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

  Problems: 1) (in Ban₁) Which Banach algebras A have A** commutative?
2) What other symmetric (and closed ?) monoidal categories have interesting dualizing objects C? For which is ** symmetric? Or, at least, for which will commutative monoid A have A** commutative? A topos, and C = Ω? AbGp, and C = the circle group? Posets? Simplicial sets? Topological spaces? R-modules? For what C?
3) The ‡ procedure endows A** with a counterpart to each finitary ⊗-operation ⊗ⁿA → A an algebra A may have, and associativity is one equation that A** inherits. Unit laws are others. Not commutativity. How about von Neuman quasi-inverse identities x x⁺ x = x? In additive symmetric monoidal categories, are Jordan or Lie algebra identities preserved? Idempotence of every element (Booleanness)? If not always, then when?
4) If ∃ well-behaved coproducts, there’s a free monoid monad A ···> Mon(A) = ⊕_n≥0 (⊗ⁿA).
Is λ_A: Mon(A**) → (Mon(A))** a distributive law λ: Mon·** → **·Mon (á la Beck) between ** and Mon? Here λ_A|⊗ⁿA** = (inj_n)** · ⊗^‡: ⊗ⁿ(A**) → (⊗ⁿA)** → (Mon(A))**.