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By O.U. Schmidt

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Let w = r(Upa). Then w :S: p, so w, and then Wi, operator commutes with p. By definition, Uw(Upa) = Upa, so Uwl(Upa) = o. Therefore UWl/lpa = UwlUpa = 0, which implies UCWI/lp)la = a. Thus Uwvp1a = a, which implies w V pi ~ r(a). Now w V pi ~ r(a) V pi follows. Since w :S: p = (pi) I, then w is orthogonal to pi, so we have w + pi = W V pi ~ r(a) V p'. 23) with e = pi and f = r(a) V p'. 24) , and thus completes the proof. 0 2. 54 JBW-ALGEBRAS O'-weakly closed hereditary subalgebras °: ; A Jordan subalgebra B of a JB-algebra A is hereditary if (a E A and b E B with a ::; b) implies a E B.

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 complete CR(X) (A 39). We then study the lattice of projections of a JBW-algebra, and establish a 1-1 correspondence of projections and a-weakly closed hereditary subalgebras, and a correspondence of a-weakly closed ideals with central projections.

An ordered Banach space A is monotone complete if each increasing net {b a } which is bounded above has a least upper bound b in A. ) A bounded linear functional a on a monotone complete space A is normal if whenever ba / b, then a(b a ) ~ a(b). 2. Definition. A lEW-algebra is a JB-algebra that is monotone complete and admits a separating set of normal states. In [AS], normal positive linear functionals on a von Neumann algebra were defined first (A 90) , and then general normal linear functionals were defined to be linear combinations of normal positive linear functionals.

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