# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
The real rank zero property of crossed product. (English) Zbl 1116.46057

Let $\left(A,G,\alpha \right)$ 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 $A{⋊}_{\alpha }G$ to the space $T{\left(A\right)}_{{\alpha }^{*}}$ of $\alpha$-invariant trace states on $A$. The following results are proven:

(1) If $\stackrel{^}{G}$ is connected and $A{⋊}_{\alpha }G$ 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 $A{⋊}_{\alpha }G$ onto $A$.

(2) If $R$ is a homeomorphism, then $A{⋊}_{\alpha }G$ has real rank zero iff all products between unitaries in ${U}_{0}\left(A\right)$ and in the commutator subgroup generated by ${C}_{c}\left(G,A\right)\cap {U}_{0}\left(A{⋊}_{\alpha }G\right)$ can be approximated in $A{⋊}_{\alpha }G$ 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 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)] .

##### MSC:
 46L55 Noncommutative dynamical systems 46L05 General theory of ${C}^{*}$-algebras 46L35 Classifications of ${C}^{*}$-algebras 46L40 Automorphisms of ${C}^{*}$-algebras
##### Keywords:
trace state space; crossed product; real rank zero