[Paper Review] Crossed products of nuclear C*-algebras by free groups and their traces

Timothy Rainone, Christopher Schafhauser|arXiv (Cornell University)|Jan 22, 2016

Advanced Operator Algebra Research22 references6 citations

TL;DR

This paper establishes that reduced crossed products of approximately subhomogeneous (ASH) C*-algebras of real rank zero by free groups are matricial field (MF) if and only if they are stably finite, with all traces on these algebras admitting finite-dimensional approximations. The key contribution is a K-theoretic characterization linking the MF property to the absence of non-zero positive elements in the K₀-group under the action of the free group, extending earlier results on AF-algebras and resolving a special case of the C*-algebra version of Connes's Embedding Problem.

ABSTRACT

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 $A times_λ \mathbb{F}_r$ is MF for any action $α$. Appealing to a result of Ozawa, Rørdam, and Sato, we show that discrete groups of the form $G times \mathbb{F}_r$ 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.

Motivation & Objective

• To determine when reduced crossed products of ASH-algebras of real rank zero by free groups are matricial field (MF) algebras.
• To characterize the MF property via K-theoretic dynamics, specifically the action of the free group on the K₀-group of the algebra.
• To show that all traces on such crossed products are MF, extending the tracial approximation framework to non-AF settings.
• To contribute to Elliott's Classification Program by verifying the MF property for a broad class of stably finite, nuclear C*-algebras satisfying the UCT.
• To resolve a special case of the C*-algebra version of Connes's Embedding Problem for crossed products with free groups.

Proposed method

• Use classification techniques and K-theoretic dynamics to analyze the action of the free group on the K₀-group of the ASH-algebra.
• Apply the Choi-Effros Lifting Theorem and Voiculescu’s noncommutative Weyl-von Neumann Theorem to relate MF and quasidiagonal properties in the nuclear setting.
• Leverage the permanence of real rank zero and A𝕋-algebra structure under tensoring with the universal UHF algebra 𝒬.
• Utilize the Gauge Invariant Uniqueness Theorem and Kirchberg’s Slice Lemma to prove isomorphisms between tensor products and Cuntz-Pimsner algebras.
• Establish equivalence between the MF property and the absence of non-zero positive elements in the K₀-group fixed under the group action.
• Apply recent results from the classification program, including Tikuisis-White-Winter’s quasidiagonality of traces on UCT-satisfying nuclear C*-algebras.

Experimental results

Research questions

• RQ1 When is the reduced crossed product of an ASH-algebra of real rank zero by a free group MF?
• RQ2 What is the precise K-theoretic condition on the action of the free group that characterizes the MF property in such crossed products?
• RQ3 Do all traces on these crossed products admit finite-dimensional approximations (i.e., are they MF)?
• RQ4 Can the MF property for these crossed products be established via classification-theoretic techniques and K-theoretic dynamics?
• RQ5 Does the resolution of the MF property for this class of algebras confirm a special case of the C*-algebra version of Connes’s Embedding Problem?

Key findings

• The reduced crossed product $ A \rtimes_\lambda \mathbb{F}_r $ is MF if and only if it is stably finite, for $ A $ an ASH-algebra of real rank zero.
• The MF property is equivalent to the condition that the subgroup of $ \mathrm{K}_0(A) $ generated by $ \{x - \alpha_s(x) \mid x \in \mathrm{K}_0(A), s \in \mathbb{F}_r \} $ contains no non-zero positive elements.
• Every trace on $ A \rtimes_\lambda \mathbb{F}_r $ is MF, meaning it admits a trace-preserving finite-dimensional approximation into the ultrapower of the universal UHF algebra.
• If $ A $ is a separable, simple, unital, nuclear, monotracial C*-algebra satisfying the UCT, then $ A \rtimes_\lambda \mathbb{F}_r $ is MF for any action of $ \mathbb{F}_r $ on $ A $.
• Discrete groups of the form $ G \rtimes \mathbb{F}_r $ with $ G $ amenable have MF reduced group C*-algebras, as a consequence of the main result and a result by Ozawa, Rørdam, and Sato.