# On two problems concerning topological centers

Eli Glasner

Arxiv ID: 0710.2625•Last updated: 3/3/2021

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.

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