The C∗-algebra of a minimal homeomorphism of zero mean dimension

Almost Finiteness and the Small Boundary Property

“…We expect that the conditions in the Toms-Winter conjecture are actually satisfied by all of the simple crossed products in the statement of Theorem 9.4. This is known to be the case when G = Z by the work of Elliott and Niu [13].…”

“…By Urysohn's lemma there is a continuous function g i,j,c : X → [0, 1] which is zero on the complement of cV i and one on W i,j,c . Given 1 ≤ j ≤ J and c ∈ C i,j and writing α t for the automorphism of C(X) that composes functions with the transformation x → t −1 x, we then define the functioñ and so the function u sgi,j,c u −1 s −g i,j,c has norm at most 1/Q by (12) and (13). Since these functions for different i, j and c have pairwise disjoint supports, we deduce that…”

“…3 Nuclearity is automatic in this case since the acting group is amenable. and tower decompositions have long played an important role in unraveling the structure of C * -crossed products, notably in the case of Z-actions [39,31,30,44,13] and in the various theories of dimension for group actions [19,41,42,25], the novelty in the concept of almost finiteness is its dovetailing of topology with measure-theoretic phenomena within the general setting of amenability. Indeed almost finiteness not only relates to the Ornstein-Weiss tower theorem via analogy but also naturally calls for the direct application of Ornstein-Weiss tiling methods towards C * -algebraic ends, as illustrated by the proof of Z -stability in the free minimal case [25].…”

Regularity of Villadsen algebras and characters on their central sequence algebras

“…Furthermore, it follows from (5), by induction, that for any 1 ≤ n < m and an arbitary vector bundle η over X n , we have ϕ * n,m (η) ∼ = µ * m,n (η) ⊕ (n + 1)rank(η)ζ n+1 ⊕ · · · ⊕ σ(m) (n + 1)! rank(η)ζ m , (8) where µ m,n := π 1 n+1 • · · · • π 1 m : X m → X n . Moreover, we find that lim m→∞ σ(m) (n + 1)!λ(m) = lim m→∞ 1 (n + 1)!m = 0.…”

“…In certain situations, a unique tracial state is sufficient to conclude regularity and even classifiability by the Elliott invariant. For instance, Elliott and Niu showed in [8] that if X is a compact metrizable Hausdorff space and σ is a minimal homeomorphism of X such that the dynamical system (X, σ) is uniquely ergodic, i.e., C(X) ⋊ σ Z has a unique tracial state, then C(X) ⋊ σ Z is Z-stable and classifiable (this is not automatic, see [11]). Similarly, as proven by Niu (see […”