Inductive limit automorphisms of the irrational rotation algebra

Abstract

It is shown that the flip automorphismU↦U ^*,V↦V ^* of the irrational rotation algebra A_θ is an inductive limit automorphism. Here, the algebra A_θ is generated by unitariesU, V satisfyingVU=e^2πiθ UV, where θ is an irrational number. Recently, Elliott and Evans proved that A_θ can be approximated by unital subalgebras isomorphic to a direct sum of two matrix algebras over$C(\mathbb{T})$, the algebra of continuous functions on the unit circle. This is the central result which they used to obtain their structure theorem on A_θ; namely, that A_θ is the inductive limit of an increasing sequence of subalgebras each isomorphic to a direct sum of two matrix algebras over$C(\mathbb{T})$. In their proof, they devised a subtle construction of two complementary towers of projections. In the present paper it is shown that the two towers can be chosen so that each summand of their approximating basic building blocks is invariant under the flip automorphism and, in particular, that the unit projection of the first summand is unitarily equivalent to the complement of the unit of the second by a unitary which is fixed under the flip. Also, an explicit computation of the flip on the approximating basic building blocks of A_θ is given. Further, combining this result along with others, including a theorem of Su and a spectral argument of Bratteli, Evans, and Kishimoto, a two-tower proof is obtained of the fact established by Bratteli and Kishimoto that the fixed point subalgebra B^θ (under the flip) is approximately finite dimensional. Also used here is the fact that B^θ has the cancellation property and is gifted with four basic unbounded trace functionals. The question is raised whether other finite order automorphisms of A_θ (arising from a matrix in SL(2,ℤ)) are inductive limit automorphisms - or evenalmost inductive limit automorphisms in the sense of Voiculescu.