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(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
(iX)*: X*** → X*
“=” Banach space X* with compact (weak X)-topology on its unit disk,
A* = |A^| = underlying group of compact character group A^;