Theorem 11.1 ([Uniform property Gamma]). Let A be a simple, separable, unital, infinite-dimensional nuclear C^* -algebra with nonempty tracial-state space and strict comparison as in Definition 15 ([Strict comparison]). For every integer …

For m=1 , take p_1=1 . Suppose m≥2 , and choose the Gaussian quantiles -∞=q_0<q_1<⋯<q_ m-1 <q_m=+∞ so that the consecutive intervals I_i between q_ i-1 and q_i each have γ -measure 1/m . The assignment of the finite endpoints is immaterial because γ has no atoms. Fix a finite set H of positive contractions in A and a tolerance ε>0 . Choose θ>0 so small that 2θ<ε, (17θ)/(16)<ε^2. Choose pairwise disjoint open intervals about the finitely many quantiles q_1,…,q_ m-1 whose union U has γ(U)<θ . Interpolating linearly across these intervals gives continuous functions f_1,…,f_m:→[0,1] with ∑_ i=1 ^m f_i=1, f_i=1_ I_i on U. At a transition around q_i , only f_i and f_ i+1 vary, with one decreasing from one to zero and the other its complement. In particular, |∫f_i\,dγ-1m|<θ, ∫(f_i-f_i^2)^2\,dγ<(θ)/(16). The second estimate uses (u-u^2)^2≤1/16 for 0≤u≤1 , and the integrand vanishes outside U . Keep these functions fixed. Apply Lemma 32 ([Uniform weak convergence with a fixed coefficient]) to every pair (a,f_i) with a∈H , and to (1,(f_i-f_i^2)^2) for each i . For all sufficiently large N , all of these finitely many convergence errors are less than θ . Set v_i=f_i(s_N) . These are positive contractions in D∩A' and satisfy ∑_i v_i=1 . Equation gives _ τ∈Λ |τ(av_i)-τ(a)/m| , <2θ<ε (a∈H),; v_i-v_i^2_ 2,Λ ^2 , =_ τ∈Λ τ((f_i-f_i^2)^2(s_N)) <(17θ)/(16)<ε^2. Thus arbitrarily accurate finite sets of trace-splitting and projection tests can be met by m positive contractions that already commute with A and sum to one. Choose a countable norm-dense family (a_k)_ k≥1 in the positive unit ball of A . Apply Lemma 17 ([Tracial diagonal principle]) to the m variables constrained to be positive contractions, with the countable tests p_i^2-p_i_ 2,Λ , =0 , , (1≤i≤m),; * ∑_ i=1 ^m p_i-1 _ 2,Λ , =0,; [p_i,a_k]_ 2,Λ , =0 , , (1≤i≤m, k≥1),; _ τ∈Λ |τ(a_kp_i-a_k/m)|, =0 , , (1≤i≤m, k≥1). The preceding finite construction verifies the hypotheses of that lemma. In this application the diagonal variables are only the m positive contractions v_i ; the elements s_N are not diagonal variables. In particular all required lifts lie in fixed norm balls. The sum-to-one equation is included among the tests, regardless of which positive contractive lifts of a trial tuple are chosen. The resulting positive contractions p_i are projections, commute with A , and sum to one, because the uniform tracial 2 -norm detects zero in D and commutation extends by norm density and linearity. They are pairwise orthogonal: for each i , 0=p_i(1-p_i)p_i =∑_ j≠i p_ip_jp_i =∑_ j≠i (p_jp_i)^*(p_jp_i), and positivity forces p_jp_i=0 for j≠i . The trace equations extend from the dense family to every positive contraction, uniformly over Λ , and then by scaling and linearity to every element of A . This proves Equation, which is uniform property Γ as in Definition 18 ([Uniform property Γ ]).