C^*-exactness and property A for group actions

2024/07/23 by Nishikawa, Hiroto
#46L05 #51F99 #FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2407.16130

Abstract

For an action of a discrete group Γ on a set X, we show that the Schreier graph on X has property A if and only if the permutation representation on ℓ₂X generates an exact C^*-algebra. This is well known in the case of the left regular action on X=Γ as the equivalence of C^*-exactness and property A of its Cayley graph. This also generalizes Sako's theorem, which states that exactness of the uniform Roe algebra C^*_u(X) characterizes property A of X when X is uniformly locally finite.

Related

• Exactness from Proper Actions
• Non-K-exact uniform Roe C*-algebras
• Exactness vs C*-exactness for certain non-discrete groups
• Uniform Local Amenability implies Property A
• On the invariant uniform Roe algebra