Consider an arbitary positive semidefinite operator ρ, acting on ℂ^{A} ⊗ ℂ^{B} ⊗ ℂ^{C}, for A,B,C finite. Also, let P be an orthogonal projector on ℂ^{B} ⊗ ℂ^{C} . For the sake of concision, I will write R = 1_{ℂA} ⊗ P ; this of course is also an orthogonal projector. Consider the completely positive transformation

M(ρ) = (1 − R) ρ (1 − R) + R ρ R .

As R is an orthogonal projector, it is easy to show that || M(ρ) ||_{2} ≤ || ρ ||_{2} . This is because we may represent ρ as matrix in a basis consisting of the eigenvectors of R; if we divide ρ into block according to rows/columns representing vectors in the image or the kernel of R, the effect of the map M is to set the non-diagonal blocks to zero.

I am interested in how the map M may similarly affect the operator 2 norm of reduced operators on ℂ^{A} ⊗ ℂ^{B}. So I would like to know:

Is it also true that || tr_{C}( M(ρ) ) ||_{2} ≤ || tr_{C}(ρ) ||_{2} — where tr_{C} is the trace operator acting on ℂ^{C}, taken in tensor product with 1_{ℂA} ⊗ 1_{ℂB} ?