Extended rotation algebras: Adjoining spectral projections to rotation algebras

Denote by A(theta) the rotation algebra corresponding to the rotation 2 pi theta. The C*-algebra B-theta generated by A(theta) together with certain spectral projections of the canonical unitary generators is studied. The C*-algebra B-theta is shown to have a unique tracial state and to be nuclear provided that theta is irrational. Moreover, we study the ideal structure of the C*-algebra B-theta. In particular, it is shown that B-theta 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 nu) nor the corresponding commutative sub-C*-algebra associated to nu contains non-zero minimal projections. In the second part of the paper, we study the extended rotation algebra B-theta generated by the spectral projections (one for each unitary) corresponding to the half-open interval from 0 to theta. (The spectral projections for each half-open interval from n theta to (n + 1)theta are then included for each integer n.) Using simplicity of B-theta for theta irrational, the natural field of C*-algebras on the unit circle with fibres B-theta 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 B-theta is an AF algebra with K-0(B-theta) congruent to Z + theta Z for generic theta, in the sense of Baire category, with the class of the unit being 1 is an element of Z.