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The real rank zero property of crossed product. (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 $A\rtimes_\alpha G$ to the space $T(A)_{\alpha^*}$ of $\alpha$-invariant trace states on $A$. The following results are proven: (1) If $\widehat{G}$ is connected and $A\rtimes_\alpha G$ has real rank zero, then $R$ is an affine homeomorphism with inverse $\Phi$ given by $\Phi (\tau)=\tau\circ \phi$, $\tau\in T(A)_{\alpha^*}$, where $\phi$ denotes the canonical conditional expectation from $A\rtimes_\alpha G$ onto $A$. (2) If $R$ is a homeomorphism, then $A\rtimes_\alpha G$ has real rank zero iff all products between unitaries in $U_0(A)$ and in the commutator subgroup generated by $C_c (G,A)\cap U_0(A\rtimes_\alpha G)$ can be approximated in $A\rtimes_\alpha G$ 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 {\sl P. de la Harpe} and {\it G. Skandalis} [Ann. Inst. Fourier 34, No. 1, 241--260 (1984; Zbl 0521.46037)] and by a theorem of {\it K. Thomsen} [Publ. Res. Inst. Math. Sci. 31, No. 6, 1011--1029 (1995; Zbl 0853.46037)] .

46L55Noncommutative dynamical systems
46L05General theory of $C^*$-algebras
46L35Classifications of $C^*$-algebras
46L40Automorphisms of $C^*$-algebras
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