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.

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 order-continuous 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 set-theoretic 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.

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.

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^-q_n(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.

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.)

A topological group G is c-compact 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 h-complete if for every continuous homomorphism j: G® H, j(G) is closed in H. Clearly, every closed subgroup of a c-compact group is h-complete. Inspired by this, we call G hereditarily h-complete if every closed subgroup of G is h-complete. In this talk, we will present results on the open-map properties of hereditarily h-complete 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.

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 (L₄[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.

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, non-zero 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 L₁(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.

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 L⁰ ( 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), 1-15) that every Polish abelian group is contained in a suitable factor-group of L⁰ ( 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), 407-417): 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.

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 non-locally compact Polish groups which, in this context, has an effect of anti-amenability. 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.

Consider a free group F with the pro-C topology (where C is a pseudovariety of finite groups e.g. p-groups, solvable groups, or nilpotent groups). It is known that any pro-C closed finitely generated subgroup is (geometrically) a free factor in a pro-C 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 pro-C group. As a consequence, we know that the converse fails for nilpotent groups (or any join-reducible pseudovariety) and for any pseudovariety satisfying a non-trivial 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.

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 used-any 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.

A space X is Lindelof-S 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.

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 Hahn-Mazurkiewicz Theorem, rim-metrizability and perfect normality. In this talk, the recent results on Suslinian continua will be presented.

Among metric continua the locally connected continua are the continuous images of arcs by the Hahn-Mazurkiewicz Theorem. Among non-metric continua the continuous images of arcs form a very restricted class.

Let X be a a continuum and G an upper semi-decomposition 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.

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 CW-complex, 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.