Extended rotation algebras: Adjoining spectral projections to rotation algebras

Abstract

Denote by Aθ the rotation algebra corresponding to the rotation 2πθ. The C*-algebra 𝔹θ generated by Aθ together with certain spectral projections of the canonical unitary generators is studied. The C*-algebra 𝔹θ is shown to have a unique tracial state and to be nuclear provided that θ is irrational. Moreover, we study the ideal structure of the C*-algebra 𝔹θ. In particular, it is shown that 𝔹θ is simple if neither the commutative sub-C*-algebra generated by the spectral projections of u in question (assumed to be a set invariant under Ad v) nor the corresponding commutative sub-C*-algebra associated to v contains non-zero minimal projections. In the second part of the paper, we study the extended rotation algebra 𝔹θ generated by the spectral projections (one for each unitary) corresponding to the half-open interval from 0 to θ. (The spectral projections for each half-open interval from nθ to (n + 1)θ are then included for each integer n.) Using simplicity of 𝔹θ for θ irrational, the natural field of C*-algebras on the unit circle with fibres 𝔹θ is shown to be continuous at irrational points. This field is lower semicontinuous on the whole circle. Much more useful is an upper semicontinuous field which is obtained by desingularizing this field at rational points on the circle. The fibres of the desingularized field at rational points are certain (computable) type I C*-algebras. Using this new field, we are able to show that 𝔹θ is an AF algebra with K0(𝔹θ) ≅ ℤ + θℤ for generic θ, in the sense of Baire category, with the class of the unit being 1 ∈ ℤ.