A proof of Schafer's conjecture for infinite-dimensional forms admitting composition

Abstract

is necessarily a separable finite-dimensional alternative algebra, and that Q is a product of irreducible factors of the generic norm. In [Z2] he showed such an algebra is necessarily alternative, is separable when finite-dimensional, and in this case (using results of Professor N. Jacobson [4]) that Q is a product of irreducible factors of the generic norm, so the problem reduced to proving finite-dimensionality. This was a classical result if Q was of degree 2; such an algebra was a composition algebra, and these could be explicitly constructed [2] and shown to have dimension 1,2,4, or 8. There is little hope of generalizing this method to Q of arbitrary degree, since it would involve an algorithm for constructing all separable alternative algebras. Schafer’s approach was more akin to Professor I. Kaplansky’s original proof [6] of the degree 2 case. Kaplansky linearized the basic identity to show first that the algebra was alternative, next that it was algebraic, and finally that it was semisimple. At this juncture results on alternative rings were invoked to conclude that the algebra was finite-dimensional. In this paper we will apply a similar procedure to normed algebras [S]. Replacing linearization techniques by differential operators, we will show first that such an algebra is a noncommutative Jordan algebra (Lemma l), next that it is algebraic (Lemma 2), and finally (Theorem 1) that is a direct sum of a finite number of simple ideals which are either