Topological Centres Everywhere
by
M. Neufang
Carleton University

In 1951, R. Arens showed that for any Banach algebra A, there are two canonical ways of extending the product in A to its bidual. It turns out that these products coincide for some class of algebras -- called Arens regular -- which contains all C^*-algebras, but in general are different. A natural way of measuring the (non-) regularity of A is to consider the sets of elements in the bidual for which left (resp. right) multiplication with respect to both products is the same, called the first (resp. second) topological centre. I will present a unified approach that I developed in order to determine these centres for various algebras arising in Abstract Harmonic Analysis, such as the group algebra L_1(G) and the measure algebra M(G) for a locally compact group G. The latter solves a conjecture made by Ghahramani and Lau in 1994. I shall also show how these techniques can be applied to answer a question on automatic boundedness of module homomorphisms on von Neumann algebras raised by Hofmeier and Wittstock in 1997. I will furthermore present: - an application to semigroup compactifications related to the work of Filali, Lau, Protasov, and Pym; - the first example of a Banach algebra which equals its first but not its second topological centre, which is in close connection to the recent work of Dales and Lau; - a tensor product version of the topological centre in the framework of operator spaces. The talk aims at offering a panorama view of the colourful circle of ideas around the 'topological centre' -- a view which we owe to John Pym to a great extent.

Date received: May 26, 2003