The structure of primitive quadratic Jordan algebras

Abstract

In this paper we give a complete description of primitive quadratic Jordan algebras following the classification of strongly prime quadratic Jordan algebras (K. McCrimmon and E. I. Zel'manov, Adv. Math. 69, No. 2 (1988), 133–222). The proof is based in two results of independent interest: we show that every P. I. primitive Jordan algebra is simple and unital with nonzero socle and prove that associative tight (*-tight) envelopes of special primitive Jordan algebras are also primitive (*-primitive). As a consequence of the latter fact we see that an associative algebra A is one-sided primitive if and only if A + is primitive, and an associative algebra A with involution * is *-primitive if and only if an ample subspace H 0 ( A , *) is primitive.