On the Gelfand–Naimark Axiom ||a* a||=‖a*‖ ‖a‖

Let A be a (possibly non-associative) complex Banach algebra and let * be a conjugate-linear vector space involution on A such that ‖ a*a‖=‖ a*‖ ‖ a ‖ for every a∈A. We prove the following facts: If A has an approximate unit bounded by 1 and consisting of *-invariant elements, and if dim (A)≠2, then A is alternative, and * is an algebra involution. If A is alternative, and if * is an algebra involution, then * is an isometry, and hence A has an approximate unit bounded by 1 and consisting of *-invariant elements. Fact 1 seems to us to be new even if A is associative, whereas Fact 2 becomes a generalization to alternative algebras of a well-known result in the associative setting.