The Af Structure of Non Commutative Toroidal Z/4z Orbifolds

For any irrational θ and rational number p/q such that q|qθ−p| < 1, a projection e of trace q|qθ−p| is constructed in the the irrational rotation algebra Aθ that is invariant under the Fourier transform. (The latter is the order four automorphism U 7→ V 7→ U, where U, V are the canonical unitaries generating Aθ .) Further, the projection e is approximately central, the cut down algebra eAθe contains a Fourier invariant q × q matrix algebra whose unit is e, and the cut downs eUe, eV e are approximately inside the matrix algebra. (In particular, there are Fourier invariant projections of trace k|qθ−p| for k = 1, . . . , q.) It is also shown that for all θ the crossed product Aθ ⋊Z4 satisfies the Universal Coefficient Theorem. (Z4 := Z/4Z.) As a consequence, using the Classification Theorem of G. Elliott and G. Gong for AH-algebras, a theorem of M. Rieffel, and by recent results of H. Lin, we show that Aθ ⋊Z4 is an AF-algebra for all irrational θ in a dense Gδ . 1991 Mathematics Subject Classification. 46L80, 46L40, 46L35.