Approximately norm-unital products on [formula omitted]-algebras, and a non-associative Gelfand–Naimark theorem

Abstract

We describe the non-associative products on a C ⁎ -algebra A which convert the Banach space of A into a Banach algebra having an approximate unit bounded by 1, and determine among them those which are associative. As a consequence, if such a product p satisfies p ( a , b ) □ = p ( b □ , a □ ) and ‖ p ( a □ , a ) ‖ = ‖ a ‖ 2 , for all a , b ∈ A and some conjugate-linear vector space involution □ on A, then p is associative. The proof of the above result involves also a new Gelfand–Naimark type theorem asserting that non-associative C ⁎ -algebras (defined verbatim as in the associative case, but removing associativity) are alternative if and only if they have an approximate unit bounded by 1.