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^;
  X** is the underlying Banach space of the **-algebra (i_X*)*: X**** → X**
freely generated by X (left adjoint to forgetful **-Alg(Ban₁) → Ban₁), just as
  A** = |βA| is the underlying group of the Abelian compact T₂ group freely generated by A (Bohr compactification of A (left adjoint to forgetful AbGp(KT₂) → AbGp)).
  Has Arens’s ‡-procedure been noticed for multilinear maps of Abelian groups?
  Or, at least, Arens multiplications on |βR| for associative rings R (⊗-monoids in AbGp)?
  Has it been observed Z** = |βZ| admits a ring-like multiplication?