Abstract view
An AF Algebra Associated with the Farey Tessellation


Published:20081001
Printed: Oct 2008
Abstract
We associate with the Farey tessellation of the upper
halfplane an
AF algebra $\AA$ encoding the ``cutting sequences'' that define
vertical geodesics.
The EffrosShen 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, 11A55, 11B57, 46L55, 37E05, 82B20 show english descriptions
General theory of $C^*$algebras Continued fractions {For approximation results, see 11J70} [See also 11K50, 30B70, 40A15] Farey sequences; the sequences ${1^k, 2^k, \cdots}$ Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20] Maps of the interval (piecewise continuous, continuous, smooth) Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
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
