[Paper Review] Purely infinite C*-algebras arising from crossed products

Mikael Rørdam, Adam Sierakowski|arXiv (Cornell University)|Jun 7, 2010

Advanced Operator Algebra ResearchMathematics19 citations

TL;DR

This paper establishes sufficient conditions for crossed product C*-algebras to be purely infinite, particularly when a discrete exact group acts on a commutative C*-algebra. It proves that every countable non-amenable exact group admits a free, amenable, minimal action on the Cantor set such that the resulting crossed product is a Kirchberg algebra in the UCT class.

ABSTRACT

We study conditions that will ensure that a crossed product of a C*-algebra by a discrete exact group is purely infinite (simple or non-simple). We are particularly interested in the case of a discrete non-amenable exact group acting on a commutative C*-algebra, where our sufficient conditions can be phrased in terms of paradoxicality of subsets of the spectrum of the abelian C*-algebra. As an application of our results we show that every discrete countable non-amenable exact group admits a free amenable minimal action on the Cantor set such that the corresponding crossed product C*-algebra is a Kirchberg algebra in the UCT class.

Motivation & Objective

• To identify conditions under which the reduced crossed product of a C*-algebra by a discrete exact group is purely infinite.
• To investigate the role of paradoxicality in the spectrum of an abelian C*-algebra in determining pure infiniteness of the crossed product.
• To construct, for any countable non-amenable exact group, a free, amenable, minimal action on the Cantor set such that the associated crossed product is a Kirchberg algebra in the UCT class.
• To extend known results on purely infinite C*-algebras arising from dynamical systems by focusing on non-simple cases and projections in the algebra.

Proposed method

• Use the ideal property (IP) to ensure projections separate ideals in the C*-algebra, enabling analysis of ideal structure in the crossed product.
• Apply the notion of essential freeness of the group action on the spectrum of the abelian C*-algebra to control the structure of the crossed product.
• Characterize pure infiniteness via the property that every non-zero positive element in the C*-algebra is properly infinite in the crossed product, under exactness and separability assumptions.
• Utilize the concept of G-paradoxical subsets of the spectrum to show that characteristic functions of such sets become properly infinite projections in the crossed product.
• Construct a separable, G-invariant C*-subalgebra of ℓ∞(G) generated by projections, containing specific sets M and M′, to model the dynamical system.
• Use the bijective correspondence between ideals in the crossed product and G-invariant ideals in the C*-algebra to reduce the problem to a minimal, free, amenable action on a compact Hausdorff space with no isolated points (i.e., the Cantor set).

Experimental results

Research questions

• RQ1 Under what conditions is the reduced crossed product of a C*-algebra by a discrete exact group purely infinite?
• RQ2 How does G-paradoxicality of a clopen subset of the spectrum of a commutative C*-algebra relate to the proper infiniteness of its characteristic function in the crossed product?
• RQ3 Can every countable non-amenable exact group admit a free, amenable, minimal action on the Cantor set such that the crossed product is a Kirchberg algebra in the UCT class?
• RQ4 What structural properties of the C*-algebra and group action ensure that every non-zero projection becomes properly infinite in the crossed product?
• RQ5 To what extent can the K-theory of such crossed products be controlled in the construction?

Key findings

• The reduced crossed product $ A \rtimes_r G $ is purely infinite if and only if every non-zero positive element in $ A $ is properly infinite in the crossed product, provided $ G $ is discrete and exact, $ A $ is separable with the ideal property, and the action on $ \widehat{A} $ is essentially free.
• For $ A = C(X) $ with $ X $ the Cantor set, $ A \rtimes_r G $ is purely infinite if and only if every non-zero projection in $ A $ is properly infinite in the crossed product, under the same assumptions.
• If a clopen subset $ E \subset X $ is $ G $-paradoxical in a clopen-respecting way, then $ 1_E $ is a properly infinite projection in $ C(X) \rtimes_r G $, and a partial converse holds.
• Every countable discrete non-amenable exact group admits a free, amenable, minimal action on the Cantor set such that the crossed product is a Kirchberg algebra in the UCT class.
• The crossed product $ C(X) \rtimes_r G $ is nuclear and belongs to the UCT class because it arises from an amenable groupoid action, and its simplicity follows from the minimality of the action.
• The K-theory of the resulting crossed product is not computed, but it is plausible that it can be designed to be trivial, implying isomorphism to $ \mathcal{O}_2 $ via the Kirchberg–Phillips classification theorem.