The arens closure of a nonassociative algebra

Given a nonassociative algebra A and an Arens pair A₁, A₂, for A, we identify a subalgcbra ? of A₂ with i (A) ? A ? A₂ and show that ? better reflects the algebraic structure ot A, in parti-cular. any multilinear identity satisfied by ? is also satisfied by ? Hence, ? is commutative or Lie when A is and Jordan when A is a Jordan algebra of characteristic not 2 or 3. Also, we list examples (1) where ? = End_D(V) for A a primitive, associative algebra with commuting ring D and irreducible faithful module V,(2) where ? is the norm closure of A in the arens algebra of all bounded functionals of the bounded functionals for a normed algebra A and (3) where ? is the Arens algebra of all bounded functionals of the bounded functionals with A again normed. Note that dif-ferent Arens closures can arise form the same choice of A, A₁, , A₂ since ? is determined by A, A₁, A₂ and subspaces A₃ ? A₂ ^*, A₄,?A₃ ^*.