Atomicity Thresholds for Normal Channel Compatibility

We classify closure properties of normal channel compatibility on spatial von Neumann tensor products. Among normal channel pairs, compatibility is point–ultraweakly closed exactly when one source is atomic or the output is atomic abelian. Its operator- and completely bounded-norm closures each equal the point–ultraweak closure exactly when one source is atomic or the output is a product of finite-dimensional matrix algebras. Explicit models distinguish the two thresholds. For a σ-finite diffuse algebra in standard representation, the complementary inclusion pair has distance at least 1/2 from normally compatible pairs in both norms. If the algebra is also an injective factor, the pair nevertheless lies in the point–ultraweak closure.