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  "body": "\n## Summary\n\nAn approximately finite-dimensional (AF) C\\*-algebra is a C*-algebra that is the closure of an increasing sequence of finite-dimensional C\\*-subalgebras.\nFinite-dimensional C\\*-algebras are finite direct sums of matrix algebras, so an AF algebra is built from matrices by a limit process.\nBratteli described AF algebras by diagrams, and Elliott classified them by their ordered K_0 groups.\nExamples are the compact operators, the UHF algebras such as the CAR algebra, and $C(X)$ for the Cantor set $X$.\n\n## Explanations\n\n### L1 human\n\nStart with a matrix algebra, put it inside a bigger matrix algebra, then a bigger one, and so on, and take the closure of the union.\nThe resulting algebra is usually infinite-dimensional, but every element is a limit of matrices of finite size.\nFor example, $M_2 \\subset M_4 \\subset M_8 \\subset \\cdots$, with each matrix $a$ sent to the block matrix $\\operatorname{diag}(a, a)$, gives the CAR algebra used for fermions in physics.\n",
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