On the Relationship between Jordan Algebras and Their Universal Enveloping Algebras

Abstract

The relationship between JW-algebras (resp. JC-algebras) and their universal enveloping von Neumann algebras (resp. $C^{\ast }$ -algebras) can be described as significant and influential. Examples of numerous relationships have been established. In this article, we established a relationship between the set of split faces of the state space (resp. normal states) of a JC-algebra (resp. a JW-algebra) and the set of split faces of the state space (resp. normal states) of its universal enveloping $C^{\ast }$ -algebra (resp. von Neumann algebra), and we tied up this relationship with the correspondence between the classes of invariant faces, closed ideals, and central projections of these Jordan algebras and of their universal enveloping algebras.