


Operator Algebras, Operator Spaces and Harmonic Analysis
Org: Ken Davidson and Brian Forrest (Waterloo) [PDF]
 MONICA ILIE, Lakehead University, Thunder Bay, Ontario
On completely bounded Fourier algebra homomorphisms
[PDF] 
We present a survey of some recent results regarding completely
bounded homomorphisms of the Fourier algebra of a locally compact
group G. We discuss the description as well as the w^{*}w^{*}
continuity of such homomorphisms.
This talk is based on separate work with Nico Spronk and with Ross
Stokke.
 EBERHARD KANIUTH, University of Paderborn, 33095 Paderborn, Germany
Stable rank and real rank of group C^{*}algebras
[PDF] 
The stable rank and the real rank of a C^{*}algebra A have been
introduced by Rieffel and Brown and Pedersen, respectively, as
noncommutative analogues of the complex and the real dimension of
topological spaces. In fact, for the algebra of continuous functions
on a compact Hausdorff space X one has RR(C(X)) = dimX and a
similar formula for sr(C(X)). Both of these ranks (in particular
the condition RR(A)=0) have played a significant role in the
classification of C^{*}algebras, and there is now an extensive
literature available, especially on group C^{*}algebras. The talk
will focus on recent work with R. J. Archbold (Aberdeen, Scotland) and
will discuss formulæ for the ranks of group C^{*}algebras C^{*}(G)
for various classes of locally compact groups G, such as nilpotent
groups, motion groups and crystal groups.
 DAVID KRIBS, University of Guelph
On graph algebras and related things
[PDF] 
In this talk I will discuss recent joint work with Elias Katsoulis on
the class of operator algebras associated with directed graphs. Time
permitting, I will also outline a unified framework for studying both
graph algebras and the causal history formalism from quantum gravity.
 ANTHONY LAU, University of Alberta, Edmonton, AB T6G 2G1
Separation and Extension Properties of Positive Definite
Functions on Locally Compact Groups
[PDF] 
Let G be a locally compact group and P(G) be the set of continuous
positive definite functions on G, i.e., all continuous
functions f: G®C such that Sl_{i}[`(l)]_{j} f(x_{i} x^{1}_{j}) ³ 0 for any l_{1}, ...,l_{n} Î C and x_{1},...,x_{n} Î G. A closed subgroup H
of G is called separating if for each x Î G\H, there exists
f Î P(G) such that f(x) ¹ 1, and f(h)=1 for all
h Î H; H is called extending if for each f Î P(H) there
exists [(f)\tilde] Î P(G) such that [(f)\tilde] extends f. G is
said to have the separation property (respectively extension property)
if each closed subgroup H of G separating (respectively
extending). In this talk I shall discuss some recent results with
Eberhard Kaniuth on the separation and the extension properties for
locally compact groups and their subgroups.
 LAURENT MARCOUX, Department of Pure Mathematics, University of Waterloo,
Waterloo, Ontario N2L 3G1
Amenable, abelian operator algebras
[PDF] 
Suppose that A is a Banach algebra and that M
is a Banach space which is also a bimodule over A. If the
action of A on M is continuous, then we say
that M is a Banach bimodule over A. For these
bimodules, the dual space M^{*} is automatically a Banach
bimodule over A via the actions a ·j(m) : = j(m ·a) and j·a(m) : = j(a ·m)
for all a Î A, m Î M and j Î M^{*}. A derivation of an algebra A
into a bimodule M is a map d which satisfies
d(ab) = a ·d(b) + d(a) ·b for all a,b Î A. Examples include the inner derivations
d_{m} (a) = a ·m  m ·a for m Î M fixed.
Finally, A is said to be amenable if all
derivations of A into dual Banach bimodules M
are inner.
In this talk we shall discuss the problem of similarity of abelian,
amenable algebras of operators on a Hilbert space H to
C^{*}algebras.
 PAUL MUHLY, University of Iowa
Models and Representations
[PDF] 
In this talk, I will discuss recent work with Baruch Solel concerning
canonical models for representations of tensor algebras. Very
roughly, we have developed a model theory for completely contractive
representations of tensor algebras that is an almost exact analogue of
the model theory for single contraction operators developed by
Sz.Nagy and Foia s, and others.
 MATTHIAS NEUFANG, Carleton University, Ottawa, Ontario K1S 5B6
Operator analogues of various algebras arising in abstract
harmonic analysis
[PDF] 
In the framework of his concept of an amenable representation,
M. B. Bekka has introduced an operator version of the C^{*}algebra
LUC(G) of bounded left uniformly continuous functions on a locally
compact group G. This C^{*}subalgebra of B(L_{2}(G)), denoted by
X(L_{2}(G)), contains both LUC(G) and the compact operators on
L_{2}(G). Using my convolution type product of trace class operators
on L_{2}(G), I present a structural analysis of X(L_{2}(G)).
