Compatible Peirce decompositions of Jordan triple systems - MSP

96 KEVIN MCCRIMMON
usual to simply refer to S itself as a structural transformation.)
The inner structural S are those in the subspace Instrl (J, M) =
L(M, J) + L(J, M) (note by (0.4)L(#, m)* = L(m, a?)), the inner derivations
are Inder (J, M) = Instrl (J, M) Π Der (J, ilf). This includes
all sums of standard inner derivations D(x, m) = L{x, m) — L(m, x),
and coincides with these if 1/2 eΦ.
Even for unital Jordan algebras, the derivations from a direct
sum are not simply the sums of derivations defined on the individual
pieces: the derivations on the pieces can be glued together to form
a global derivation only if they are suitably orthogonal.
PROPOSITION 7.3. A linear transformation S: J —> M of a Jordan
triple J = J ι EB EB J n into a bimodule M is structural iff it has
the form S — S x φ 0 S n for S*: J* —> ikf satisfying the following
relations (for x if y u z t eJ* with i, j, kΦ):
(i ) each Si is structural: Si e Strl (J i9 M)
(ii) P{x i)Sny 5)
(iii) {XiSf( Vj)z k}
(iv) {x tSϊ(y t)z k} - {x ty tS k(z h)}
S is a derivation iff each Si is a derivation, S? = — S M are precisely the S =
for Sii Ji-+ M. The structural condition (7.1) for all x, y eJ reduces
to the following conditions on the spanning elements x if y if z { e J ύ:
(7.1a) Smxjyj) + P(Xi)Sf(y ά) = {x.
(7.1b) ^({^7/^}) + {XiSf( yj)z k} = {xMjSM} +
for all i, j, Jc with %Φk. Condition (7.1a) for j = i is the condition
(i) that Si be structural on J> for j Φ i, by orthogonality J* l J^
the condition (7.1a) reduces to (ii). Similarly, by orthogonality (7.1b)
reduces to (iii) if j Φ i,k and to (iv) if j = i. •
To see that the requisite orthogonality is not automatic, consider
the following
EXAMPLE 7.4. Let J = Φz ffl Φw be a trivial Jordan triple
system and M = Φm 0 Φ^ the bimodule determined by p(z) =
/o(«, w) = Z(β, 2) = l(w, w) = i(w, 2) = 0, ί(z, w)m = n, l(z, w)n = 0.
Here the split null extension J=JφM may be imbedded in M 4 (Φ) +
via z -»£ 12 , w ~> β 23 , m -> e 3i , n —> β u . Then {zwM} = Φn Φ 0, so {Ji/yM"}
is not necessarily 0 in a direct sum J = Ji EB EB J n.
Thus orthogonality of Ji, J^ in J does not imply orthogonality