Compatible Peirce decompositions of Jordan triple systems - MSP

100 KEVIN MCCRIMMON
By (7.8) this is equivalent to S preserving the individual Peirce
spaces
(7.90
or even just to preserving the tripotents,
(7.9")
Notice that if J is unital (if = {e}) then /= J 2(e) and unitality just
means S(J)aM 2(e) or S(e)eM 2(e) (and dually for S*):
PROPOSITION 7.10. A linear transformation S from a direct
sum J = J ± EB SB J n of locally unital Jordan triple systems into
a bimodule M is structural (resp. a derivation) iff it has the form
S = S o φ S λ φ φ S n where S o is inner (resp. standard inner
derivation) and S*: J* —> M are locally unital and structural (resp.
derivations). In particular, any S is the sum of an inner S Q and
a locally unital S\
Proof. Sufficiency is easy: S is structural or a derivation iff
S' = S-S 0 is, and the condition (7.5) on S'^S^φ φSL is
automatic (both sides vanish by orthogonality because Sfd/0 eM 2($?i) 9
S k(z k) e M 2( ifj by local unitality).
For necessity we apply the Straightening Proposition 7.8 to
the compatible family if = g^ U U ξ? n (recalling (2.3)). •
We can use these results straightening structural transformations
to prove additivity of the cohomology groups H\ just as we
used liftings of tripotents to straighten linear lifts and thereby
prove additivity of H 2
. Recall that there are two slightly different
first cohomology groups
H\J, M) - Der (J, M)/Inder (J, M)
H\J, M) = Strl (/, ikf)/Instrl (J, M) .
7.11. DIRECT DECOMPOSITION THEOREM FOR H 1
. If Ju , Jn are
locally unital Jordan triple systems relative to covers ifx , ifn and M is a bimodule for the direct sum J = J EB EB e/ , then
1 π
H\J, M) s H\J U Md Φ Φ H\J n , M n )
H\J, M) s H\J 19 M,) Φ φ H\J n> M n )
where M t = M u + M ί0 + M Q0 = f\ j
Proof. As in (6.2) we begin by imbedding @H\J iy M t) in