Let be a -dynamical system with a unital -algebra and a discrete Abelian group. The paper under review studies the canonical continuous affine restriction map from the trace state space of the crossed product to the space of -invariant trace states on . The following results are proven:
(1) If is connected and has real rank zero, then is an affine homeomorphism with inverse given by , , where denotes the canonical conditional expectation from onto .
(2) If is a homeomorphism, then has real rank zero iff all products between unitaries in and in the commutator subgroup generated by can be approximated in by unitaries with finite spectrum. This result is further strengthened when is an inductive limit of “non-elementary” simple -algebras of real rank zero.
Important technical tools in the proofs are provided by the determinant associated to a trace by P. de la Harpe and G. Skandalis [Ann. Inst. Fourier 34, No. 1, 241–260 (1984; Zbl 0521.46037)] and by a theorem of K. Thomsen [Publ. Res. Inst. Math. Sci. 31, No. 6, 1011–1029 (1995; Zbl 0853.46037)] .