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/(2k-1) for some positive integer k > 1,
then for each function f in Lp there exists a function F
in Lp 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 Lp
spaces has a unique majorant of minimal Lp-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 non-discrete and abelian then the measure algebra of G has a
non-zero point derivation. We show that the measure algebra of every
non-discrete group G has a non-zero 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 Ap(E) of the Herz algebra
If the subset E of T supports a synthesizable
p-pseudofunction T, then the restriction Ap(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 Ap(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 non-Arens 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 non-Arens 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 non-Arens 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 Ja(G) denote the
set of all closed left ideals J in L1(G) which have the form
J=[L1(G)*(de -m)]- where m is an absolutely continuous
probability measure on G. We explore the order structure of
Ja(G) when Ja(G) is ordered by inclusion.
When G is connected and amenable, every nonempty family
F Í Ja(G) admits both a minimal and a
maximal element, in particular, every ideal in Ja(G)
contains an ideal that is minimal in Ja(G). Furthermore,
every chain in Ja(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
- 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 non-compact connected semi-simple 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
Hilbert-Schmidt 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 set-up.
- 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 non-Haudorff case. In
the Hausdorff case, this representation takes place on a natural Hilber
module over C0(X) where X is the space of units of the groupoid.
In the non-Hausdorff case, a natural locally compact space Y, which
contains X as a dense subset, is constructed. The representation
takes place on a Hilber C0(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, B-module. Then
T = [
becomes a triangular Banach algebra when equipped with an
appropriate norm. A Banach algebra is said to be n-weakly
amenable if all derivations from that algebra into its n-th dual
space are inner. In this talk we investigate the n-weak amenability
of triangular Banach algebras in terms of the n-weak 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 Ap(G)
Let BpM(G) be the pointwise multiplier algebra of the
Figa-Talamanca-Herz algebra Ap(G) equipped with the multiplier norm
on BpM(G). We will prove that BpM(G) is a dual Banach space.
Properties of BpM(G) such as the density of Ap(G) in BpM(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=L2(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
well-known theorems of Brainerd/Edwards and Curtis/Figà-Talamanca.
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(L2(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 Figà-Talamanca-Herz algebras
Let G be a locally compact group. We use the canonical operator
space structure on the spaces Lp(G) for p Î [1,¥]
introduced by G. Pisier to define operator space analogues OAp(G)
of the classical Figà-Talamanca-Herz algebras Ap(G). If p Î (1,¥) is arbitrary, then Ap(G) Ì OAp(G) such that the
inclusion is a contraction; if p = 2, then OA2(G) @ A(G) as
Banach spaces spaces and, for amenable G, the identity from A(G) to
OA2(G) is a complete contraction. We show that OAp(G) is a
completely contractive Banach algebra for each p Î (1,¥), and
that OAq(G) Ì OAp(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 OAp(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 semi-discrete tight frames on Rn
Given A Î GLn(R), we show that there exists a function
w Î L2(R) whose dilates by powers of A and translates
form a semi-discrete frame on L2(Rn) if and only if
|det(A)| ¹ 1. If in addition, A lies in the exponential group
of GLn(R) then there exists w Î L2(Rn)
whose dilates by exponents of A and translates form a continuous
frame on L2(Rn) 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
In the monograph ``L1-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 L1(G)
has a right (resp. left) approximate identity (ea) with
a*ea (resp. ea*a) converging to a uniformly for
a in compact sets of J if and only if J is approximately
complemented in L1(G).