Compatible Peirce decompositions of Jordan triple systems - MSP

98 KEVIN MCGRIMMON
S and S* preserve Peirce spaces relative to e> S(J k (e)) + S*(J k (e))(Z
M k (e) for k = 2, 1, 0, iff S and S* preserve e in the sense that S(e),
S*(e) e M 2 (e). Always S'= S - {L(m lf e) + L(e, m,)} has S\e), S'*(β) 6
M 2 (e) so we can modify S by an inner multiplication to obtain S'
preserving Peirce spaces. If S = D is a derivation then L(m u e) +
L(e, mj = D(m lf e) is a standard inner derivation.
Proof. If S(e) = m = Σ w,, S*(e) = m* = Σ ^f for m*, m? 6
Λfi(β) then by (7.1) S(P(e)e) + P(e)S*(e) = {S(e)ee} implies (m 2 + m^
m 0) + m 2* = 2m 2 + m 2, hence m 0 = 0 and m$ = m 2. Dually m* = 0
and m 2 = m 2. This establishes (i).
For XiβJiie) the linearized varsion of (7.1) yields S({eeXi}) +
icJ = {a?iβjS(β)} + {eeS(Xi)} 9 hence ΐS(^) + {emtXi} + {emtxt} =
+ {m^α J + L(e, e)S(Xt). But L(e, m?) = L(mF, e)=L(m2 f β) by
(i) and (1.4) so
{L{e, e) ~ U}S(Xi) = {emfίcj — {m^Xi} .
For ΐ = 1 taking components in M2, Mo yields (iv). For i = 0 we
get the expression (iii) for EβiXo), where S(xo)eMo + Mλ since by
(7.1) P(e)S(x0) = -S*(P(e)a? 0) + {S*(βKe} = 0. For ΐ = 2 we get the
expression (ii) for EβiXz), where S(x2)eM2+M1 since by (7.1) S(x2) =
S(P(e)x2) = -P(e)S*(xt) + {S(e)x2e} e P(e)M + {eJM}.
If J is locally unital then S{J)^S(^ J2{e,)) c.^ Afβ(ef)+Λfι(e