


General Topology and Topological Algebra / Topologie générale et algèbre topologique (Org: Ilijas Farah, York University and/et Vladimir Pestov, University of Ottawa)
 MAX BURKE, University of Prince Edward Island
Renormings of C(X) and Borel measurability of separately
continuous functions

Lebesgue proved that every separately continuous function f:R×R® R is a pointwise limit of
continuous functions. We examine the Borel measurability of separately
continuous functions f: X×Y® R defined on a
product of topological spaces. When X is compact, Borel
measurability of f can sometimes be deduced from the existence of a
pointwise Kadec renorming for C(X), i.e., a norm equivalent to the
sup norm for which the norm topology and the topology of pointwise
convergence coincide on the unit sphere. Some of the results in this
talk are part of joint work with W. Kubis and S. Todorcevic.
 ILIJAS FARAH, York University
Between von Neumann's and Maharam's problems

In 1937 J. von Neumann asked whether every ccc, weakly distributive,
complete Boolean algebra carries a strictly positive countably
additive measure.
This was the first attempt on finding an abstract characterization of
measure algebras. A few years after von Neumann, D. Maharam asked
whether every algebra that carries a strictly positive
ordercontinuous submeasure is a measure algebra. Maharam's
problem reoccurred in the early seventies as the Control
Measure Problem for vector measures. A negative answer to
Maharam's problem would imply a negative answer to von Neumann's
problem.
Unlike von Neumann's problem, the answer to Maharam's problem is
absolute: it is not sensitive to the choice of settheoretic
axioms.
I will present some recent results on the relation between these two
problems, as well as some applications.
For example, the only simply definable ideals that satisfy an analogue
of Fubini's theorem are the ideal of sets of first category and the
ideal of null sets for some Borel measure.
 NEIL HINDMAN, Howard University, Washington, DC 20059, USA
Almost disjoint large subsets of semigroups

There are several notions of largeness in a semigroup (S,·)
which originated in topological dynamics and have simple
characterizations in terms of the algebraic structure of the Stone
Cech compactification bS of S. These include central
sets (members of a minimal idempotent), piecewise syndetic
sets (sets whose closure meets the smallest ideal of bS),
thick sets (sets whose closure contains a left ideal of bS), and syndetic sets (sets whose closure meets every left
ideal of bS). The latter three notions also have simple
elementary descriptions in terms of S.
The central sets are especially interesting because they are partition
regular (meaning that when a central set is divided into finitely many
pieces, one of those pieces must be central) and are guaranteed to
contain a substantial amount of combinatorial structure. We
investigate the extent to which these large sets may be split into
almost disjoint families of large sets. For example, we show that if
A is an alphabet with A = k ³ w, S is the free
semigroup on the alphabet A, there is some almost disjoint
collection of m subsets of A, and P is any of the first three
properties, then any subset of S with property P can be split into
m almost disjoint subsets each having property P. On the other
hand, while any syndetic subset of S can be split into infinitely
many pairwise disjoint syndetic sets, there is no collection of
k^{+} almost disjoint syndetic subsets of S.
 WOJCIECH JAWORSKI, Carleton University, Ottawa, ON K1S 5B6
Contractive and weakly contractive authomorphisms of locally
compact groups

An authomorphism t of a topological group N is called weakly
contractive modulo a compact subgroup H if t(H)=H and there
exists a dense subset D Í N and an increasing sequence
{q_{n}}_{n=1}^{¥} Í N such that D Í È_{k=1}^{¥} Ç_{n=k}^{¥} t^{qn}(U) for every
neighbourhood U of H. If D=N and q_{n}=n, t is called
contractive modulo H. When N is locally compact and t is
weakly contractive modulo H, there always exists a compact subgroup
K Ê H such that t is contractive modulo K. Weakly
contractive authomorphisms arise in a natural way in the study of
dissipation of convolution powers of a probability measure and the
above result implies universal dissipation of such convolution powers
in every noncompact group.
 DAN KUCEROVSKY, University of New Brunswick at Fredericton
Unusual ideals in the multipliers of C(X)ÄK

