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Crossed products of nuclear C*-algebras by free groups and their traces

We study the matricial field (MF) property for certain reduced crossed product C*-algebras and their traces. Using classification techniques and induced K-theoretic dynamics, we show that reduced crossed products of ASH-algebras of real rank zero by free groups are MF if and only if they are stably finite. We also examine traces on these crossed products and show they always admit certain finite dimensional approximation properties. Combining these results with recent progress in Elliott's Classification Program, it follows that if $A$ is a separable, simple, unital, nuclear, monotracial C*-algebra satisfying the UCT, then $A \rtimes_λ \mathbb{F}_r$ is MF for any action $α$. Appealing to a result of Ozawa, Rørdam, and Sato, we show that discrete groups of the form $G \rtimes \mathbb{F}_r$ with $G$ amenable admit MF reduced group C*-algebras In the process, some new permanence properties of MF algebras are obtained which are of independent interest. In particular, minimal tensor products of MF algebras are again MF provided one of the factors is exact.