Given a nonassociative algebra A and an Arens pair A1, A2, for A, we identify a subalgcbra A¯ of A2 with i (A) ⊂ A ⊂ A2 and show that A¯ better reflects the algebraic structure ot A, in parti-cular. any multilinear identity satisfied by A¯ is also satisfied by A¯ Hence, A¯ 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 A¯ = EndD(V) for A a primitive, associative algebra with commuting ring D and irreducible faithful module V,(2) where A¯ 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 A¯ 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 A¯ is determined by A, A1, A2 and subspaces A3 ⊂ A2*, A4,⊂A3*.
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