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For any irrational theta and rational number p/q such that q|qtheta-p|<1, aprojection e of trace q|qtheta-p| is constructed in the the irrational rotationalgebra A_theta that is invariant under the Fourier transform. (The latter isthe order four automorphism U mapped to V, V mapped to U^{-1}, where U, V arethe canonical unitaries generating A_theta.) Further, the projection e isapproximately central, the cut down algebra eA_theta e contains a Fourierinvariant q x q matrix algebra whose unit is e, and the cut downs eUe, eVe areapproximately inside the matrix algebra. (In particular, there are Fourierinvariant projections of trace k|qtheta-p| for k=1,...,q.) It is also shownthat for all theta the crossed product A_theta rtimes Z_4 satisfies theUniversal Coefficient Theorem. (Z_4 := Z/4Z.) As a consequence, using theClassification Theorem of G. Elliott and G. Gong for AH-algebras, a theorem ofM. Rieffel, and by recent results of H. Lin, we show that A_theta rtimes Z_4 isan AF-algebra for all irrational theta in a dense G_delta.Comment: 35 page

The AF structure of non commutative toroidal ℤ/4ℤ orbifolds