# Applications of Elliott's theorem concerning the classification of AF-algebras

An AF-algebra is a $C^*$-algebra which is the inductive limit of an inductive sequence of finite-dimensional $C^*$-algebras.

Elliott's theorem concerning the classification of AF-algebras says that two AF-algebras $\mathfrak{A}$ and $\mathfrak{B}$ are isomorphic as $C^*$-algebras if and only if $(K_0(\mathfrak{A}),K_0(\mathfrak{A})^+,\Gamma(\mathfrak{A}))$ and $(K_0(\mathfrak{B}),K_0(\mathfrak{B})^+,\Gamma(\mathfrak{B}))$ are isomorphic as scaled ordered groups, where $\Gamma(\mathfrak{A})$ denotes the dimension range, i.e. the elements of $K_0(\mathfrak{A})^+$ given as equivalence classes of projections in $\mathfrak{A}$.

Although this result is interesting and beautiful on its own, I would like to know whether there are interesting applications that can be understood by and might be interesting for students who are familiar with basic K-theory for $C^*$-algebras. Of course I'm also interested in more advanced applications or situations where Elliott's theorem provides insights which are hard to obtain otherwise.

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