M. L. RACINE
MARTINDALE'S THEOREM (). Let (0, *) be an associativ e algebra
with involution. Assume that 1 = / e., n 3, where the e. are orthogonal
• - i
1 - 1 r k k k
idempotents such that e,
M(G, *), G.e.ci= G and e. =J a,, b a., for some
k k k k
i ^ i , a e e ne.3 a = (a .)* and b.
M(G. *) . Then any homomorphism of
ij i ] ji i] J
W(G, *) into an algebra B , where B is a (unital) associativ e algebra, ha s a
unique extension of G into B.
If (G, *) is central simple over a field $ and the degree of G 3,
extending the bas e field if necessary , we may apply Martindale's Theorem.
In particular we obtain that M(G, *) generates G; an automorphism of G is
either an automorphism of G or an antiautomorphism of G. If G/$ is central
simple, the automorphisms of M(G, *) are inner automorphisms of
G : a - c a c with c c * € $ 1 . If G/r is central simple, r / § a quadratic
field extension, the automorphisms of W(G, *) are automorphisms of G/$,
which commute with *; thes e need not be inner. (See , p . 248, where
the arguments are als o valid in characteristic 2. )
§2. Orders in Jordan Algebras.
Let K be the quotient field of a Dedekind domain o. If v is a finite
dimensional K-vector space , an o-lattice L of V is a finitely generated
o-module L C As with KL = V. An element x of a Jordan algebra over K is
said to be integral if it satisfie s a monic integral polynomial, that is with
coefficients in o. From now on in this chapter $ will be a finite dimension -
al Jordan algebra over K. We note that (4) and (5) imply that o[x], a s an