On two problems concerning topological centers


Abstract. 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 ⊂ `∞(Γ) denote the subalgebra of distal functions on Γ and let D = ΓD = |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. 1

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