Vol. 15, No. 3, 1965

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Norms and noncommutative Jordan algebras

Kevin Mor McCrimmon

Vol. 15 (1965), No. 3, 925–956

Roughly speaking, a norm on a nonassociative algebra is a nondegenerate form Q satisfying Q(Mxy) = m(x)Q(y) for all x, y in the algebra where Mx is a linear transformation having something to do with multiplication by x and where m is a rational function; taking Mx = Lx or Mx = Ux = 2Lx2 Lx2 we get the forms Q satisfying Q(xy) = Q(x)Q(y) or Q(Uxy) = Q(x)2Q(y) investigated by R. D. Schafer. This paper extends the known results by proving that any normed algebra A is a separable noncommutative Jordan algebra whose symmetrized algebra A+ 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”.

Mathematical Subject Classification
Primary: 17.40
Received: 7 April 1964
Published: 1 September 1965
Kevin Mor McCrimmon