Approximately finite-dimensional $C^*$-algebras with projective Hilbert modules, their Bratteli diagrams, and $K_0$-groups

This paper is devoted to the homological classification of approximately finite-dimensional $C^*$-algebras. The algebras in this class for which there exists at least one non-trivial Hilbert module and those for which there exists at least one faithful Hilbert module are described. The description is given in terms of the Bratteli diagrams of the algebras in question and in terms of the ordered $K_0$-groups of these algebras.