ON m-STABLE COMMUTATIVE POWER- ASSOCIATIVE ALGEBRAS

A commutative power-associative algebra A of characteristic >5 with an idempotent u may be written1 as the supplementary sum ^=^4„(l)+4u(l/2)+^4u(0) where 4U(X) is the set of all xx in A with the property xx« =Xxx. The subspaces Au(l) and .4K(0) are orthogonal subalgebras, [AU(1/2)]2QAU(1)+AU(0) andAu(K)Au(l/2) C4„(l/2)+^4u(l—X) forX=0, 1. The algebra A is called w-stable if 4u(X)-4„(l/2)C.4u(l/2) and is called stable if it is w-stable for every idempotent element u of A. A. A. Albert has shown in [3] that a simple commutative powerassociative algebra A of degree > 1 over its center F with characteristic prime to 30 is a Jordan algebra if and only if it is stable. Moreover, it is known that every simple algebra of degree >2 is a Jordan algebra. Thus there remains the problem of determining the nonstable simple algebras of degree two. There do exist simple algebras of characteristic p>5 which are not Jordan algebras [3; 4]. Of course, these algebras are not stable, although they may be w-stable for some idempotent u. In this paper we shall obtain the following result.