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

  For A a monoid in symmetric monoidal category, write • (not μ) for its bilinear multiplication map A × A → A (Arens’s suggestive/ambiguous product notation).
  Using a and b as variables from A, φ as variable from A*, and S and T as variables from A**, we have the following stages towards the Arens multiplication on A**:

•: A × A → A	(a, b) ··> a • b	a • b = •(a, b)
•†: A* × A → A*	(φ, a) ··> [b ··> φ(a • b) = (φ •† a)(b)]	φ •† a = φ(a • –)
•††: A** × A* → A*	(T, φ) ··> [a ··> T(φ •† a) = (T •†† φ)(a)]	T •†† φ = T(φ •† –)
•‡ = •†††: A** × A** → A**	(S, T) ··> [φ ··> S(T •†† φ) = (S •††† T)(φ)]	S •‡ T = S(T •†† –)

  Thus, (ev_a •^‡ T)(φ) = ev_a(T •^†† φ) = (T •^†† φ)(a) = T(φ •^† a) = T(φ(a • –)), and (S •^‡ ev_b)(φ) = S(ev_b •^†† φ) = S(φ(– • b)). (Indeed, ev_b •^†† φ = φ(– • b)