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”.
