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

Abstract

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.

35 pages

References in corpus (4)

• Crossed products by finite cyclic group actions with the tracial Rokhlin property
• Moduli spaces of maximally supersymmetric solutions on noncommutative tori and noncommutative orbifolds
• Compactification of M(atrix) theory on noncommutative toroidal orbifolds
• Classification of simple $C^{\ast }$-algebras of tracial topological rank zero

Cited by in corpus (5)

• Crossed products by finite cyclic group actions with the tracial Rokhlin property
• Non-linear sigma-models in noncommutative geometry: fields with values in finite spaces
• K-theory of Rotation Algebra Crossed Products by Amalgamated Products of Finite Cyclic Groups
• Isomorphism and Morita equivalence classes for crossed products of irrational rotation algebras by cyclic subgroups of $S{L}_2(\mathbb{Z})$
• K-Theory of Approximately Central Projections in the Flip Orbifold