Approximately norm-unital products on C⁎-algebras, and a non-associative Gelfand–Naimark theorem

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.