Approximately finite-dimensional C*-algebra

Summary

An approximately finite-dimensional (AF) C*-algebra is a C*-algebra that is the closure of an increasing sequence of finite-dimensional C*-subalgebras. Finite-dimensional C*-algebras are finite direct sums of matrix algebras, so an AF algebra is built from matrices by a limit process. Bratteli described AF algebras by diagrams, and Elliott classified them by their ordered K_0 groups. Examples are the compact operators, the UHF algebras such as the CAR algebra, and $C(X)$ for the Cantor set $X$.