The multipliers C(X) ÄK are the strictly continuous maps from
X to B(H). One would expect that the only ideals of the
multipliers might be those that come from closed sets in X or from
the trace ideals. We find an example of an "unexpected" ideal that
does not come from either of these cases.
We study some of the properties of this ideal, and find that it is
projectionless in a certain weak sense. One corollary of our work is
that the corona algebra
M ( C(X) ÄK ) / ( C(X) ÄK ) is in general not simple, even though the point
evaluations give the classic Calkin algebra, which is simple.
(This is joint work with P. W. Ng.)
 GÁBOR LUKÁCS, Universität Bremen, Germany
Hereditarily hcomplete groups

A topological group G is ccompact if for every topological
group H, the projection p_{H} : G ×H ® H maps
closed subgroups of G ×H to closed subgroups of H; G is
hcomplete if for every continuous homomorphism j: G® H, j(G) is closed in H. Clearly, every closed
subgroup of a ccompact group is hcomplete. Inspired by this, we
call G hereditarily hcomplete if every closed subgroup of
G is hcomplete. In this talk, we will present results on the
openmap properties of hereditarily hcomplete groups with respect
to large classes of groups. A theorem on the (total) minimality of
subdirectly represented groups will also be presented, followed by
numerous applications. In the sequel, some open problems will be
formulated.
 MICHAEL MEGRELISHVILI, BarIlan Unversity, Israel
Relatively minimal subgroups

We say that a topological subgroup X of G is relatively
minimal in G if every coarser Hausdorff group topology of G
induces on X the original topology. We show that every normed space
V is a relatively minimal subgroup in some topological group
G : = Heis(V), where by Heis(V) we denote the generalized
Heisenberg group based on V. As an application we obtain the
following result which answers a question of A. Shtern.
Theorem: There exists a topological group G (namely, G : = Heis
(L_{4}[0,1])) such that
(a) G is a topological subgroup of Is(V) for some
reflexive Banach space V, where Is(V) is endowed with the
strong (equivalently, weak) operator topology.
(b) Weakly continuous unitary representations of G in
Hilbert spaces do not separate points of G.
 MATTHIAS NEUFANG, Carleton University, School of Mathematics and Statistics,
1125 Colonel By Drive, Ottawa, ON K1S 5B6
Decomposability of von Neumann algebras and applications

The decomposability number of a von Neumann algebra M, denoted by
dec(M), is defined to be the greatest cardinality of a family of
pairwise orthogonal, nonzero projections in M. This is a very
natural invariant since a von Neumann algebra is determined by its
projections.
In this talk, I shall focus on those von Neumann algebras whose
preduals are function spaces/Banach algebras on a locally compact
group G, such as the group algebra L_{1}(G), the measure algebra
M(G), the Fourier algebra A(G), etc. I will show that, for these
von Neumann algebras, the exact value of dec(M) can be expressed in
terms of two dual cardinal invariants of the underlying group G: the
compact covering number k(G) and the least cardinality b(G) of an
open basis at the identity of G.
It turns out that the decomposability number reveals intriguing links
between topology, harmonic analysis and Banach algebra theory. I shall
present applications reaching from semigroup compactifications over
the topological centre problem to Kac algebras. Furthermore, I shall
discuss in more detail the intimate relation between decomposability
and Mazur's property and property (X) of higher cardinal level; here,
measurable cardinals play a crucial role.
This is joint work with Zhiguo Hu.
 VLADIMIR PESTOV, University of Ottawa  Université d'Ottawa
Some results and questions about topological subgroups of the
unitary group

What topological groups are isomorphic with subgroups of the unitary
group U(H) of a suitable infinite dimensional Hilbert space H,
equipped with the strong operator topology? This is still not well
understood, and an intrinsic description of such groups is
lacking. One of such topological groups of considerable interest is
the group L^{0} ( X,U(1) ) of measurable maps from the unit
interval into the circle rotation group. It was observed recently by
the speaker and Su Gao (Fund. Math. 209(2003), 115) that
every Polish abelian group is contained in a suitable factorgroup of
L^{0} ( X,U(1) ), and we discuss some consequences (e.g. a
new proof of a theorem by Sid Morris and the speaker (J. Group Theory
3(2000), 407417): every Polish abelian group embeds into a
monothetic group) and related open questions. This development is
closely linked to free (abelian) topological groups, some of which may
themselves be embeddable into the unitary groups.
 SLAWOMIR SOLECKI, University of Illinois  UrbanaChampaign
Free groups and small subsets of Polish groups

