Compatible Peirce decompositions of Jordan triple systems - MSP

COMPATIBLE PEIRCE DECOMPOSITIONS OF JORDAN TRIPLE SYSTEMS 97
in all extensions: the orthogonality may be "fortuitous". However,
there is an important case when orthogonality is "intrinsic", persisting
in all extensions:
{JiJjJ} = 0 if Jί9 Jά are orthogonal and one is locally unital
since for example if Jt has cover g^ then J; = J2(&i) and by orthogonality
JjdJά&i), so that {JtJjJ} c {J2J0J} = 0 by (2.2) (PI).
Thus if all Jt are locally unital, (7.3ii-iv) reduce to (ii) P(xi)S*(yί) =
0, (iii). {xSt(Vό)zk} = 0, plus
(7.5) {XiSΐίvM = {x iyiS k(z k)} .
Now in Lemma 7.7 we will see Sf(y/) eM 2{^ 5) + M^ 5), while
x i9 ^eJoί^ ), so (ii), (iii) follow automatically from Peirce orthogonality
P(J 0)M 2 = P{J 0)M 1 = 0, and (7.3ii-iv) reduce to (7.5). However,
(7.5) is not vacuous (both sides identically zero).
EXAMPLE 7.6. Let J = Φe1 ffl Φe2 be the locally unital Jordan
triple system (even unital Jordan algebra) with cover g 7
= {el9 β2}, and let M = Φm be the bimodule determined by p(et) = l(eif eά) = 0,
Vifiu e
o) " Kfiit e
i) — I- Here J = J0 Λf imbeds in iί2(Φ) via ^ —> e«,
m-^e 12
+ e 21, and M = MΊ(ej) — Mi(β 2). Then the map S(ae ι + /3e 2) =
(α + β)m is structural with £* = S, and jD(αe! + /3e 2) = (α — /5)m is
a derivation, but S(J 1) = S(J 2) = D(J 1)=D(J 2) = Φm so that {J^S^)} =
{JiS*(J t)J d} = Φm Φ 0, and condition (7.5) is not vacuous. Note that
S = L(m, e x + e 2) and JD = L(m, e t) — L(e l9 m) are inner. •
This example suggests that the troublesome nonorthogonality
of the pieces Su -—,Sn in (7.3) is due to an inner multiplication.
This is in fact the case: once we remove a suitable inner multiplication,
the remaining pieces St map Jt into ikf2 (i^) and hence are
automatically orthogonal. We need to investigate in more detail
the interaction of a structural S with a family of tripotents.
we consider a single tripotent.
First
STRAIGHTENING LEMMA 7.7. If S e Strl (J, M) and eeJ is tripotent,
then
( i ) S(e) = m 2 + m 19 S*(e) = m? + m* for m 2 = m 2 =P(e)m 2 ,
m i9 mf e M t (e)
(ii) S(J 2 (e)) c M 2 (e) + M L (e) with Eβ(x 2 ) = {m.ex,}
(iii) S(J 0 (e)) c M 0 (e) + MM with Eβfa) - {emfx 0 }
(iv) SiJM) c MM + MM + Moie) with E 2 S(x 1 ) = {em?x 1 }, E 0 S(x ί ) =
// J is locally unital with cover & then
S(J)czM 2(t?)