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

  In a monoidal setting, there’s a ⊗^‡: <X**, Y**, ..., Z**> → (X ⊗ Y ⊗ ... ⊗ Z)** arising from the universal multilinear ⊗: < X, Y, ..., Z> → X ⊗ Y ⊗ ... ⊗ Z and, when μ: <X, Y, ..., Z> → A, we have (naturality in A) μ^‡ = μ** · ⊗^‡: <X**, Y**, ..., Z**> → (X ⊗ Y ⊗ ... ⊗ Z)** → A**.
  One may also view ⊗^‡ and μ^‡ as maps X** ⊗ Y** ⊗ ... ⊗ Z** → (X ⊗ Y ⊗ ... ⊗ Z)** and X** ⊗ Y** ⊗ ... ⊗ Z** → A** — i.e., Arens’s work exhibits monoidal structure on ** .
  In a symmetric monoidal setting, one can ask: is ** symmetric monoidal? That is, where τ_AB: A⊗B → B⊗A denote the symmetry isomorphisms, and μ: A⊗B → C is given, will ((μ·τ_BA): B⊗A → A⊗B → C)^‡·τ_A**B**: A**⊗B** → B**⊗A** → C** = μ^‡: A**⊗B** → C**?
  Particularly, focussing only on ⊗: <A, B> → A⊗B (and abusing notation), have we ((⊗·τ_BA)^‡·τ_A**B**: A**⊗B** → B**⊗A** → (A⊗B)** = ⊗^‡: A**⊗B** → (A⊗B)**?
  Or, when A = B and μ · τ_AA = μ, have we μ^‡·(τ_A**A**) = μ^‡? What if C = A as well, and μ makes A a (commutative) monoid? — “It depends ... :” Yes, if A = C(X) (X KT₂) in Ban₁.
  No (example later) for some other commutative Banach algebras (though always μ^‡ makes A** an associative monoid (with unit of A, if available, as unit)).