Lemma 19 ([A lower rank bound preserved by averaging]). Fix lower data y,g,η as in --. If a finite rational average satisfies \|∑_αλα yv_α^*g(H_α(s))v_αy-y²\| <²/2, then for every extended trace on Q , d_τ((y-)₊) ≤2\|y\|²²\, …

Write the displayed average as T and let p=1_ (,∞) (y) in the bidual. The norm estimate gives pTp≥(^2/2)p . Since the v_α are contractions and 0≤g(t)≤1_ (η,∞) (t) , positivity and traciality give (^2)/(2)τ(p) ≤\|y\|^2∑_αλ_α d_τ((H_α(s)-η)_+). The sum is exactly the rank on the right of , by the normalized averaging convention. Here is a justification when τ is not densely finite. If the right side is infinite, there is nothing to prove. If it is finite, discard zero coefficients. Every support of g(H_α(s)) then has finite rank. The norm estimate gives (y^2-^2/2)_+ T , and T is Cuntz below the finite direct sum of the g(H_α(s)) , by its finite row factorization. Since p is the support of (y^2-^2)_+ , it has finite rank and belongs to the normal trace's finite part. One may initially use a slightly smaller spectral projection inside p . The displayed computation is now a computation for the induced normal trace on this finite part. Let the additional cut slack decrease to zero. This gives the inequality for p by normal monotone convergence and proves the assertion for every extended weight.