MINIMAL DYNAMICS AND $\mathcal{Z}$-STABLE CLASSIFICATION

Abstract

Let X be an infinite compact metric space, α : X → X a minimal homeomorphism, u the unitary that implements α in the transformation group C*-algebra C(X) ⋊αℤ, and [Formula: see text] a class of separable nuclear C*-algebras that contains all unital hereditary C*-subalgebras of C*-algebras in [Formula: see text]. 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 [Formula: see text], we prove that (C(X) ⋊αℤ) ⊗ Mq∞is tracially approximately [Formula: see text] if there exists a y ∈ X such that the C*-subalgebra (C*(C(X), uC0(X\{y}))) ⊗ Mq∞is tracially approximately [Formula: see text]. If the class [Formula: see text] consists of finite dimensional C*-algebras, this can be used to deduce classification up to tensoring with [Formula: see text] 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 C(X) ⋊αℤ or C*(C(X), uC0(X\{y})), 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.