Jordan algebras of self-adjoint operators

Edward G. Effros, Erling Størmer
1967 Transactions of the American Mathematical Society  
1. Introduction. A Jordan algebra of self-adjoint operators on a Hubert space, or simply, a J-algebra, is a real linear space of such operators closed under the product A o B=^(AB + BA). A JC-algebra, respectively, a JW-algebra, is a uniformly closed, respectively, weakly closed /-algebra (we show in §3 that a-weakly closed /-algebras are weakly closed). In a recent paper [4], D. Topping has shown that many of the techniques used in the study of self-adjoint algebras of operators are applicable
more » ... tors are applicable to /-algebras. We continue in this direction, proving that various problems are simplified by passing to the second dual. We begin by showing that the second dual 91** of a /-algebra 9Í is isometric to a JW-algebm. We then use 9Í**, together with the second dual of the C*algebra [91] generated by 91 to investigate the uniformly closed Jordan ideals in 91. If 9 is such an ideal we prove that 9Í/8 is isometrically isomorphic to a /-algebra. We also show that if [S] is the C*-algebra generated by 8, then $ = [$] n 91, and [S] is an ideal in [91] . We conclude with a characterization of the uniformly closed Jordan ideals in 9t. The simplified proof of this result was suggested to us
doi:10.1090/s0002-9947-1967-0206733-x fatcat:de5nahnmnnax5ksqx2kpxo6w6q