Property of the spatial projectivity in the class of CSL-algebras with atomic commutant

This work continues to study spatial homological properties of, generally speaking, non-selfadjoint, reflexive operator algebras in a Hilbert space $H$. A “lattice” criterion of spatial projectivity of an algebra $A$ (i.e. the projectivity of $H$ as left Banach $A$-module) is obtained in the class of indecomposable CSL-algebras: the existence of immediate predesessor of $H$ as element of the lattice of invariant subspaces. Also, the direct product of indecomposable CSL-algebras $A_\alpha$, $\alpha\in\Lambda$, is a spatial projective algebra iff the algebra $A_\alpha$ is spatial projective for every $\alpha$.