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# An AF Algebra Associated with the Farey Tessellation

Published:2008-10-01
Printed: Oct 2008
• Florin P. Boca
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## Abstract

We associate with the Farey tessellation of the upper half-plane an AF algebra $\AA$ encoding the cutting sequences'' that define vertical geodesics. The Effros--Shen AF algebras arise as quotients of $\AA$. Using the path algebra model for AF algebras we construct, for each $\tau \in \big(0,\frac{1}{4}\big]$, projections $(E_n)$ in $\AA$ such that $E_n E_{n\pm 1}E_n \leq \tau E_n$.
 MSC Classifications: 46L05 - General theory of $C^*$-algebras 11A55 - Continued fractions {For approximation results, see 11J70} [See also 11K50, 30B70, 40A15] 11B57 - Farey sequences; the sequences ${1^k, 2^k, \cdots}$ 46L55 - Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20] 37E05 - Maps of the interval (piecewise continuous, continuous, smooth) 82B20 - Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs