ANNOUNCEMENT

2004 Canadian Symposium on Abstract Harmonic Analysis

to be held at the

University of Western Ontario

May 17-18, 2004

Organized by

B. Forrest, Z. Hu, T. Miao, P. Milnes (milnes at uwo.ca)

Photos

The schedule of talks  starts at 9:00 am, May 17 and ends at 4:10 pm, May 18. One hour talks will be given by Jean-Paul Pier and John Pym.

Conference venue:  The talks will take place in Middlesex College 105,
near top-centre of the campus map.   (Here is a slightly different, but printable map ; here is a
map of the relevant part of London .)

Travel to London and UWO:  See   http://jdc.math.uwo.ca/directions.php   
(prepared by a member of my department) and UWO's   
http://communications.uwo.ca/western/about_directions.html
For those going to COSY: Brian Forrest <beforres@math.uwaterloo.ca> will organise a van to take people to Waterloo on May 18, if sufficient interest is expressed; contact him as soon as possible if you are interested. Also, the Via Rail and Greyound websites say that, on May 18, there are trains from London to Kitchener at 5:25 and 8:35 pm (5/4 hr trip) and a bus at 5:15 pm (3/2 hr trip).

Accommodation:
    One accommodation option is Essex Hall, click on 'Bed and Breakfast' at
http://www.uwo.ca/hfs/cs/    (where you make all the arrangements yourself).
Others can be arranged.
    At Essex Hall, we have a block of 40 rooms reserved until April 16, when any
unclaimed rooms become available to the public at large. When making an on line
reservation for Essex Hall, you must choose one of

Economy Rate (no breakfast, no cleaning)   $29.00,      and
Full Service Rate (continental breakfast, daily clean)  $43.50.

However, you can change your choice when you arrive.  You can also choose the
economy rate and then go to the breakfast nook in Essex Hall and pay à la carte. 
The breakfast at Essex Hall is 'continental',
 - 1 bread item (toast, bagel, cereal, muffin),
 - 1 fruit item,
 - coffee, tea, juice, milk.

A hot breakfast is available from 7:30 am at Lucy's in Somerville House,
which is on the way to Middlesex College from Essex Hall (see the map).
You can also look at    http://www.has.uwo.ca/hospitality/eateries/eateries.cfm   

The Conference Dinner will be held at the Dragon Court Restaurant
in the evening of May 17 ($15/person).

The organizers acknowledge support from the Fields Institute.

Announcement in the CMS Notes February/04, p. 42.

SCHEDULE OF TALKS
ABSTRACTS OF TALKS
will appear below, as they become available.

Abstracts:

Jean-Paul Pier, Harmony fosters Beauty and Amenability.

John Pym, Some uses of second duals in harmonic analysis.

Cédric Delmonico, Atomisation process for convolution operators on some locally compact groups.
Abstract: Let G be a locally compact group and 1 < p < infty. A p-convolution operator on G is a bounded operator on L^p(G) which is commuting with the left translation. The set of all p-convolution operators on G, denoted by CVp(G), is a Banach algebra. There is a injective homomorphism of the space of the bounded measures of G into CVp(G), defined by the convolution of the functions on L^p(G) with bounded the measure.
    We say there is an atomisation process for the convolution operators on G if there is a net (L_alpha) of endomorphisms of CVp(G) which is converging to id such that L_alpha(T) have discrete support, for all alpha. Moreover, we want to keep the control of the norm and the support of L_alpha(T). In his thesis,
N. Lohoué have proved the existence of such atomisation process on locally compact abelian groups
using structure theorems. I will give a constructive proof of existence of atomisation process for a large class of locally compact groups. In particular, we can approximate bounded mesures with discrete mesures, on R^n, T^n, O(2) or the Heisenberg group.

Jorge Galindo, Interpolation sets and some of the harmonic analysis they should involve.

