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Let $A = \overline{\bigcup A_n}$, $B = \overline{\bigcup B_n}$ be AF algebras with the same Bratelli diagram. Then there is an isomorphism $\phi : A \rightarrow B$.

Let $x \in \bigcup A_n$ (say, $x \in A_k$ for some $k$). Is it always the case that $\phi(x) \in \bigcup B_n$ ?

vagnard
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1 Answers1

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If you look at the proof of Proposition III.2.7 in Davidson's C$^*$-algebras by Example, the isomorphism is constructed first between $\bigcup A_n$ and $\bigcup B_n$, and only then extended to the closures (using that it is isometric).

Martin Argerami
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