Let $C(K)$ be the algebra of continuous functions on Cantor set. Is it possible to prove that $C(K)$ forms an AF-algebra without Bratteli diagram?
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Sure. Regard $K$ as $\{0,1\}^N$ and let $E_n$ for $n\in N$ be the functions in $C(K)$ that depend only on the first $n$ components.
answered Jan 7, 2021 at 18:09
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1$\begingroup$ The $E_n$ are an increasing sequence of subalgefbras with dense union, and for each $n$, $E_n$ is algebra isometric to $\ell_\infty^{2^n}$. $\endgroup$ Jan 9, 2021 at 5:23
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$\begingroup$ It is completely clear now. Thanks. $\endgroup$ Jan 9, 2021 at 6:20