Almost finiteness for general etale groupoids and its applications to stable rank of crossed products

Abstract

We extend Matui's notion of almost finiteness to general etale groupoids and show that the reduced groupoid C*-algebras of minimal almost finite groupoids have stable rank one. The proof follows a new strategy, which can be regarded as a local version of the large subalgebra argument. The following three are the main consequences of our result. (i) For any group of (local) subexponential growth and for any its minimal action admitting a totally disconnected free factor, the crossed product has stable rank one. (ii) Any countable amenable group admits a minimal action on the Cantor set all whose minimal extensions form the crossed product of stable rank one. (iii) For any amenable group, the crossed product of the universal minimal action has stable rank one.

Minor revision, 25 pages, to appear in IMRN

References in corpus (7)

• Tilings of amenable groups
• Folner tilings for actions of amenable groups
• Crossed products of the Cantor set by free minimal actions of Z^d
• The comparison property of amenable groups
• Minimal ambient nuclear C*-algebras
• Construction of minimal skew products of amenable minimal dynamical systems
• A class of simple C*-algebras with stable rank one

Cited by in corpus (11)

• Complete descriptions of intermediate operator algebras by intermediate extensions of dynamical systems
• The comparison property of amenable groups
• The type semigroup, comparison and almost finiteness for ample groupoids
• Almost elementary étale groupoids
• On tracial $\mathcal{Z}$-stability of simple non-unital C*-algebras
• Classification of tiling $C^{\ast }$-algebras
• Invariant ergodic measures and the classification of crossed product $C^{\ast }$-algebras
• Amenable actions on ill-behaved simple C*-algebras
• Stable rank of $\mathrm{C}(X)⋊\Gamma$
• Cantor combinatorics and almost finiteness
• Comparison and Simplicity of Commutator Subgroups of Full Groups