Compatible Peirce decompositions of Jordan triple systems - MSP

COMPATIBLE PEIRCE DECOMPOSITIONS OF JORDAN TRIPLE SYSTEMS 91
Proof. Penico solvability means S n (K) = 0 for some n, where
S(K) = P(J)P(K)J + P(K)J•+ L(K, K)J, S\K) = S{S n ^(K)). By
induction it suffices to consider the case S(K) = 0 of a trivial ideal.
By 5.8 we can left e to e, then fe Ji(e) to feJ^e). Since / covers
/, P(/)e" = L(f, f)e-e = 0 implies P(/)β - z 0 , L(f,f)e ~ e = z 2 lie
in if, SO the result follows from
LEMMA 5.11. If 1/2 eΦ and e, f are tripotents which are collinear
modulo a trivial ideal K and have
fe J&), P{f)e = z 0, L(/, f)e-e = z 2 (z.eK
then /' = / — 1/2 2 X /or ^ = {efz 0} is a tripotent collinear with e and
congruent modulo K to f. If g is a tripotent with eeJ^g), fsJj(g)
then f remains in J 3-(g).
Proof. Clearly /' is congruent modulo zxeK to /, it remains
in Jx{e) since z1eJ1{e), and if eeJt(g) f feJά(g) then by (1.2) zx =
L(e, f)P(f)eeJj(g) so /' remains in J9 (g). We must show (i) /' is
tripotent, (ii) eeJ^f), i.e., {f'f'e} = e.
By triviality of K we have /' 3
= f - 1/2{P(/) + L(/,/)K=/-
1/2{P(/) + L(f, f)}L(e, f)z0 = / - 1/2 st = /' since {P(/) + L(ff f)}L(e,
f)z0 - {P(P(/>, /) + L{e,f)L{f,f) + L({ffe}> f) - Ufi, {///})}«o (by
(0.2), (0.5)) = {P(z0, f) + L(e, f)L(f, f) + L(e + z2, /) - 2L(e, /)}« 0 =
L(e, f){L(f, f)-I}z0 (by triviality of K)=L{e, f)z0 because L(/, /)« 0=
Uf,f)P(f)e = P(P(f)f,f)e (by (0.2)) = 2P(/)β = 2β 0. Thus /' is
tripotent.
Again by triviality, {f'f'e} = {#β} - 1/2[{A*} + {^/β}] = e + s 2 -
l/2{P(e, /) + Uβ, f)}L(e, f)z 0 - e since {P(β, /) + L(e, f)}L(e, f)z 0 -
{Ufi, e)P(f) + P(e, P(f)e) + L(P(e)f,f) + 2P(e)P(f)}z 0 (by (0.7), (0.6)) =
{L(e, e)P(f) + P(β, βo) + 0 + 2P(e)P(/)}^ 0 = {L(e, β) + 2P( β )}P(/)^ 0 (by
triviality of iί again) - 2{I + P(e)}P(/) 2 e (as L(e, e) = 21 on J 2 (e)) =
2{P(/) 2 P(β) + P(β)P(/) 2 }β = 2{P({ffe}) + P(P(fYe, e) - L(f, /)P(β)L(/,
/)}β (by (0.8)) = 2{P(β + z 2 ) + 2P(/) 2 - L{f,ff)e (always P(e){aAe} =
{b&e}) = 2{(e + 2z 2 ) + 2P(/) 2 β - L(/, f)e - 2P{f) 2 e] (by triviality and
(0.6)) = 2{(e + 2z 2 ) - (β + 2J 2 )} = 2^. Thus eeJ^f) is collinear with
/'. D D
However, it is not clear that this argument can be extended
to show a whole collinear (or orthogonal-collinear) family can be
lifted to one of the same type.
REMARK 5.12. The fact that collinear e, f have been lifted to
tripotents β, / with feJ^e) does not imply collinearity eeJ^f),
hence the modification /' in the above proof is really necessary.