Compatible Peirce decompositions of Jordan triple systems - MSP

COMPATIBLE PEIRCE DECOMPOSITIONS OF JORDAN TRIPLE SYSTEMS 65
Collinearity in JT{A)
From any associative algebra A we can form a Jordan triple
system JT(A) on the linear space A by
P(x)y — xyx .
Here an element x is tripotent iSxxx = x, i.e., # 8 = x. In this ease
e = a; 2
is an ordinary associative idempotent, and e# = xe = #, Thus
x lies in the unital Peirce subalgebra eAe and is a "square root of
unity" therein. Examples of collinear tripotents are the matrix
units x = E12, y — E1Z or & = i£ 12 + i? 21, V = Eu + E91. The latter
example is quite general, since we have
1.15. COLLINEARITY THEOREM FOR JT(A). TWO nonzero tripotents
x, y in JT(A) are collinear iff there is a subalgebra B = M Z(Φ) of
A with x = E ί2 + E 21, y ^ E 1Z + E Z1.
Proof. Tripotence means x* = x, y z
= y and collinearity means
xyx = 2/#2/ = 0> x*V + 2/^ 2
= l/> y 2
% + ί»2/ 2
= «• Then x 2
y 2
—(y—yx 2
)y—
y(y — x 2
y) — y z
% 2
, so a direct calculation shows
e n = x 2 2/ 2 = 2/V β 22 = αjί/ 2 a; e S5 — yx 2 y
form a complete family of matrix units, hence yield an isomorphism
of M Z(Φ) into A by E iά -> e