Schafhauser-Tikuisis-White Problem XXI: tracial real rank zero implies uniform property Gamma

Theorem 1.1: for $A$ unital, separable, simple, infinite-dimensional, nuclear and stably finite with $T(A)\neq\emptyset$, if $B^\omega$ has real rank zero then there is a projection $p\in B^\omega\cap B'$ with $\lambda(px)=\lambda(x)/2$ for all $x\in B$ and all limit traces $\lambda$, so $A$ has uniform property $\Gamma$; it also identifies all traces on $B$. It does not show that real rank zero of $B^\omega$ holds in general, and Toms (2026) has shown nuclearity alone does not give uniform property $\Gamma$.

Theorem 1.1: for $A$ unital, separable, simple, infinite-dimensional, nuclear and stably finite with $T(A)\neq\emptyset$, if $B^\omega$ has real rank zero then there is a projection $p\in B^\omega\cap B'$ with $\lambda(px)=\lambda(x)/2$ for all $x\in B$ and all limit traces $\lambda$, so $A$ has uniform property $\Gamma$; it also identifies all traces on $B$. It does not show that real rank zero of $B^\omega$ holds in general, and Toms (2026) has shown nuclearity alone does not give uniform property $\Gamma$.