On the convergence of ordered sets of projections

Abstract

E. R. Lorch has shown [3, p. 223 ] that a uniformly bounded, naturally ordered sequence of projections in a reflexive Banach space converges to its supremum in the strong topology of operators. In this note it is shown that a uniformly bounded, naturally ordered set of projections in a Banach space X having a weak x-cluster point for each x CX converges to its supremum in the strong topology of operators. The method of proof is substantially different from that of Lorch. A bounded operator E, acting in a Banach space X, is a projection if E2 =E. Letting I denote the identity projection, E(X) and (I-E) (X) are strongly closed manifolds. The projections in X have a natural order: E?<E2 if ElE2=E2E, -EB. If { a } is a directed set, and a, < a2 implies Ea,, <Ea2, then BEa} is naturally ordered. If Y is a subset of X, w( Y) will denote the weak closure of Y. The span of Y, denoted by sp(Y), is the strongly closed linear manifold generated by Y. Suppose {EBa} is a set of projections in X. A projection E such that E(X) =sp(UaEa(X)) and (I-E)(X) =fna(I-Ea)(X) is the supremum of {Ea, }, and is denoted by UaEa. If { Ea } is naturally ordered, y, is a weak x-cluster point of { Ea} if yx G faw( { E:(x) I j _ a }). The central result of this note follows: