2024/06/14 by Kopsacheilis, Grigoris, Liao, Hung-Chang, Tikuisis, Aaron +1
#Dynamical Systems (math.DS) #FOS: Mathematics #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2406.09808
We prove that, for a free action α\colon G \curvearrowright X of a countably infinite discrete amenable group on a compact metric space, the small boundary property is implied by uniform property Γ of the Cartan subalgebra (C(X) ⊆ C(X) \rtimesαG). The reverse implication has been demonstrated by Kerr and Szabó for free actions, from which we obtain that these two conditions are equivalent. We moreover show that, if α is also minimal, then almost finiteness of α is implied by tracial Z-stability of the subalgebra (C(X) ⊆ C(X) \rtimesαG). The reverse implication is due to Kerr, resulting in the equivalence of these two properties as well. As an application, we prove that if α\colon G \curvearrowright X and β\colon H \curvearrowright Y are free actions and α has the small boundary property, then α× β\colon G × H \curvearrowright X × Y has the small boundary property. An analogous permanence property is obtained for almost finiteness in case α and β are free minimal actions.