minimal homeomorphisms of the
Cantor set (as in the work of GiordanoPutnamSkau). In particular, we
can give two minimal diffeomorphisms of the 3torus (S^{1})^{3} which
are not orbit equivalent but whose crossed product C^{*}algebras are
isomorphic. We can also give two minimal diffeomorphisms, one of S^{3}×S^{1} and one of S^{5} ×S^{1}, whose crossed product
C^{*}algebras are isomorphic; since the manifolds don't even have the
same dimension, orbit equivalence is clearly impossible.
 CHRISTIAN SKAU, Norwegian University of Science and Technology(NTNU), Norway
Toeplitz flows and Choquet simplicesusing Bratteli diagrams
and dimension groups to establish a connection

Toeplitz flows,defined via arithmetical progressions,form a
distinguished class of minimal dynamical systems. It was shown by
T. Downarowicz in 1991 (a preliminary result had been obtained by
S. Williams in 1984) by a long and arduous argument that every Choquet
simplex can be realized as the set of invariant probability measures of
a Toeplitz flow. We will present a more conceptual proof, extending
Downarowicz result, using Bratteli diagrams and the basic theory of
dimension groups. We will simultanously obtain a characterization of
the K0 groups associated to Toeplitz flows.
 KEITH TAYLOR, University of Saskatchewan, Saskatoon, Saskatchewan
A connection between projections in the L^{1}algebra of a
locally compact group and wavelets

We discuss a strong connection between projections (selfadjoint
idempotents) in L^{1}(G) for the group G formed as a semidirect
product of the reals acting on an ndimensional vector space on the
one hand and functions which generate continuous frames on the other
hand.

