A unital JB-algebra is a Jordan algebra A with identity together with a complete norm satisfying, for all a, bA,(i) (a2b) a = a2(ba),(ii) a2 = a2,(iii) abab,(iv) a2 + b2a2, b2.(It should be noted that axiom (iii) is a consequence of (ii) and (iv).) Such spaces have been studied by several authors (3, 6, 11), and as a consequence their structure is now quite well understood. Many of the results of these papers, while relying on the existence of an identity for their proofs, can be formulated for algebras which lack this property. C*-algebra theory and operator theory abound in examples of spaces which fail to be unital JB-algebras only in this one respect, and this motivates the study of the general case undertaken in this note.
