On Two Problems concerning Topological Centers

Let Γ be an infinite discrete group and βΓ its Čech-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 βΓ \ Γ, left multiplication Lp : βΓ → βΓ is not Borel measurable. Next assume that Γ is abelian. Let D ⊂ l(Γ) denote the subalgebra of distal functions on Γ and let D = Γ = |D| denote the corresponding universal distal (right topological group) compactification of Γ. Our second result is that the topological center of D (i.e. the set of p ∈ D for which Lp : D → D is a continuous map) is the same as the algebraic center and that for Γ = Z, this center coincides with the canonical image of Γ in D.