Crossed products of nuclear \(\mathrm C^{\ast}\)-algebras and their traces
This interesting paper studies the Blackadar-Kirchberg Question (BKQ), which is a C*-analogue of the Connes embedding conjecture for von Neumann algebras. More precisely, it is investigated when certain C*-algebras are matricial field (MF), a property that asks when the C*-algebra embeds into a norm ultraproduct of matrix algebras. The BKQ asks whether every stably finite C*-algebra is MF. The main results of the paper prove that certain crossed products or group C*-algebras are MF. They also deal with tracial versions of the MF and prove results in this direction for the same classes of C*-algebras. A characterization is given of when a (reduced) crossed product by a free group is MF. This characterization can be phrased both in terms of stable finiteness or in terms of $K$-theory. In particular, it is proved that a crossed product of the form \(A\rtimes_r \mathbb{F}_r\) is MF whenever \(A\) is separable, simple, unital, nuclear and monotracial satisfying the UCT and \(\mathbb{F}_r\) is a free group of rank~\(r\). The (tracial) MF property is also investigated for (reduced) C*-algebras of certain semidirected products of groups of the form \(G\rtimes \mathbb{F}_r\) for an amenable group \(G\) or \(H\rtimes G\) for a locally finite group \(G\) and a group f\(H\) or which \(C^*_r(H)\) has the MF property.
scientific article; zbMATH DE number 7044289
scientific article