Abstract: Two types of questions related to the existence of interpolation sets, an object of study in Harmonic Analysis for more than 80 years, will be examined  in this talk: which  groups admit interpolation sets in abundance and which have none at all (the ubiquity and existence problems). While trying to capture some of the general features, we shall focus on two types of interpolation sets: I_0-sets and Sidon sets or, what is the same, interpolation sets for the  the algebra of  almost periodic functions and interpolation sets for the Fourier-Stieltjes algebra.
    With the aid of well-known theorems of Rosenthal, and Bourgain, Fremlin and Talagrand it is easy to see that a Polish group G has no X-interpolation  set (X being the uniform closure  of the above algebras of functions) if and only if the spectrum of this algebra is Rosenthal compact (i.e. contained in the set of functions of thefirst Baire class on some Polish space). These topological arguments tie  the existence of interpolation sets with the structure of some weakly almost periodic compactifications (the Bohr and  Eberlein compactifications) and of spaces of matrix coefficients of unitary representations. While this will provide reasonable characterizations of locally compact groups with no I_0-sets, it remains unclear which locally compact groups have no Sidon sets.
    The ubiquity problem will be put in connection with Rosenthal's ell^1-theorem and  a version of this theorem in the setting of locally compact groups will be presented. This links the ubiquity problem for Sidon and I_0-sets with the kind of uniformity the  group receives from some of its (weakly) almost periodic compactifications and with the existence of  unitary representations generating von Neumann algebras with  specificfeatures.

Fereidoun Ghahramani, More on approximate amenability.
ABSTRACT.In this talk first I will review some background from the paper, F.Ghahramani and R.J.Loy, Generalized notions of amenability, J.Functional Analysis, 208(2004)229-260.Then I intend to present more recent developments on approximate amenability such as:
 1. The equivalence of the three notions, approximate amenability, weak* approximate amenability and approximate conractibility;
 2. approximate amenability of the James algebra;
 3. Approximate weak amenability of every Symmetric Segal algebra on an amenable group.

Colin C. Graham, epsilon-Kronecker sets in discrete abelian groups.
(Based on joint work with K.E. Hare and T.W. Körner)
Abstract: Let 0 < epsilon < 2.  A closed subset E of the discrete  abelian group  Gamma  is
``epsilon-Kronecker'' if for every  continuous function f: E --> T there exists  x in the dual of  Gamma  with  |<gamma,x>  - f(gamma)| < epsilon for all gamma in E.  These sets have the following properties when epsilon < sqrt 2:
  (i). E is I_0 (and the interpolating discrete measures can be chosen in
any open subset when the dual of  Gamma  is connected).
  (ii). The step length in an epsilon-Kronecker set goes to infinity.
  (iii). Previously known results about cluster points of E - E  and E + E
from Hadamard setsare extended to sums and differences of  epsilon-Kronecker sets.
  (iv).  Sums of epsilon-Kronecker sets are  U_0 sets if the number of terms  is small relative
to epsilon and can be all of the group if the number of terms is large.
  (v).  For each epsilon > 0, there are epsilon-Kronecker sets in the natural
numbers that are not finite unions of Hadamard (lacunary) sets.
    Additionally,
  (vi). When epsilon < 2, E does not contain arbitrarily long arithmetic progressions.

E. E. Granirer, The Fourier -Herz -Lebesgue algebras  Arp = Ap intersection Lr.

N. Hindman, Discrete n-tuples in beta(D).

Abstract: Everyone learns in the crib that any infinite subset of a Hausdorff space contains an infinite discrete subspace. In the course of investigating some Ramsey Theoretic results,
we established the following theorem:
    Let X be a  Hausdorff topological space and assume that for each i,j  in  omega, one has x_{i,j} in X such that  x_{i,j} neq x_{l,m} whenever (i,j) neq (l,m).  There exists an  infinite subset M in omega such that x_{i,j} : i,j in M  is discrete if and only if {y in X:  some injective sequence in X converges to y} is finite.  In particular, if X = beta(D) for an infinite discrete space D, there exists an infinite subset M in omega such that {x_{i,j} : i,j in M} is discrete.
    After consulting with several experts we were surprised to find out that the above theorem appears to be new. This result raised the following question:
    Let k in N. For which Hausdorff spaces X is it true that whenever Gamma is an injective function
from omega^k to X, there must exist some  M in omega^{omega} such that Gamma[M^k] is discrete?
    For  k = 1 the answer is ``all of them''.  And for k = 2, the theorem tells us that the answer is ``those for which there are only finitely many points which are the limit of injective sequences''.  For larger values of k we are not able to answer this question.  However, we do show that the answer includes beta(D) for infinite discrete spaces D.

