On $C$-compact orthogonally additive operators

We consider $C$-compact orthogonally additive operators in vector lattices. After providing some examples of $C$-compact orthogonally additive operators on a vector lattice with values in a Banach space we show that the set of those operators is a projection band in the Dedekind complete vector lattice of all regular orthogonally additive operators. In second part of the article we introduce a new class of vector lattices, called $C$-complete, and show that any laterally-to-norm continuous $C$-compact orthogonally additive operator from a $C$-complete vector lattice to a Banach space is narrow, which generalizes a result of Pliev and Popov.