Crossed products of UHF algebras by some amenable groups (Q1977504)

Let \( C^* \)-algebra \( A \) be UHF and \( \alpha:G\to \text{ Aut}(A) \) a representation of an amenable group \( G \). The principal result is that if \( G=\mathbb Z^n \), there exists a \(*\)-monomorphism \( p:A\rtimes_\alpha\mathbb Z^n\to B\) where \( B \) is AF (i.e., the crossed product is AF embeddable) and \( \rho_*: K_0(A\rtimes_\alpha\mathbb Z^n)\to K_0(B) \) is also injective. Together with an imprimitivity theorem of \textit{P. Green} [see Cor. 2.8. J. Funct. Anal. 36, 88-104 (1980; Zbl 0422.46048)] this entails a generalization. If \( G\in\Gamma \) (resp. \( G\in\Gamma_{fg} \)) then \( A\rtimes_\alpha G \) is quasidiagonal (resp. AF-embeddable). Here \( \Gamma \) (resp. \( \Gamma_{fg} \)) is the class of separable l.c. groups \( G \) which have a finite series of subgroups \( H_1\subset H_2\subset\dots\subset H_n=G \) such that \( H_1 \) is discrete and abelian (resp., moreover, finitely generated) and, for \( i=1,\dots,n-1 \), \( H_i \) is a closed normal subgroup of \( H_{i+1} \) with \( H_{i+1}/H_i \) compact. The proof relies on the work by \textit{A. Kishimoto} [``Automorphisms of \(C^*\)-algebras with the Rohlin property'', J. Oper. Theory]. The main tools are \( K \)-theory and the Rohlin property of automorphisms.