Lie Structure in Operator Algebras

Abstract

Let M be a von Neumann algebras, α — involutive *-anti-automorhism on M. Consider Mα(±1) = {x ∈ M : α(x) = ±x} the spectral subspaces of α. It follows that M α (+1) is a Jordan algebra with respect to the sym¬metrized product x • y = ½(xy + yx) and M α (- 1) is a Lie algebra with pespect to the brackets [x,y] = xy - yx. Robinson and Stormer [RS] have initiated the classification of pairs of Jordan and Lie algebras which can occur in this manner by examining the case M = B(H) the algebra of all bounded linear operators on a complex Hilbert space H, i.e. when M is a type I factor. In this chapter we shall continue the study of Lie algebras M α (-1) for arbitrary factors M.