Uniform Property Γ

Abstract

Abstract We further examine the concept of uniform property $\Gamma $ 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 II$_1$ factors, we establish the equivalence of uniform property $\Gamma $, 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 $\Gamma $ 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.