




Abstract Harmonic Analysis / Analyse harmonique abstraite (Anthony Lau and Keith Taylor, Organizers)
 LARRY BAGGETT, University of Colorado, Boulder, Colorado 80309, USA
An abstract formulation of wavelet theory

We present an abstract formulation for an ``internal affine structure''
on a Hilbert space in terms of a unitary representation of a special
kind of group. Given such an affine structure, natural definitions of
wavelet subspace and multiresolution analyses can easily be made. Some
of the ``magical'' formulas of wavelet theory are clarified from this
abstract point of view.
 BRIAN FORREST, Department of Pure Mathematics, University of Waterloo,
Waterloo, Ontario N2L 3G1
The Fourier algebra as an operator space

As the predual of a von Neumann algebra, the Fourier algebra
A(G) of a locally compact group G has a natural operator
space structure. In this talk we will show how the operator
space structure can play an important role in studying the
Fourier algebra of a nonabelian group.
 JOHN FOURNIER, Department of Mathematics, University of British Columbia,
Vancouver, British Columbia V6T 1Z2
The peculiar cases of the lower majorant property

Use the symbol [^(f)](n) for the coefficients in the complex form of
the Fourier series of a function f on the interval [p, p).
It is known that if p = 2k/(2k1) for some positive integer k > 1,
then for each function f in L^{p} there exists a function F
in L^{p} that majorizes f in the sense that [^(F)](n) ³ [^(f)](n) for all n, but that also satisfies F_{p} £ f_{p}. The
existence proofs for such paradoxical majorants do not provide
constructions of them, but each function in one of these peculiar L^{p}
spaces has a unique majorant of minimal L^{p}norm. We add slightly to
the classical existence proof and a modern proof, and
get some interesting restrictions on the form of the minimal majorant.
We also do this for the Schatten norms on matrices.
 FEREIDOUN GHAHRAMANI, Department of Mathematics, University of Manitoba, Winnipeg,
Manitoba R3T 2N2
Point derivations and amenability of measure algebras

Let G be a locally compact group. In 1972 B. E. Johnson proved that
the group algebra of G is an amenable Banach algebra if and only if
G is an amenable group. The question of amenability of the measure
algebras was left open. In 1976 G. Brown and W.Moran proved that if
G is nondiscrete and abelian then the measure algebra of G has a
nonzero point derivation. We show that the measure algebra of every
nondiscrete group G has a nonzero point derivation. This in
particular resolves the questions about weak amenability and
amenability of the measure algebrs. This is joint work with
H. G. Dales and A. Ya. Helemskii.
 COLIN GRAHAM, Bowen Island
Arens Regularity for quotients A_{p}(E) of the Herz algebra

If the subset E of T supports a synthesizable
ppseudofunction T, then the restriction A_{p}(E) to E of the
Herz algebra on T is not Arens regular. The proof is direct,
by brute force, and does not show the existence of Day points (see
below) for A_{p}(E).
 EDMOND GRANIRER, Department of Mathematics, University of British Columbia,
Vancouver, British Columbia V6T 1Z2
A few remarks on quotients of the Fourier Algebra

Some functional analytic properties of quotients of the Fourier and the
Herz algebras connected with the nonuniqueness of invariant means on
the space of pseudomeasures will be discussed.
 ZHIGUO HU, University of Windsor, Windsor, Ontario N9B 3P4
Inductive extreme nonArens regularity of the Fourier algebra

Let G be a locally compact group and let A(G) be the Fourier
algebra of G. In this talk, we will show that A(G) is inductive
extreme nonArens regular in the sense that A(G) is an inductive
union of a family of Fourier algebras such that this family has an
inductively compatible extreme nonArens regularity.
 WOJCIECH JAWORSKI, School of Mathematics and Statistics, Carleton University,
Ottawa, Ontario K1S 5B6
Probability measures on almost connected amenable locally
compact groups and some related ideals in group algebras

