In mathematics, an approximately finite-dimensional (AF) C*-algebra is a C*-algebra that is the inductive limit of a sequence of finite-dimensional C*-algebras. Approximate finite-dimensionality was first defined and described combinatorially by Ola Bratteli. Later, George A. Elliott gave a complete classification of AF algebras using the K0 functor whose range consists of ordered abelian groups with sufficiently nice order structure.
The classification theorem for AF-algebras serves as a prototype for classification results for larger classes of separable simple amenable stably finite C*-algebras. Its proof divides into two parts. The invariant here is K0 with its natural order structure; this is a functor. First, one proves existence: a homomorphism between invariants must lift to a *-homomorphism of algebras. Second, one shows uniqueness: the lift must be unique up to approximate unitary equivalence. Classification then follows from what is known as the intertwining argument. For unital AF algebras, both existence and uniqueness follow from the fact the Murray-von Neumann semigroup of projections in an AF algebra is cancellative.
The counterpart of simple AF C*-algebras in the von Neumann algebra world are the hyperfinite factors, which were classified by Connes and Haagerup.
In the context of noncommutative geometry and topology, AF C*-algebras are noncommutative generalizations of C0(X), where X is a totally disconnected metrizable space.