Approximately finite-dimensional C*-algebra

Remarks

Every finite-dimensional C*-algebra is isomorphic to a direct sum $M_{n_1}(\C) \oplus \cdots \oplus M_{n_k}(\C)$ of matrix rings. An AF algebra in this sense is separable. Some authors call a nonseparable C*-algebra AF when every finite subset can be approximated by elements of a finite-dimensional C*-subalgebra.

Example: the compact operators $K(H)$ on a separable infinite-dimensional Hilbert space $H$ with orthonormal basis $(e_k)$. Let $A_n$ be the operators that map $\operatorname{span}(e_1, \ldots, e_n)$ into itself and vanish on its orthogonal complement; then $A_n \cong M_n(\C)$ and $A_n \subseteq A_{n+1}$. The union is dense because the finite-rank operators are dense in $K(H)$, and $P_n T P_n \to T$ for compact $T$, where $P_n$ is the projection onto the span.

Non-example: $C([0, 1])$ is not AF. A finite-dimensional commutative C*-algebra is spanned by projections, and the only projections in $C([0,1])$ are $0$ and $1$, since $[0,1]$ is connected. So every finite-dimensional C*-subalgebra lies in the constants $\C 1$, which are not dense.