We solve the isomorphism problem for essential unital C^*-algebra extensions of the form 0 → K ⊕ K → E π Mₙ ⊗ C(T) → 0. We then relate these to analogs of the Effros Shen AF algebras for rational numbers. This involves C^*-algebras constructed from categories of paths built from certain nonsimple continued fraction expansions.
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Jack Spielberg (2024) studied this question.
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