Compatible Peirce decompositions of Jordan triple systems - MSP

94 KEVIN MCCRIMMON
pairs over arbitrary Φ.
Proof. We begin by imbedding 0 H\Ji9 Af«) in H\J,M). We
have a linear map @C\Jif Af,) -> C 2
(J, M) sending 0ft to p = 0ft
defined as /t>(α; 6) = Σ pi(at; δ,). This induces φC^Ji, Λf«) -> JEΓV, Af).
To characterize the kernel of this map, assume p = δgeB\J,M)
for some linear g: J—>M. Using the Peirce decomposition Af=0ikfifc of (2.10) relative to the g^ , we can write the restriction of g to
Ji as g = ®g for g : Ji->M . Applying the Peirce projection
ih jk jk
E'u + E + E of M on Af, to the relation p(a ; 6,) = d£r(α,; δ,) for
i0 oo t
α,, δ, e J, = J we get by the Peirce relations ft(α*; δ,) = p(α,)(flr,, +
iί9
Λo)(&t) + ϊ(α,, 6ί)(5r + flTio)(δi) - (flr« + 0*0 + gw)(P(a )bi) so that ft = δflr,
£ί t 9
for flr, = g + ^ + 0oo J< ->Λf Af. Indeed, δ^(α; b)=p(a)g(b) + l(a, b)g(a)-g(P(a)b) =
{Σ P(α*) + Pfa, α*)} Σ flΓy(δy) + Σ ί(α,, fcjflr^) - Σ ΰiiP^h) (by orthogonality
and local unitality of J J, An J) = Σ{P.(^)Λ(W + ϊ( u α t>
&«)&( £Γ t 2 (/, M) is precisely ®B\J Af,), so we
i9
haye an induced imbedding of φjff 2 (J,, Af
Ji-^Ji-+0; we must work with the linear space Ju instead of the
subsystem Jt .) The cocycle p associated to J via this lift σ is of
the desired form 0ft since p(a; b) = P(σ(a))σ(b) - σ(P(a)b) =