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

  The Argentinian Ignacio Zalduendo had a particularly simple example [1991] (though it relies on that fragment of the Axiom of Choice given by the Hahn-Banach Theorem): start with the convolution algebra A = l¹(Z), multiplication being given by (a • b)_n = _{p, q:}Σ_p+q=n (a_p·b_q ). The dual space A* is just l^∞(Z), and here we define σ as follows:

σn =

{

–1 (n < 0)   0 (n = 0) . +1 (n > 0)

Now the subspace of l^∞(Z) consisting of those bounded sequences a for which both lim_n→–∞ a_n and lim_n→+∞ a_n exist is closed, and the functionals L^– and L⁺ assigning those limits are continuous, so use H.-B. to extend them to all of l^∞(Z).
  Of course, L^–(σ) = –1 and L⁺(σ) = +1. More importantly, by Zalduendo’s calculations, (L^– • σ)_n = –1 and (L⁺ • σ)_n = +1 for all n, so that   (L^– • L⁺)(σ) = L^–(L⁺ • σ) = lim_n→–∞ (..., +1, ...) = +1, whereas   (L⁺ • L^–)(σ) = L⁺(L^– • σ) = lim_n→+∞ (..., –1, ...) = –1.
  Thus L^– • L⁺ ≠ L⁺ • L^–, and, while the convolution multiplication on l¹(Z) is commutative, Arens’s multiplication on (l¹(Z))** is not, so ** on Ban₁ isn’t symmetric.