C.M.S. • Fredericton, NB • June 2010
Meditations on Arens Multiplication
   F.E.J. Linton • Wesleyan Univ. Math/CS Emeritus

  Write T for the usual (compact, T₂) circle group, C = |T| for its underlying (discrete) Abelian group. The Bohr compactification βA of discrete Abelian group A “is” the left adjoint to | |: AbGp(KT₂) → AbGp. Its (discrete Abelian) character group (βA)^ is AbGp(KT₂)(βA, T) ~ AbGp(A, |T|) = AbGp(A, C) = A*. And it itself is (Pontryagin) the (compact, Abelian) character group of that: βA = (A*)^. Thus A** = |(A*)^| = |βA|.
  Parallels with the Ban₁ situation:
  X* = underlying Banach space of **-algebra (i_X)*: X*** → X*
“=” Banach space X* with compact (weak X)-topology on its unit disk, just as
  A* = |A^| = underlying group of compact character group A^;