Z. Hu, A "Kakutani-Kodaira theorem" for von Neumann algebras on locally compact groups.
 
Abstract. Let G be a locally compact group. In this talk, we present a decomposition theorem for the
von Neumann algebras L_infty(G) and VN(G). We show how the exploration for such a unified
approach for this dual pair of Kac algebras has led to a generalized Kakutani-Kodaira theorem on the
local structure of G.  

Monica Ilie, Completely bounded Fourier algebra homomorphisms.
Abstract: Given a locally compact group G, the Fourier and Fourier-Stieltjes algebra are defined as spaces of coefficient functions associated with  continuous unitary representations of $G$. In the same
time, they can also be looked at as preduals of certain von Neumann algebras and, consequently, they have a natural operator space structure.
    It is known that, for abelian groups, there is a deep connection between Fourier algebra homomorphisms and piecewise affine maps between their underlying groups. We explore this fact from the point of view of operator spaces, for general locally compact groups. In joint work with Nico Spronk we extend a result of B. Host to the class of locally compact groups that are amenable as discrete. This represents a  continuation of previous work done by the speaker in the discrete case.

Wojciech Jaworski, Ergodic and mixing probability measures on locally compact  groups.
Abstract: A probability measure mu on a locally compact groups G is called ergodic (resp., mixing) if for every varphi in L^1_0(G) the sequence \frac1n\sum_{i=1}^n\varphi *\mu^i (resp.,  varphi *\mu^n) converges to zero in L^1(G). In general, mixing is strictly  stronger than ergodicity. We will discuss a long outstanding conjecture that under a certain natural condition these two concepts are equivalent. 

A.T. Lau, The centre of the second conjugate algebra of the Fourier algebra for
infinite products of groups.
Abstract: In this talk, I shall report on a recent joint work with Viktor Losert on the centre of the second conjugate algebra of the Fourier algebra A(G) of a locally compact group G for an infinite product of locally compact groups.

Mehdi Sangani Monfared, On a class of Banach algebra extensions of the generalized Fourier algebras that split strongly.
Abstract: In 1983, Lau introduced a new class of Banach algebras (later called Lau algebras)
and studied in details certain direct sums of  these algebras. In this talk we show
that how persuing the same ideas will lead to a class of strongly splitting extensions
of the generalized Fourier algebras A_p(G).  Among other things we discuss the
spectrum, bounded approximate identities, minimal idempotents, and the Jacobson radical
of these extensions.

M. Neufang, Aspects of quantization in abstract harmonic analysis.
ABSTRACT: Our aim is to give an overview of our recent contributions to various aspects of 'quantization' in abstract harmonic analysis -- where this (very fashionable) word has two meanings for us:
1. the use of operator space theory in the study of classical objects in harmonic analysis;
2. the investigation of non-commutative counterparts of these objects arising through the replacement of functions by operators.
    We shall illustrate this programme by discussing the following topics (throughout, G denotes a locally compact group).
1.1 Amenability theory.  Z.-J. Ruan showed that the canonical operator space structure of the Fourier algebra A_2(G)  is crucial for understanding its amenability properties -- by proving that A_2(G) is operator amenable if and only if the group G is amenable (this equivalence fails in the realm of classical amenability, as shown by B.E. Johnson). We extend Ruan's theorem to the case of the Figa-Talamanca--Herz algebras A_p(G), for p in (1, infty), endowed with an appropriate operator space structure. The construction of the latter requires the concept of non-hilbertian column operator spaces, as developed recently by A. Lambert and G. Wittstock.
     This is joint work with A. Lambert and V. Runde.
1.2 Representation theory. We present a common representation-theoretical framework for various
Banach algebras arising in abstract harmonic analysis, such as the measure algebra M(G) and the completely bounded multipliers of the Fourier algebra M_{cb} A_2(G). The image algebras are intrinsically characterized as certain normal completely bounded bimodule maps on B(L_2(G)). The study of our representations reveals intriguing properties of the algebras, and provides a very simple description of their Kac algebraic duality.
    Part of this is joint work with Z.-J. Ruan and N. Spronk.
2.1 Quantized convolution. The space T(L_2(G)) of trace class operators is usually considered as the non-commutative version of  L_1(G) on the level of Banach spaces.  We show that this analogy may be extended to the Banach algebra level by
introducing a new product on T(L_2(G)) which parallels convolution of
functions.
2.2 Harmonic operators. In the context of the representation model as described in 1.2, we study an operator version of the classical Choquet-Deny equation (for a fixed probability measure on G), the solutions of which we refer to as harmonic operators. We show that the space of harmonic operators is naturally equipped with the structure of a non-commutative von Neumann algebra, and completely describe its structure as a W*-crossed product over the classical harmonic functions.
    This is joint work with C. Cuny and W. Jaworski.

