JBW-algebras

Abstract

JBW-algebras are the Jordan analogs of von Neumann algebras (also known as W^*algebras), and include the self-adjoint part of von Neumann algebras as a special case. In this chapter we will develop basic facts about JBW-algebras. We begin with the definition and the relevant topologies. Then we introduce an abstract notion of range projection, and a spectral theorem for JBW-algebras, derived from the spectral theorem for monotone completeC_R(X)(A 39). We then study the lattice of projections of a JBW-algebra, and establish a 1-1 correspondence of projections and u-weakly closed hereditary subalgebras, and a correspondence of u-weakly closed ideals with central projections. We show that the bidual of a JBalgebra is a JBW-algebra. Then we use the bidual to prove that JB-algebras are the same as commutative order unit algebras, to show that a unital order isomorphism of a JB-algebra is a Jordan isomorphism, and to show that skew order derivations are in fact Jordan derivations. We prove that every JBW-algebra has a unique predual consisting of the normal linear function-als. Then we develop some basic facts about JW-algebras (a-weakly closed subalgebras of β(H)_sa)and we finish this chapter with an order-theoretic characterization of the maps a H {pap} for projectionsp.