Compatible Peirce decompositions of Jordan triple systems - MSP

COMPATIBLE PEIRCE DECOMPOSITIONS OF JORDAN TRIPLE SYSTEMS 71
β = βiu f— e 22> Jiz = Φβn)' Then K = Φe + J n is an ideal which is
not an ideal direct summand, yet it is covered (even spanned) by
e and all e t = e + z t for some finite basis {zj for J 12 . These e* are
idempotents but are not compatible with e: {eeβi} = 2β + z t ί J 2 (ei)
since 2 £ g Λfe).
Orthogonal families
We may regard (2.2) as an analogue for triple systems of the
Peirce decomposition relative to a single idempotent g 7
. If we have
mutually orthogonal families g\, •••, S? n (for example, if g^ consists
of compatible tripotents from J* in a direct sum J — Jx S ffl «/"„)
we have the following triple system analogue of the Peirce decomposition
relative to orthogonal idempotents g\, , &n. 2.5. PEIRCE DECOMPOSITION RELATIVE TO ORTHOGONAL COMPATIBLE
FAMILIES. If g? = g^ U U g" n is ίfee ^wio^ o/ mutually orthogonal
compatible families g^, •••, g^ n of tripotents, then the Jordan triple
system J has orthogonal Peirce decomposition
for
Ju - J*(&i) - Λ(^) n Π Jl&ϊ)
^oo — ΓΊ «/o( ^ί)
i
n Jxί^ ) = ^(ίf,) n J^^ ) n n
The Peirce decompositions relative to & and g 7 * are recovered by
== Σ JiU JJ&) = Σ Jii + Σ ^0, e7 0 (^) - J oo
Σ Λ
0
Jo(^i) = Σ Jst
j k i
The Peirce spaces multiply according to the following rules. A
product is zero unless its indices can be linked or linked through 0,
(2 6) i W « = {J«JM = o if {k, 1} n [i, j} = 0
or {k, 1}