On two problems concerning topological centers

Abstract

Let G be an infinite discrete group and bG its Cech-Stone compactification. Using the well known fact that a free ultrafilter on an infinite set is nonmeasurable, we show that for each element p of the remainder bG G, left multiplication L_p:bG \to bG is not Borel measurable. Next assume that G is abelian. Let D \subset \ell^\infty(G)$ denote the subalgebra of distal functions on G and let G^D denote the corresponding universal distal (right topological group) compactification of G. Our second result is that the topological center of G^D (i.e. the set of p in G^D for which L_p:G^D \to G^D is a continuous map) is the same as the algebraic center and that for G=Z (the group of integers) this center coincides with the canonical image of G in G^D.