Moreover, the dual X(L_{2}(G))^{*} naturally becomes a (completely
contractive) Banach algebra. The latter admits a completely isometric
representation as completely bounded operators on B(L_{2}(G)); as
such, it gives new insight into the representation theoretical
programme (carried out by M. Neufang, Z.J. Ruan and N. Spronk) of
studying algebras in abstract harmonic analysis as subalgebras of CB( B(L_{2}(G)) )such as the measure algebra and the
completely bounded multipliers of the Fourier algebra. Finally, the
investigations of X(L_{2}(G))^{*} lead to what may be considered as a
noncommutative analogue of the measure algebra. Extensions of the
results to locally compact quantum groups are also discussed.
This is joint work with Z.J. Ruan.
 DAVID PITTS, University of Nebraska, Lincoln, NE 68588
Isomorphisms for Triangular Subalgebras of C^{*}Diagonals
[PDF] 
Renault, extending a definition of Kumjian, defines a
C^{*}diagonal to be a pair (C,D), where C is a
unital C^{*}algebra and D Í C is a unital abelian
C^{*}subalgebra satisfying:
(a) every pure state of D has a unique extension to a
pure state of C,
(b) [`span] {v Î C: v Dv^{*} Èv^{*} D
v Í D} = C, and
(c) the unique conditional expectation E : C
® D (existence is implied by (a)) is faithful.
A normclosed subalgebra A Í C is
triangular if AÇA^{*} = D.
In this talk, I will discuss the following result:
Theorem
For i=1,2 suppose (C_{i},D_{i}) are C^{*}diagonals and A_{i} Í C_{i} is triangular. Then any bounded isomorphism
q: A_{1} ® A_{2} is completely bounded with
q_{cb} = q.
Let B_{i} Í C_{i} be the C^{*}subalgebra of C_{i}
generated by A_{i}. It turns out that B_{i} is the
C^{*}envelope of A_{i}. Thus, when q is isometric, the
theorem implies that q extends to a *isomorphism of
B_{1} onto B_{2}. This provides a new proof for, and an
extension of, a result of Muhly, Qiu and Solel.
 ZHONGJIN RUAN, University of Illinois at UrbanaChampaign
Group C^{*}algebras and Related Harmonic Analysis
Properties
[PDF] 
In this talk, I will discuss some properties on group C^{*}algebras
and their connections to Fourier algebras and FourierStieltjes
algebras.
 VOLKE RUNDE, University of Alberta
Amenability of the Fourier algebra in the cbmultiplier norm
[PDF] 
H. Leptin proved that a locally compact group G is amenable if and
only if its Fourier algebra A(G) is has a bounded approximate
identity. On the other hand, there are nonamenable groups, such as
F_{2}, the free group in two generators, that have an
approximate identity that is bounded with respect to the cbmultiplier
norm on A(G). Later, Z.J. Ruan improved Leptin's theorem by
showing that G is amenable if and only if A(G) is operator
amenable. In this talk, which is based on joint work with
B. E. Forrest and N. Spronk, we show that the completion of
A(F_{2}) in the cbmultiplier norm is operator amenable.
 ROGER SMITH, Texas A&M University, College Station, TX 77843, USA
The Pukanszky invariant in group factors
[PDF] 
In 1960, Pukanszky associated to each maximal abelian selfadjoint
subalgebra (masa) in a finite factor an isomorphism invariant which is
a subset of the extended natural numbers. The primary examples of
factors arise from groups, and abelian subgroups give examples of
masas. In this talk I will describe how to compute the Pukanszky
invariant, and show how this can be used to give new examples of the
invariants that can arise.
This is joint work with Allan Sinclair.
 NICOLAAS SPRONK, University of Waterloo, Waterloo Ontario
The Spine of a FourierStieltjes Algebra
[PDF] 
Let G be a locally compact group, A(G) be its Fourier algebra and
B(G) its FourierStieltjes algebra. If G is abelian, with
Pontraygin dual group [^(G)], the B(G) is isometrically
isomorphic to the measure algebra M([^(G)]). A subalgebra of
M([^(G)]) was developed independantly by J. Taylor and J. Inoue in
the '70s, which comprised of all "maximal group algebras" inside of
M([^(G)]); this was called the spine of M([^(G)]).
We develop the spine of B(G) for any locally compact group G. It
is comprised of all of the "maximal Fourier algebras" inside of
B(G). More precisely, if t is any group topology on G which
is coarser than the ambient topology, and for which the completion
G_{t} is locally compact, we obtain a copy of A(G_{t}) in
B(G), and the sum of all of these algebras is the spine. If we
restrict ourselves to what we call nonquotient topologies, we may
even realise the spine as a direct sum. This algebra admits an
appealing structure as a graded Banach algebra, graded over a lattice
semigroup. As such we can compute its Gelfand spectrum, which in
turn is a semigroup, the spine compactification of G. I
will illustrate some examples with Lie groups.
This represents part of my joint work with Monica Ilie.

