Compatible Peirce decompositions of Jordan triple systems - MSP

COMPATIBLE PEIRCE DECOMPOSITIONS OF JORDAN TRIPLE SYSTEMS 101
ίϊ\J, M). We have a natural imbedding φS, -> S of φ Strl (J if M t)
in Strl (J, M) (the orthogonality condition (7.5) being automatically
satisfied thanks to S S(J), Sf(J) cAf, = Πw M 0(%Ί) and J ό = J*(&i)
by hypothesis on S s). This induces a linear map φ Strl (J if Mi) —>
H\J,M).
To characterize the kernel of this map, we need to know when
S— φSi is inner on J= EBe/ί, i.e., a sum of L(x kk, m kι), L(n lk, y kk)
for various elements # fc/c, y kk cJ k — J kk, m kU n lk e M kl the Peirce spaces
(2.5) relative to the given orthogonal covers g^, •••, c H\J, M) has kernel precisely φlnstrl (Jiy Mi) and thus induces an imbedding of φJΪ 1
^, Mt) ~ {φStrl (Jif Mi)}/
{φlnstrl (Ji9 Mi)} in ffV, M).
By (7.10) this imbedding is surjective: if S e Strl (/, M) then S
is congruent modulo Instrl (J, Jkf) to a sum S1 φ φ Sn of locally
unital S where by local unitality of S with respect to g^ we
if t
have S Sϊ mapping J if r
t = J2 (g7 < ) into ikί^^) = ikί^ cil^, so S,6
Strl (J Mi). Thus we have a natural isomorphism @H\J M ) ~
if i9 t
8\J, M).
The same argument, mutatis mutandis, shows @H\J M ) =
U t
D
Just as in (6.3), we can apply this to semisimple Jordan pairs