Norms and noncommutative Jordan algebras
Kevin McCrimmon · Pacific Journal of Mathematics · 1965
Roughly speaking, a norm on a nonassociative algebra is a nondegenerate form Q satisfying Q(M x y) = m(x)Q(y) for all x, y in the algebra where M x is a linear transformation having something to do with multiplication by x and where m is a rational function; taking M x = L x or M x -U x = 2L 2 X -L x % we get the forms Q satisfying Q{xy) = Q(x)Q(y) or Q(U x y) = Q(x) 2 Q(y) investigated by R. D. Schafer.This paper extends the known results by proving that any normed algebra % is a separable noncommutative Jordan algebra whose symmetrized algebra 9ϊ+ is a separable Jordan algebra, and that the norm is a product of irreducible factors of the generic norm.As a consequence we get simple proofs of Schafer's results on forms admitting associative composition and can extend his results on forms admitting Jordan composition to forms of arbitrary degree q rather than just q = 2 or 3. We also obtain some results of M. Koecher on algebras associated with ω-domains.In the process, simple proofs are obtained of N. Jacobson's theory of inverses and some of his results on generic norms.The basic tool is the differential calculus for rational mappings of one vector space into another.This affords a concise way of linearizing identities, and through the chain rule and its corollaries furnishes methods not easily expressed "algebraically".Algebras having some sort of "norm" have appeared in various investigations.R. D. Schafer proved in [12] that any algebra 31 with a nondegenerate form Q admitting associative composition Q(xy) -Q(x)Q(y) is a separable alternative algebra.In [11] he proved that if 31 is commutative and has a form Q of degree 2 or 3 admitting Jordan composition Q(U x y) = Q(xfQ(y), where U x -2L\ -L X 2, then it is a separable Jordan algebra.In the applications of Jordan algebras to several complex variables [9] M. Koecher considered domains in a real vector space on which a positive homogeneous real-analytic function ω was defined satisfying ω(H x y) = det H x ω(y), where H x was essentially the Hessian of log ω at x.He associated with such an ω-domain a real εemisimple Jordan algebra 3t in which H x = ~U~λ.In all these cases the algebra was a separable noncommutative Jordan algebra and the norm Q (or ώ) was essentially a product of the irreducible factors of the generic norm of 21.