Using left Haar null sets, a measure theoretic notion of smallness of
subsets of Polish groups, we extend to new situations results, due to
Weil and Christensen, on continuity of homomorphisms which are assumed
to be merely universally measurable. These earlier results used Haar
measure zero sets on locally compact groups and Haar null sets on
Abelian Polish groups. Apart from applying a different notion of
smallness, our result differs from the earlier theorems in its
connection with the algebraic structure of the group, particularly,
with amenability. This connection turns out not to be accidental. We
isolate a condition involving free subgroups of nonlocally compact
Polish groups which, in this context, has an effect of
antiamenability. We use it to show that on a large class of Polish
groups (including S_{¥}, Homeo([0,1]), products of
groups containing discrete free subgroups, and others) left Haar null
sets do not form an ideal and do not have the Steinhaus property with
respect to universally measurable sets. This implies that no smallness
notion considered so far can be used to prove continuity of
universally measurable homomorphisms on such groups.
 BENJAMIN STEINBERG, Carleton University, 1125 Colonel By Drive, Ottawa, ON K1S 5B6
On the profinite monoid of closed subsets of a profinite group

Consider a free group F with the proC topology (where C is a pseudovariety of finite groups e.g. pgroups, solvable
groups, or nilpotent groups). It is known that any proC
closed finitely generated subgroup is (geometrically) a free factor in
a proC open subgroup. We show that the converse holds
precisely when a certain relatively free profinite monoid embeds in
the profinite monoid of closed subsets of a free proC group.
As a consequence, we know that the converse fails for nilpotent groups
(or any joinreducible pseudovariety) and for any pseudovariety
satisfying a nontrivial group identity. We also obtain a simple to
state sufficient condition for the converse to hold in terms of wreath
products; in particular it holds for the pseudovariety of finite
groups whose order divides a (fixed) infinite supernatural number. Our
techniques use the geometry of profinite graphs. Applications to
finite monoid/automata theory will be mentioned.
 JURIS STEPRANS, York University, Toronto
Translation Invariants of Group Actions

Cardinal invariants such as the least number of meagre sets required
to cover the real line are now quite well understood. This particular
invariants is quite robust in the sense that its definition does not
depend on the version of the reals usedany Polish space will
yield the same number. A version of this invariant for group actions
will be defined and it will be shown that this number is more
sensitive to the space used.
 PAUL SZEPTYCKI, York University, Toronto, ON
A new class of LindelofS spaces

A space X is LindelofS if X has a cover C by
compact sets and there is a countable family N which is a
countable network at each C Î C. If we demand that the
compact sets in C all be of weight £ k for some
infinite cardinal k we obtain a class of spaces that we denote
LS( £ k). This class of spaces was recently introduced
by O. Okunev. We study some basic properties of these spaces and of
the subclass of compact LS( £ k)spaces.
 MURAT TUNCALI, Nipissing University
On Suslinian Continua

A continuum is a compact connected Hausdorff space and a continuum is
said to be Suslinian, if it does not contain an uncountable collection
of disjoint subcontinua. Recently, various properties of Suslinian
continua have been studied in relation to problems concerning
generalizations of the HahnMazurkiewicz Theorem, rimmetrizability and
perfect normality. In this talk, the recent results on
Suslinian continua will be presented.
 ED TYMCHATYN, University of Saskatchewan, Saskatoon, SK
Lifting paths on quotient spaces

Among metric continua the locally connected continua are the
continuous images of arcs by the HahnMazurkiewicz Theorem. Among
nonmetric continua the continuous images of arcs form a very
restricted class.
Let X be a a continuum and G an upper semidecomposition of X
such that each element of G is the continuous image of an arc. If
the quotient space X/G is the continuous image of an arc we give
conditions under which X is itself the continuous image of an arc.
We give examples to show the extent to which our conditions are
necessary.
This represents joint work with D. Daniel, J. Nikiel, L. B. Treybig
and M. Tuncali.
 VESKO VALOV, Nipissing University, 100 College Drive, P.O. Box 5002,
North Bay, ON P1B 8L7
Extension dimension of maps between metrizable spaces

Extension dimension, introduced by Dranishnikov, is a unification of
the covering dimension and the cohomological dimension. The
fundamental problem, studied in this theory, is the possibility to
extend a map defined on a closed subset of a given space X with
values in a CWcomplex, over the whole space X. Many classical
dimension results have been reexamined and their far reaching
generalizations were established. In our talk some extension results
concerning maps between metrizable spaces are going to be discussed.