Given a locally compact group G let J_{a}(G) denote the
set of all closed left ideals J in L^{1}(G) which have the form
J=[L^{1}(G)*(d_{e} m)]^{} where m is an absolutely continuous
probability measure on G. We explore the order structure of
J_{a}(G) when J_{a}(G) is ordered by inclusion.
When G is connected and amenable, every nonempty family
F Í J_{a}(G) admits both a minimal and a
maximal element, in particular, every ideal in J_{a}(G)
contains an ideal that is minimal in J_{a}(G). Furthermore,
every chain in J_{a}(G) is necessarily finite. A natural
generalization of these results to almost connected amenable groups is
discussed. Our proofs use results from the theory of boundaries of
random walks.
 EBERHARD KANIUTH, University of Paderborn, Paderborn, Germany
Hardy's uncertainty principle for Lie groups

Roughly speaking, the classical uncertainty principles say that a
nonzero function on the real line and its Fourier transform cannot
simultaneously be concentrated. One possibility of measuring
concentration is to consider the decay of a function at infinity. An
important result making this precise is Hardy's theorem (1933).
Naturally, there has been some effort to prove Hardy like theorems for
various connected Lie groups. Specifically, analogues of Hardy's
theorem have been shown for the Heisenberg groups, for Euclidean motion
groups, for Cartan motion groups and for SL(2,R). Very recently and
most notably, analogues of Hardy's uncertainty principle were
established for general noncompact connected semisimple Lie groups
with finite centre and for arbitrary connected and simply connected
nilpotent Lie groups. In all these cases the decay condition for the
Fourier transform has to be replaced by a similar condition on the
HilbertSchmidt norm of the operator valued Fourier transform on the
dual space of the underlying group. The talk will present some of
these results and will focus on explaining the setup.
 MAHMOOD KHOSHKAM, Department of Mathematics and Statistics, University of
Saskatchewan, Saskatoon, Saskatchewan
Regular representations of groupoids

The left regular representation of a locally compact groupoid is
constructed in the Hausdorff as well as in the nonHaudorff case. In
the Hausdorff case, this representation takes place on a natural Hilber
module over C_{0}(X) where X is the space of units of the groupoid.
In the nonHausdorff case, a natural locally compact space Y, which
contains X as a dense subset, is constructed. The representation
takes place on a Hilber C_{0}(Y)module.
 LAURENT MARCOUX, University of Alberta, Edmonton, Alberta
Weak amenability of triangular Banach algebras

Let A and B be unital Banach algebras and
M be a Banach A, Bmodule. Then
T = [
]
becomes a triangular Banach algebra when equipped with an
appropriate norm. A Banach algebra is said to be nweakly
amenable if all derivations from that algebra into its nth dual
space are inner. In this talk we investigate the nweak amenability
of triangular Banach algebras in terms of the nweak amenability of
A and B and their action upon M.
 TIANXUAN MIAO, Department of Mathematics, Lakehead University, Thunder Bay,
Ontario P7B 5E1
On the multiplier algebra of A_{p}(G)

Let B_{p}^{M}(G) be the pointwise multiplier algebra of the
FigaTalamancaHerz algebra A_{p}(G) equipped with the multiplier norm
on B_{p}^{M}(G). We will prove that B_{p}^{M}(G) is a dual Banach space.
Properties of B_{p}^{M}(G) such as the density of A_{p}(G) in B_{p}^{M}(G)
will be discussed.
 MATTHIAS NEUFANG, Department of Mathematical Sciences, Edmonton, Alberta T6G 2G1
Abstract harmonic analysis and quantized functional analysis

Among the most interesting objects in abstract harmonic analysis are
the measure algebra M(G) and the dual LUC(G)^{*} of the bounded left
uniformly continuous functions on a locally compact group G, endowed
with the Arens product. Adopting the point of view of operator spaces,
we present the construction of an isometric representation of these
algebras in the algebra CB(B(H)) of completely bounded
operators on B(H), where H=L_{2}(G), which extends earlier work of
E. Størmer and F. Ghahramani.
The talk gives an overview of the various fruitful interactions between
abstract harmonic analysis and quantized functional analysis in this
framework. The central features are operator analogues of the
wellknown theorems of Brainerd/Edwards and Curtis/FigàTalamanca.
Further investigating the structure of the image algebras arising from
our representation, we obtain bicommutant theorems for the latter which
parallel von Neumann's classical bicommutant theorem in the setting of
completely bounded operators on B(H).
Our main tools are factorization theorems for families in the left
LUC(G)^{*}modules LUC(G), L_{¥}(G) and
B(L_{2}(G)), respectively, that are of interest on their own
right. These yield results about automatic normality, on the one hand,
thus moreover providing a completely new approach to the topological
center problem for L_{¥}(G)^{*} and LUC(G)^{*} (which is closely
related to the (bi)commutant problems mentioned above), and on
automatic continuity, on the other hand, answering in particular a
question raised by G. Wittstock in 1997.
 VOLKER RUNDE, University of Alberta, Edmonton, Alberta
Operator FigàTalamancaHerz algebras

