On the Structure and Tensor Products of JC-Algebras

Abstract

Norm closed (or weakly closed) Jordan algebras of self-adjoint operators on a Hilbert space were initially studied by Topping, Effros, and Stormer [15], [4], [12], [13] . These works are very “spatial”, in that the algebras are considered in one given representation. The introduction of their abstract counterparts, the JB- and JBW- algebras, has led to an increased interest in this subject. The author hopes this paper will support the view that a more “space-free” approach is fruitful, even if only the “concrete” algebras are under study. In accordance with this view, a “ JC -algebra” in this paper will mean a normed Jordan algebra over the reals, which is isometrically isomorphic to a norm closed Jordan algebra of self-adjoint operators. Some of the results in this paper are closely related to, or rewordings of, results in the above-mentioned papers. However, I feel that the present approach is sufficiently different to be of interest in itself. In particular, many of the technical difficulties associated with earlier approaches are avoided.