C.M.S. • Fredericton, NB • June 2010

**Meditations on Arens Multiplication**

F.E.J. Linton • Wesleyan Univ. Math/CS Emeritus

F.E.J. Linton • Wesleyan Univ. Math/CS Emeritus

Write **T** for the usual (compact, T_{2}) circle group,
**C** = |**T**| for its underlying (discrete) Abelian group.
The **Bohr compactification** β*A* of discrete Abelian group *A*
“is” the left adjoint to
| |: *AbGp*(KT_{2}) → *AbGp*.
Its (discrete Abelian) character group (β*A*)^ is
*AbGp*(KT_{2})(β*A*, **T**)
__~__ *AbGp*(*A*, |**T**|) = *AbGp*(*A*, **C**) = *A**.
And it itself is (Pontryagin) the (compact, Abelian) character group of that:
β*A* = (*A**)^.
Thus *A*** = |(*A**)^| = |β*A*|.

Parallels with the*Ban*_{1} 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*^;

Parallels with the

“=” Banach space