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 sequence in II1
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.Comment: IMRN, to appear. Accepted version; 39 page