Let G be a locally compact group. We use the canonical operator
space structure on the spaces L^{p}(G) for p Î [1,¥]
introduced by G. Pisier to define operator space analogues OA_{p}(G)
of the classical FigàTalamancaHerz algebras A_{p}(G). If p Î (1,¥) is arbitrary, then A_{p}(G) Ì OA_{p}(G) such that the
inclusion is a contraction; if p = 2, then OA_{2}(G) @ A(G) as
Banach spaces spaces and, for amenable G, the identity from A(G) to
OA_{2}(G) is a complete contraction. We show that OA_{p}(G) is a
completely contractive Banach algebra for each p Î (1,¥), and
that OA_{q}(G) Ì OA_{p}(G) completely contractively for amenable
G if 1 < p £ q £ 2 or 2 £ q £ p < ¥. Finally, we
characterize the amenability of G in terms of the operator
amenability of OA_{p}(G) for arbitrary p Î (1,¥), thus
extending a result by Zh.J. Ruan for the Fourier algebra A(G).
 ECKART SCHULZ, Suranaree University of Technology, Nakhon Ratchasima, Thailand
Continuous and semidiscrete tight frames on R^{n}

Given A Î GL_{n}(R), we show that there exists a function
w Î L^{2}(R) whose dilates by powers of A and translates
form a semidiscrete frame on L^{2}(R^{n}) if and only if
det(A) ¹ 1. If in addition, A lies in the exponential group
of GL_{n}(R) then there exists w Î L^{2}(R^{n})
whose dilates by exponents of A and translates form a continuous
frame on L^{2}(R^{n}) if and only if det(A) ¹ 1.
 PETER WOOD, Wilfrid Laurier University
Invariant complementation and projectivity in A(G)

In the category of operator spaces, it is known that the Fourier
algebra A(G) is amenable if and only if G is amenable. Amenability
is a powerful property which, for our purposes, allows us to perform
``averaging arguments'' which might not otherwise be available.
For example, let I be a closed ideal in the Fourier algebra A(G).
In this talk we investigate when I is complemented. The general
approach to this question has been to construct projections onto I
which are invariant, i.e. we find P: A(G)® I such that
P(ax)=aP(x) for all a, x Î A(G). Once such a projection is
constructed, reasonable necessary conditions on I can be determined.
Historically, such projections were constructed by utilizing the
amenability of the algebra. However, the language of homological
algebra, amenability becomes one of many ``invariance'' type
properties. Another important such property is biprojectivity.
In this talk, we shall show that A(G) is biprojective in the category
of operator spaces if and only if G is discrete. Using this result,
we can show that for discrete groups, every complemented ideal I of
A(G) is complemented by an invariant projection.
 YONG ZHANG, Department of Mathematics, University of Manitoba, Winnipeg,
Manitoba R3T 2N2
Approximate identities for ideals of Segal algebras on a compact
group

In the monograph ``L^{1}Algebras and Segal Algebras'' H. Reiter has
shown that every closed ideal of a symmetric Segal algebra on a compact
group admits an approximate identity. With a different way of approach
we show that every closed ideal of a symmetric or pseudosymmetric Segal
algebra on a compact group admits an approximate identity. Moreover,
for a compact group G, a closed left (right) ideal J of L^{1}(G)
has a right (resp. left) approximate identity (e_{a}) with
a*e_{a} (resp. e_{a}*a) converging to a uniformly for
a in compact sets of J if and only if J is approximately
complemented in L^{1}(G).

