Complemented Subspaces of Cp Spaces and CSL Algebras

Abstract

In this article we consider the question of whether or not a subspace of the form Cp ∩ Alg L, where L is a commutative subspace lattice (CSL), has a complement in Cp, 1 <p < ∞. While the answer is affirmative in the case where L is a nest [8] we show that this need not be true even for completely distributive CSLs. Consequently, we consider CSLs generated by a completely distributive CSL and finitely many commuting nests and we prove that Cp ∩ Alg L has a complement in Cp if and only if U2(Cp ∩ Cq) ⊆ Cp ∩ Cq, where 1/p+1/q = 1 and U2 denotes the orthogonal projection from C2 onto C2 ∩ Alg L Moreover, we observe that this result fails for CSLs which do not satisfy the above requirement. Finally, we show that in certain cases the existence of a complement for Cp ∩ Alg L is related to the behaviour of the net {UF}F. Thus we explore the theory of integration with respect to an arbitrary CSL and we obtain results which contrast the corresponding theory for nests.