Approximately Finite Dimensional C*-algebra

Approximately Finite Dimensional C*-algebra

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 Bratteli. 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 nuclear 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 non-commutative generalizations of C0(X), where X is a totally disconnected topological space.

Read more about Approximately Finite Dimensional C*-algebra:  Examples, Elliott's Classification Program, Von Neumann Algebras

Famous quotes containing the words finite and/or dimensional:

    Sisters define their rivalry in terms of competition for the gold cup of parental love. It is never perceived as a cup which runneth over, rather a finite vessel from which the more one sister drinks, the less is left for the others.
    Elizabeth Fishel (20th century)

    I don’t see black people as victims even though we are exploited. Victims are flat, one- dimensional characters, someone rolled over by a steamroller so you have a cardboard person. We are far more resilient and more rounded than that. I will go on showing there’s more to us than our being victimized. Victims are dead.
    Kristin Hunter (b. 1931)