The WAP-compactification of a SIN-group
by
Stefano Ferri
Universidad Nacional Autonoma de Mexico
Coauthors: Dona Strauss

A semigroup compactification of a (Hausdorff) topological group G is a pair (X, f), where X is a semigroup with a compact Hausdorff topology and f:G -> X is a continuous homomorphism with dense image such that all right translations x -> xy are continuous in X and the left translations y -> f(s)y are continuous in X for all s in G. In this talk we shall be interested in two semigroup compactifications of G: the LUC-compactification, luc(G), which is the largest semigroup compactification of G (any other is a natural quotient), and the WAP-compactification, wap(G), which is the largest semigroup compactification of G in which the product is continuous in both variables separately.

In general wap(G) need not be very large, as there are examples of groups whose WAP-compactification is a singleton. However, in this talk we shall show that this is not the case when G is a SIN-group. More precisely, we shall show that, if we regard wap(G) as a quotient of luc(G) and denote the quotient map by p, then, when G is a SIN-group, there exists a dense open subset of luc(G)-G consisting of points of unicity for p of cardinality 2^2^k(G), where k(G) denotes the compact covering number of G.

Date received: May 9, 2003