Compatible Peirce decompositions of Jordan triple systems - MSP

COMPATIBLE PEIRCE DECOMPOSITIONS OF JORDAN TRIPLE SYSTEMS 93
/ = π(j) = π(j 2) + π(j x) + π(j 0) = JJ&) φ jj&) 0 j o(gf) where L is
orthogonal to %f cK, we must have π(J 2) = / 2(^) = ^, ^(Λ) =
J 0(gf) = L, π(J γ) = J^gf) = 0. Thus 0->ikf o(g 7 )-^/ o(^)-^ ί'->0 is
exact, hence by our hypothesis on L it too splits: there is a subsystem
CaJ 0(&) isomorphic under π to L. Because we have lifted
I? into J^^) and C into J o(&), they are automatically orthogonal
by Peirce orthogonality (2.2) (PI), and the sum B ffl CcJis automatically
a direct sum of triple systems isomorphic under π to K S3
L = J. Thus J = (5 0 C) © M is split. •
The argument actually shows it suffices if all but one of the
Ji is locally unital, since we never needed a cover for L.
The crucial step in the above proof was lifting a cover £? of
if to a compatible family in J, which depended on knowing H\K, M) =
0. To get a relation between H\J f M) and the H\J i9 Λf 4) in general,
we need to be able to lift the tripotents which form the local unit
of Ji. This is why we studied the problem of lifting compatible
families modulo trivial ideals in (5.8.)
Using hierarchical families we can reduce extensions of direct
sums to extensions of the pieces. We recall how the equivalence
classes of extensions of a Jordan triple system by a bimodule M
gains its algebraic structure H\J, M). If J is protective as Φmodule,
each extension has the form J = σ(J) φ M for some linear
lifting a of J into J, and is associated with a cocycle or factor set
peC\J,M)
p(a; b) = P(σ{a))σ(b) - σ(P(a)b)
which measures how far σ is from being a triple system lift. The
equivalence class of p modulo coboundaries δg e B 2
(J, M)
Sg(a; b) = p(a)g(b) + Z(α, b)g(a) - g(P(a)b)
for a linear map g:J-+M is independent of the particular a, and
the extensions of J by Mare in 1 — 1 correspondence with H\J, M) =
C 2
(J, M)/B 2
(J, M). Thus H\J, M) becomes a module over Φ [6, p.
287-288].
6.2. DIRECT DECOMPOSITION THEOREM FOR H\ IfJ=J ι m--mJ n
is protective as module over Φ (containing 1/2), and as triple system
is a direct sum of locally unital Jordan triple systems Ji possessing
hierarchical covers $? if then for any J-bimodule M
H\J, M) s Θ H\J i9 Mi)
for Mi = M u + M ί0 + Moo = f\ jΦi M 0(8 7
i) The same holds for Jordan