Uniform Property Γ

We further examine the concept of uniform property Γ for C^*-algebras introduced in our joint work with Winter. In addition to obtaining characterisations in the spirit of Dixmier’s work on central sequences in II₁ factors, we establish the equivalence of uniform property Γ, a suitable uniform version of McDuff’s property for C^*-algebras, and the existence of complemented partitions of unity for separable nuclear C^*-algebras with no finite dimensional representations and a compact (non-empty) tracial state space. As a consequence, for C^*-algebras as in the Toms–Winter conjecture, the combination of strict comparison and uniform property Γ is equivalent to Jiang–Su stability. We also show how these ideas can be combined with those of Matui–Sato to streamline Winter’s classification by embeddings technique.