Compatible Peirce decompositions of Jordan triple systems - MSP

60 KEVIN MCCRIMMON
(k = j or i = j" or & = ΐ = j or i = j, k = £ allowed)
for distinct indices i, j, fc, I, and all other unlinked products vanish.
The old unit element c has P(c)y == # y , so c remains the unit only
if j = I is trivial. Thus the algebra case H (D, D ) results from
n o
choosing the trivial automorphism j. If j is not an inner automorphism
on /, i.e., x j
is not of the form U(u)x, then Hn(D, DQ) is not
a Jordan algebra: there is no unit element u, since P(u) = U{u)P(c) —
I iff Uiu) = P(c) = j.
1Φ Compatible tripotents* An element eeJ is tripotent if
P(e)e = e; such an element determines a decomposition J = E x © £Ό Θ
2£ 0 of the identity operator on J into a direct sum of Peirce projection
operators
E (e) = P(e) 2 2 , E {e) = L(e, e) - 2P(ef ,
x
El) - B(e, e) = I - L(e, e) + P(e) 2
.
Such an operator decomposition leads immediately to a Peirce decomposition
J = Λ Θ J; θ Λ (/* = «/x. The Peirce spaces multiply according to
P(J )JjC:J _ {JiJiJkί^Ji-o+k, oτ more specifically for i = 2, 0, j =
t 2i jf
2-i
P(J t )J a - {JJJ,} = 0,
(1.2) PWΛ - 0, P(JJJ 4 c J i9 {JoJM c /,
{/,/,JJ c J if {JM} c /,, P{J X)J X c J,
We will make frequent use of the fact that multiplications L(x, y)
by elements in the same Peirce space leave all Peirce spaces invariant,
(1.3)
We also have the general rules