Lemma 32. For a separably acting properly infinite von Neumann algebra P⊂(H) and h=h^*∈(H) , (h,P')≤4_h|_P.

Put d=_h|_P . Choose a unital copy S of (ℓ^2) in P , with matrix units (e_ ij ) whose diagonal sum is I . The finite-dimensional unital algebras S_n=span e_ ij :1≤i,j≤n +(I-∑_ i=1 ^ne_ ii ) increase and generate S . Average h over (S_n) and take a weak-operator cluster point h_0 . Each average is self-adjoint, within d of h , and has scalar commutator bound at most d on P . These properties pass to the cluster point. Haar invariance at every earlier stage gives h_0∈S' . For any k , choose isometries s_1,…,s_k∈S with s_i^*s_j=δ_ ij I and ∑_i s_is_i^*=I . The unitary V:H^k→H , V(ξ_i)=∑_i s_iξ_i , identifies M_k(P) with P and h_0^ (k) with h_0 . Consequently _ h_0 |_P _ ≤d. By Lemma 1, (h_0,P')≤d , so in fact (h,P')≤2d . The stated larger constant will be convenient.