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An AF Algebra Associated with the Farey Tessellation
Published online by Cambridge University Press: 20 November 2018
Abstract
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We associate with the Farey tessellation of the upper half-plane an $\text{AF}$ algebra $\mathfrak{A}$ encoding the “cutting sequences” that define vertical geodesics. The Effros–Shen $\text{AF}$ algebras arise as quotients of $\mathfrak{A}$. Using the path algebra model for $\text{AF}$ algebras we construct, for each $\tau \,\,\in \,\,\left( 0 \right.,\left. \frac{1}{4} \right]$, projections $({{E}_{n}})$ in $\mathfrak{A}$ such that ${{E}_{n}}{{E}_{n\pm 1}}E\le \tau {{E}_{n}}$.
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