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The real rank zero property of crossed product. (English) Zbl 1116.46057

Let (A,G,α) 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 α G to the space T(A) α * of α-invariant trace states on A. The following results are proven:

(1) If G ^ is connected and A α G has real rank zero, then R is an affine homeomorphism with inverse Φ given by Φ(τ)=τφ, τT(A) α * , where φ denotes the canonical conditional expectation from A α G onto A.

(2) If R is a homeomorphism, then A α 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)U 0 (A α G) can be approximated in A α 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:
46L55Noncommutative dynamical systems
46L05General theory of C * -algebras
46L35Classifications of C * -algebras
46L40Automorphisms of C * -algebras