Stable rank of transformation group C*-algebras

It is shown that, for a free and minimal Z d \mathbb {Z}^d -action on a compact Hausdorff space X X , the transformation group C*-algebra C ( X ) ⋊ Z d \mathrm {C}(X)\rtimes \mathbb {Z}^d always has stable rank one, i.e., the invertible elements are dense. Moreover, the C*-algebra C ( X ) ⋊ Z d \mathrm {C}(X)\rtimes \mathbb {Z}^d is also shown to be classified by the Elliott invariant if, and only if, the strict order of its Cuntz semigroup is determined by the traces. That is, it satisfies the Toms-Winter conjecture. In fact, for any free and minimal Γ \Gamma -action on X X , where Γ \Gamma is a countable discrete amenable group, if ( X , Γ ) (X, \Gamma ) has the uniform Rokhlin property and Cuntz comparison of open sets, then the C*-algebra C ( X ) ⋊ Γ \mathrm {C}(X)\rtimes \Gamma is shown to have stable rank one and to satisfy the Toms-Winter conjecture.