Compatible Peirce decompositions of Jordan triple systems - MSP

COMPATIBLE PEIRCE DECOMPOSITIONS OF JORDAN TRIPLE SYSTEMS 67
forms a family of hermitian matrix units and thus yields an imbedding
H Z(Φ) —• / sending l[ii] -> e if l[ij] -> ι%. •
Collinearity in JT(A, *)
A more general method for obtaining Jordan triples T from
Jordan algebras J is through P(x)y — U(x)y* for some involution *
of J. However, there seems to be no relation between tripotents
x e JT(J, *) and idempotents in J (in general there don't seem to be
idempotents in J). In the special case where / = A +
, so P(x)y =
xy*x, an element x is tripotent iff x — a + δ for aeeAe, beeA(l — e)
satisfying αα* + 66* = e for a symmetric idempotent β (namely e =
a%c*, a = xe, 6 = a;(l — e)). Collinearity becomes complicated,
1.20. COLLINEARITY THEOREM FOR JT(A, *). Two nonzero tripotents
x, y in JT(A, *) are collinear iff £/&erβ are £wo families e n,
€22, β 33 a^cί / n, / 22, / 33 o/ symmetric orthogonal idempotents and elements
x ijf y i3' in e uAf 5j such that
x = x 12 + x 21, y = y 13 + y 31
Proof. a->a' = (\ Q) imbeds JΓ(A, *) in JΓ(B) for B=Λf 2(il),
so from (1.15) ^' - eί 2 + e 21, y' = eί s + ώ for ej,.= ^ t °Q j
\i, j = 1, 2)
^4 = (°/Q) (i, fc = 1, 3), e'u = ^Q* ?) we get the result. Π
2* Compatible Peirce decomposition* A finite family gf —
{βi, , en } of tripotents is compatible if every pair eif e3- is compatible.
Now any time we have a finite number of commuting decompositions
I — Eziβi) + E^βi) + E0 (ei) relative to eu them together to get a simultaneous decomposition
, en we can put
of the identity operator for
Σ
(i I , .i Λ )e{2,l,0} Λ
By commutativity these JSΓS are supplementary projection operators,
and hence yield a compatible Peirce decomposition.
J — © ^ (ϊi, ,ί Λ )
(2.1) « 1 . .i,)βu, 1 .θ!.