7 research outputs found
The AF structure of non commutative toroidal Z/4Z orbifolds
For any irrational theta and rational number p/q such that q|qtheta-p|<1, a
projection e of trace q|qtheta-p| is constructed in the the irrational rotation
algebra A_theta that is invariant under the Fourier transform. (The latter is
the order four automorphism U mapped to V, V mapped to U^{-1}, where U, V are
the canonical unitaries generating A_theta.) Further, the projection e is
approximately central, the cut down algebra eA_theta e contains a Fourier
invariant q x q matrix algebra whose unit is e, and the cut downs eUe, eVe are
approximately inside the matrix algebra. (In particular, there are Fourier
invariant projections of trace k|qtheta-p| for k=1,...,q.) It is also shown
that for all theta the crossed product A_theta rtimes Z_4 satisfies the
Universal Coefficient Theorem. (Z_4 := 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 A_theta rtimes Z_4 is
an AF-algebra for all irrational theta in a dense G_delta.Comment: 35 page