Die Struktur endlicher schwach abgeschlossener Jordanalgebren Teil I. Stetige Jordanalgebren

On the basis of the authors previous paper 6] the properties of the lattice of idempotents in a finite weakly closed Jordan algebra A are examined. The results then yield a unique direct decomposition A=A_d⊕A_c where A_d is a discrete and A_c is a continuous Jordan algebra. An application of Jacobsons coordinatization theorem shows A_c=H(D), D associative, since there exist sufficiently many strongly connected idempotents. Then we prove that D is a uniquely determined continuous real W^* -algebra with finite normal faithful trace.