[Paper Review] Minimal dynamics and Z-stable classification

Karen R. Strung, Wilhelm Winter|arXiv (Cornell University)|Jan 8, 2010

Advanced Operator Algebra Research24 references3 citations

TL;DR

This paper establishes a criterion for Z-stable classification of C*-algebras arising from minimal dynamical systems by showing that if a certain subalgebra tensored with a UHF algebra is tracially approximately a class of C*-algebras S, then the full crossed product tensored with the same UHF algebra is also tracially approximately S. The key result is a Z-stability classification result for transformation group C*-algebras without requiring assumptions on real rank, stable rank, or dimension of the underlying space.

ABSTRACT

Let X be an infinite compact metric space, α: X o X a minimal homeomorphism, u the unitary implementing αin the transformation group C*-algebra, and S a class of separable nuclear C*-algebras that contains all unital hereditary C*-subalgebras of C*-algebras in S. Motivated by the success of tracial approximation by finite dimensional C*-algebras as an abstract characterization of classifiable C*-algebras and the idea that classification results for C*-algebras tensored with UHF algebras can be used to derive classification results up to tensoring with the Jiang-Su algebra Z, we prove that the transformation group C*-algebra tensored with a UHF algebra is tracially approximately S if there exists a y in X such that a certain C*-subalgebra is tracially approximately S. If the class S consists of finite dimensional C*-algebras, this can be used to deduce classification up to tensoring with Z for C*-algebras associated to minimal dynamical systems where projections separate tracial states. This is done without making any assumptions on the real rank or stable rank of either the transformation group C*-algebra or the C*-subalgebra, nor on the dimension of X. The result is a key step in the classification of C*-algebras associated to uniquely ergodic minimal dynamical systems by their ordered K-groups. It also sets the stage to provide further classification results for those C*-algebras of minimal dynamical systems where projections do not necessarily separate traces.

Motivation & Objective

• To extend classification results for C*-algebras associated to minimal dynamical systems beyond the real rank zero or finite-dimensional cases.
• To establish Z-stability classification without assuming finite dimensionality, real rank zero, or stable rank conditions on the crossed product or its subalgebras.
• To provide a general framework for Z-stable classification using tracial approximation by a class S of C*-algebras.
• To bridge existing classification theorems with the Z-stability results of Toms and Winter, particularly in the context of uniquely ergodic minimal systems.
• To enable classification of transformation group C*-algebras where projections do not separate traces, using UHF stabilization and tracial approximation.

Proposed method

• Use tracial approximation by a class S of separable nuclear C*-algebras to characterize Z-stable classification of transformation group C*-algebras.
• Apply the strategy of classifying C*-algebras up to tensoring with the Jiang–Su algebra Z by first classifying after tensoring with UHF algebras.
• Leverage the fact that if a subalgebra C*(C(X), uC₀(X∖{y})) ⊗ M_{q^∞} is tracially approximately S, then the full crossed product (C(X) ⋊_α ℤ) ⊗ M_{q^∞} is also tracially approximately S.
• Utilize the result that tracially approximately S algebras with sufficient stability (e.g., approximate divisibility) imply real rank zero and hence tracial rank zero.
• Apply the classification theorem from [19] to deduce *-isomorphism between Z-stabilized algebras when their invariants are isomorphic.
• Use the UCT and the fact that UHF algebras are Z-stable to ensure compatibility with existing classification frameworks.

Experimental results

Research questions

• RQ1 Under what conditions can C*-algebras associated to minimal dynamical systems be classified up to Z-stability without assuming finite dimensionality or real rank zero?
• RQ2 Can tracial approximation by a class S of C*-algebras be used to derive Z-stable classification results for transformation group C*-algebras?
• RQ3 Is it possible to classify such C*-algebras when projections do not separate tracial states, using UHF stabilization and tracial approximation?
• RQ4 How does the structure of the subalgebra C*(C(X), uC₀(X∖{y})) influence the Z-stability of the full crossed product?
• RQ5 What is the role of UHF algebras in enabling classification results that lead to Z-stable isomorphism?

Key findings

• If the subalgebra C*(C(X), uC₀(X∖{y})) ⊗ M_{q^∞} is tracially approximately S, then (C(X) ⋊_α ℤ) ⊗ M_{q^∞} is also tracially approximately S for any class S of separable nuclear C*-algebras closed under hereditary subalgebras.
• When S is the class of finite-dimensional C*-algebras, the result implies that (C(X) ⋊_α ℤ) ⊗ M_{q^∞} has tracial rank zero, provided the subalgebra satisfies the same condition.
• The algebras A ⊗ M_{q^∞} and B ⊗ M_{q^∞} have tracial rank zero for any q, which implies that A ⊗ ℤ and B ⊗ ℤ are *-isomorphic if their invariants are isomorphic.
• The classification result holds without assumptions on the real rank, stable rank, or covering dimension of X, making it applicable to infinite-dimensional and non-regular systems.
• For uniquely ergodic minimal systems on odd spheres Sⁿ with n ≥ 3 odd, the crossed product is Z-stable and thus classified by its ordered K₀-group, even though it has no nontrivial projections.
• The result provides a missing link between existing classification theorems and the Z-stability results of Toms and Winter, particularly in the absence of finite-dimensionality assumptions.