Theorem 15.1. For every n≥2 there are orthogonal projections f₁,…,f_n∈ D∩A' summing to 1_D such that τ(f_i a)=1nτ(a) (a∈A, τ∈Λ). In particular, A has uniform property Γ .

Proposition 42. For every separable unital E⊂D and 0<c<1 , there is a projection f∈D∩E' such that τ(fx)=cτ(x) (x∈E, τ∈Λ).

Lemma 41. For every separable unital E⊂D and 0<λ<1 , there is a positive contraction e∈D∩E' such that, for τ∈Λ , |τ(ex)-λτ(x)|≤λ³/2 (x∈E_sa , \|x\|≤1), τ(e-e²)≤λ³/2 .

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Start with the separable unital algebra C^*(A,1_D) . At step j=1,…,n-1 , adjoin the previously selected projections, apply Proposition 42 with c=1/(n-j+1) , and multiply its projection by the remaining complement q . The factors commute, so this gives a projection f_j≤q . Apply to qa in the enlarged algebra. Inductively τ(qa)=(n-j+1)τ(a)/n , so τ(f_ja)=τ(a)/n . Take the final complement for f_n . These are exactly the projection identities in the definition of uniform property Γ .

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Theorem 15.1. For every n≥2 there are orthogonal projections f1,…,fn∈ D∩A' summing to 1D such that τ(fi a)=1nτ(a) (a∈A, τ∈Λ). In particular, A has uniform property Γ .

Proposition 42. For every separable unital E⊂D and 0<c<1 , there is a projection f∈D∩E' such that τ(fx)=cτ(x) (x∈E, τ∈Λ).

Lemma 41. For every separable unital E⊂D and 0<λ<1 , there is a positive contraction e∈D∩E' such that, for τ∈Λ , |τ(ex)-λτ(x)|≤λ3/2 (x∈Esa , \|x\|≤1), τ(e-e2)≤λ3/2 .

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Theorem 15.1. For every n≥2 there are orthogonal projections f₁,…,f_n∈ D∩A' summing to 1_D such that τ(f_i a)=1nτ(a) (a∈A, τ∈Λ). In particular, A has uniform property Γ .

Lemma 41. For every separable unital E⊂D and 0<λ<1 , there is a positive contraction e∈D∩E' such that, for τ∈Λ , |τ(ex)-λτ(x)|≤λ³/2 (x∈E_sa , \|x\|≤1), τ(e-e²)≤λ³/2 .