Crossed products of nuclear C⁎-algebras and their traces

Abstract

This work addresses the Blackadar–Kirchberg Question (BKQ) as it pertains to group and reduced crossed product C⁎-algebras. The BKQ asks if every stably finite C⁎-algebra admits the matricial field (MF) property. We characterize the MF property for a class of reduced crossed product C⁎-algebras and for their traces. We prove that every ordered abelian group is, in a sense, MF, and show that there is no K-theoretic obstruction to an affirmative answer to the BKQ. Using classification techniques and induced K-theoretic dynamics, we give an affirmative answer to the BKQ for reduced crossed products of AH-algebras of real rank zero by free groups. Combining our results with recent progress in Elliott's Classification Program shows that the reduced crossed product of a separable, simple, unital, nuclear, UCT, and monotracial C⁎-algebra by the free group is always MF. We also examine traces on these crossed products and show they always admit certain finite dimensional approximation properties. By appealing to a result of Ozawa, Rørdam, and Sato, we show that semidirect products of discrete amenable groups by free groups admit MF reduced group C⁎-algebras.