  We introduce a new class of operator algebras -- tracially complete
C*-algebras -- as a vehicle for transferring ideas and results between
C*-algebras and their tracial von Neumann algebra completions. We obtain
structure and classification results for amenable tracially complete
C*-algebras satisfying an appropriate version of Murray and von Neumann's
property gamma for II_1 factors. In a precise sense, these results fit between
Connes' celebrated theorems for injective II_1 factors and the unital
classification theorem for separable simple nuclear C*-algebras. The theory
also underpins arguments for the known parts of the Toms-Winter conjecture.
