22 M. L. RACINE
PROPOSITION 2. Let M be an order of P = H(G, *). Assume that £
generates G. Then M' (the o-subalgebra of G generated by M) is a
---stable order of G. If G contains maximal orders then all maximal orders
are of the form M = # nF , F a maximal ^-stabl e order of G. On the other
hand if M = # n M ' and M' is a maximal ^-stabl e order of G then M is a
maximal order of $.
PROOF. By Lemma 1 M' is an order of G. It is spanned by elements
of the form a,a_ . . . a , a, e M. Since (a,a ^ . . . a )-- = a* . . . a * a * =
1 2 r
1 2 r r 2 1
a . . . a a
M', M' is ^-stable . If Q contains maximal orders there are
no infinite chains of orders, a fortiori no infinite chains of ---stable orders.
Therefore M' C F a maximal ---stable order of G and $ n F is an order of
9. Hence if M is maximal, M = # n F and any maximal order of # is of the
form ^ n F , F a maximal ---stable order of G. If M = J n M 1 and M' is a
maximal ---stable order of G then M C M , an order of P, implies
M* C M ' and the -- maximality of M* implies M' = M ' . Hence
M C ^ n M ! ^ n M ' = M . Therefore M is a maximal order of «?.
q. e. d.
The assumption # generates G is not really restrictive since if 8 is the
subalgebra of G generated by #, the same argument a s above shows that B
is ---stable. Therefore ((B, --) is an algebra with involution and p = H(®, *).
Examples will be given of orders M = # n F, F a maximal --stable order of G,
with M not maximal. Also there exist maximal orders M with M' not
maximal ---stable. Much of the remaining of this chapter will be directed at