Jordan Algebras

Abstract

Let V, {, ,} be a Jordan triple system (real, complex or hermitian). For a ∈V, let us denote by V ^(α) the vector space V endowed with the bilinear symmetric product o_a defined by

$$ x{o_a}y = \frac{1}{2}\left\{ {xay} \right\} $$

(2.1)

then V ^(a) is a commutative (in general non-associative) algebra over I $\mathbb{R}or\mathbb{C}$. If there is no ambiguity in the choice of a, we will write simply xy instead of x o_a y. For x ∈ V, the multiplication operator L _a (x) or L(x) is defined by

$$ {L_a}(x)y = L(x)y = xy $$

(2.2)

.