*(English)*Zbl 1116.46057

Let $(A,G,\alpha )$ be a ${C}^{*}$-dynamical system with $A$ a unital ${C}^{*}$-algebra and $G$ a discrete Abelian group. The paper under review studies the canonical continuous affine restriction map $R$ from the trace state space of the crossed product $A{\u22ca}_{\alpha}G$ to the space $T{\left(A\right)}_{{\alpha}^{*}}$ of $\alpha $-invariant trace states on $A$. The following results are proven:

(1) If $\widehat{G}$ is connected and $A{\u22ca}_{\alpha}G$ has real rank zero, then $R$ is an affine homeomorphism with inverse ${\Phi}$ given by ${\Phi}\left(\tau \right)=\tau \circ \phi $, $\tau \in T{\left(A\right)}_{{\alpha}^{*}}$, where $\phi $ denotes the canonical conditional expectation from $A{\u22ca}_{\alpha}G$ onto $A$.

(2) If $R$ is a homeomorphism, then $A{\u22ca}_{\alpha}G$ has real rank zero iff all products between unitaries in ${U}_{0}\left(A\right)$ and in the commutator subgroup generated by ${C}_{c}(G,A)\cap {U}_{0}\left(A{\u22ca}_{\alpha}G\right)$ can be approximated in $A{\u22ca}_{\alpha}G$ by unitaries with finite spectrum. This result is further strengthened when $A$ is an inductive limit of “non-elementary” simple ${C}^{*}$-algebras of real rank zero.

Important technical tools in the proofs are provided by the determinant associated to a trace by *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)] .