Lateral order on complex vector lattices and narrow operators

In this paper, we continue investigation of the lateral order on vector lattices started in [25]. We consider the complexification (Formula presented.) of a real vector lattice E and introduce the lateral order on (Formula presented.). Our first main result asserts that the set of all fragments (Formula presented.) of an element (Formula presented.) of the complexification of an uniformly complete vector lattice E is a Boolean algebra. Then, we study narrow operators defined on the complexification (Formula presented.) of a vector lattice E, extending the results of articles [22, 27, 28] to the setting of operators defined on complex vector lattices. We prove that every order-to-norm continuous linear operator (Formula presented.) from the complexification (Formula presented.) of an atomless Dedekind complete vector lattice E to a finite-dimensional Banach space X is strictly narrow. Then, we prove that every C-compact order-to-norm continuous linear operator (Formula presented.) from (Formula presented.) to a Banach space X is narrow. We also show that every regular order-no-norm continuous linear operator from (Formula presented.) to a complex Banach lattice (Formula presented.) is narrow. Finally, in the last part of the paper we investigate narrow operators taking values in symmetric ideals of compact operators.