Completely positive maps in the tensor products of von Neumann algebras

Let Mᵢ and Nᵢ be von Neumann algebras and τᵢ be completely positive maps from Mᵢ to Nᵢ(i=1,2) . Then there exists a completely positive map τ₁⊗τ₂ (called the product map of τ₁ and τ₂ ) from the spatial C* -tensor product M₁⊗ₐM₂ to the spatial C* -tensor product N₁⊗αN₂ such that τ₁⊗τ₂(a⊗ b)=τ₁(a)⊗τ₂(b) .