Vladimir Pestov, Some recent links between topological transformation groups, geometric functional analysis, and Ramsey theory.
Abstract. We will survey properties of topological groups of transformations determined by oscillations of functions on phase spaces (such as the extreme amenability, or the fixed point on compacta property) and discuss their relationship with geometric functional analysis and structural Ramsey theory, paying attention to a number of concrete examples. In particular, we will discuss a reformulation of the distortion property of the Hilbert space in terms of the infinite unitary group, and show that it makes sense for great many `infinite dimensional' groups and their homogeneous spaces in the context of infinite combinatorics. We will survey some open problems.

Volker Runde, (Non-)amenability of the Fourier algebra.
Abstract: We show that the Fourier algebra A(G) of a locally compact group G is amenable if and only if G has an abelian subgroup of finite index. This talk is based on joint work with Brian Forrest.

Ebrahim Samei, Approximately local derivations.
Abstract: We initiate the study of certain linear operators from a Banach algebra A into a Banach A-bimodule X, which we call approximately local derivations. We show that when A is a C*-algebra, a Banach algebra generated by idempotents, a semisimple annihilator Banach algebra, or the group algebra of a SIN or a totally disconnected group, bounded approximately local derivations from A into X are derivations. This, in particular, extends a result of B. E. Johnson that ``local derivations on C*-algebras are derivations" and provides an alternative proof of it.

N. Spronk, On representations of measures in spaces of completely bounded maps.
Abstract: Representations of the measure algebra M(G) of a locally compact group G on spaces of operators on B(H) (itself the von Neumann algebra of bounded operators on a Hilbert space H) have been studied by Stormer, Ghahramani, Neufang, Smith and the presenter.  I plan to present on some work in progress where I study representations of certain compactifications of G in spcaes wCB(B(H)) (normal completely bounded maps on B(H)).

Ross Stokke, Amenable representations and coefficient subspaces of  Fourier-Stieltjes algebras.
Abstract:   The theory of amenable representations was first developed in 1990 by M.E.B. Bekka and has since enjoyed the  attention of many authors. I will present characterizations of amenable representations of locally compact groups, G, in terms of associated coefficient subspaces of the Fourier-Stieltjes algebra, B(G), and in terms of associated von Neumann and C^*-algebras. I will  introduce Fourier and  reduced Fourier-Stieltjes algebras associated to arbitrary representations, and will discuss amenable
representations in relation to these algebras. Some illustrative (counter-)examples will also be presented.

Keith Taylor, Smoothness of tight frame generators.
Abstract: We will explore the connection between the structure of a 'dilation' matrix and the degree of
smoothness that an associated tight frame generator may have.

Yong Zhang,  Approximate diagonals of Segal algebras.
Abstract: A proper Segal algebra is never amenable so it never has a bounded approximate diagonal. But it may have an (unbounded) approximate diagonal. We will discuss when such an approximate diagonal exists.