The Strong Property (B) for $$L_p$$ Spaces

Abstract

Given a purely non-atomic, finite measure space $$(\Omega ,\Sigma ,\nu )$$ ( Ω , Σ , ν ) , it is proved that for every closed, infinite-dimensional subspace V of $$L_p(\nu )$$ L p ( ν ) ($$1\le p<\infty $$ 1 ≤ p < ∞ ) there exists a decomposition $$L_p(\nu )=X_1\oplus X_2$$ L p ( ν ) = X 1 ⊕ X 2 , such that both subspaces $$X_1$$ X 1 and $$X_2$$ X 2 are isomorphic to $$L_p(\nu )$$ L p ( ν ) and both $$V\cap X_1$$ V ∩ X 1 and $$V\cap X_2$$ V ∩ X 2 are infinite-dimensional. Some consequences concerning dense, non-closed range operators on $$L_1$$ L 1 are derived.