Proposition 11 ([Weighted matrix blocks]). Let K⊆(A) be finite and 0<η<1 . There are contractions v₁,…,v_m∈A and nonnegative numbers λ₁,…,λ_m summing to one, chosen using only K,η , such that the map Φ:D⟶D, Φ(b)=∑_l=1^mλ_l v_lbvl^*, …

Put μ=η^2 and choose Φ by Lemma 9 ([Averaging with trace control]). In particular, the coefficient list is fixed before r . Choose 0<σ<1 , then β>0 , so that β<σ/2, m(σ+2β)<μ, (2β)/(σ) +((2+√2)β)/(σ^2)<η. This is possible by choosing σ with mσ<μ first. Scalar matrix approximation. Partition r by Lemma 2 ([Scalar compressions]), using the finite list v_l^*v_n, v_l^*xv_n (1≤l,n≤m, x∈K). Take the compression tolerance smaller than β/(2m) . For a nonzero partition piece q , set V=[√λ_l\,v_lq]_ l=1 ^m, h_q=VV^*=Φ(q), M=V^*V, N_x=V^*xV. These are contractions in their respective matrix spaces. Apply any state of qDq entrywise to M,N_x , obtaining scalar matrices G,H_x . Positivity and contractivity of a state at matrix levels give 0≤G≤1 and H_x≤1 . The compression tolerance implies M-G⊗q<β, N_x-H_x⊗q<β. Indeed, each unweighted entry differs from its scalar state value by less than twice the compression tolerance, and an m -by- m matrix of entries of norm at most c has norm at most mc . Construction of the blocks. Rotate the columns of V by a scalar unitary diagonalizing G . Retain the columns corresponding to eigenvalues at least σ , forming a row W with k columns and scalar diagonal matrix G_+ . The cutoff gives the uniform lower bound needed to normalize the retained row by an inverse square root. If k=0 , set b_q=0 . Otherwise put X=W^*W≥(σ/2)(1_k⊗q), U=WX^ -1/2 , b_q=U(G_+⊗q)U^*. Then U^*U=1_k⊗q , so b_q is a finite-spectrum positive contraction. Writing W_- for the row of deleted columns, we have W_-^*W_-≤σ+β , and therefore τ(W_-W_-^*)=(⊗τ)(W_-^*W_-) ≤m(σ+β)τ(q). The contribution is zero if no columns are deleted. Also b_q-WW^*=U(G_+⊗q-X)U^* has norm at most β , supported on a projection of trace kτ(q) . Consequently, including the case k=0 , b_q-h_q τ ≤m(σ+2β)τ(q)≤μτ(q). We verify explicitly that the compression estimate is independent of the corner and its matrix size. For positive invertible X,Y≥a1 , the resolvent formula for inverse square roots gives X^ -1/2 -Y^ -1/2 -Y π ∫_0^∞ t^ -1/2 (a+t)^2 \,dt =X-Y 2a^ 3/2 . Use Y=G_+⊗q and a=σ/2 . If H⊗q denotes the corresponding retained corner of H_x⊗q , then H≤1 , and gives W^*xW-H⊗q<β . Expanding the difference between U^*xU=X^ -1/2 (W^*xW)X^ -1/2 Y^ -1/2 (H⊗q)Y^ -1/2 and using bounds its norm by (2β)/(σ) +((2+√2)β)/(σ^2)<η. The second expression is a scalar matrix over q . The tolerances σ,β were chosen before the partition of r , so the compression bound is uniform over its pieces. Write b_q as the finite sum of its nonzero upper spectral projections, weighted by the lengths of their level intervals: b_q=∫_0^1 1_ (a,∞) (b_q)\,da =∑_ j∈J_q w_jp_j. Each p_j is the image of a scalar coordinate projection under U . It therefore has a subrow U_j satisfying , with q_j=q . Taking matrix corners gives. Summing over the partition gives . Averaged commutators. For a finite-spectrum positive contraction b=∑_s s e_s , include its kernel projection in this spectral resolution. Tracial orthogonality gives, for unitary x , ∫_0^1 [x,1_ (a,∞) (b)] τ ^2\,da , =∑_ s,t |s-t|e_sxe_t τ ^2; , ≤2√τ(b)\,[x,√b] τ ; , ≤2√ τ(b)b-xbx^* τ . For the middle inequality, factor |s-t|=|√s-√t|(√s+√t) and apply Cauchy--Schwarz to the sum. The second squared sum is at most 2∑_ s,t (s+t)e_sxe_t τ ^2=4τ(b) . The last inequality is Lemma 10 ([A square-root trace estimate]). For b=b_q , equations, , and give τ(b_q)≤(1+μ)τ(q), b_q-xb_qx^* τ ≤3μτ(q). Thus the contribution of J_q in is at most 2√3(1+μ)\,ητ(q)≤6ητ(q) . Summing over q proves the assertion.