Hikaru Awazu | Searcharxiv

arXiv2025

On the permanence properties of residually exact groups

A discrete group $\Gamma$ is called exact if the reduced group C*-algebra ${C_{\lambda}}^{*}(\Gamma)$ is exact as C*-algebras, and a discrete group $\Lambda$ is called residually exact if every nonunital element $g \in \Lambda$ admits a surjective group homomorphism from $\Lambda$ to some exact group $\Gamma$ which maps $g$ to a nonunital element of $\Gamma$. We prove the class of residually exact groups is closed under taking Green's graph products [1], double amalgamed products and special HNN extensions.

math.GR↗

arXiv2025

Amenability of group actions on compact spaces and the associated Banach algebras

For a topological group $G$, amenability can be characterized by the amenability of the convolution Banach algebra $L^1(G)$. Here a Banach algebra $A$ is called amenable if every bounded derivation from $A$ into any dual--type $A$--$A$--Banach bimodule is inner. We extend this classical result to the case of discrete group actions on compact Hausdorff spaces. By introducing a Banach algebra naturally associated with the action and adopting a suitably weakened notion of amenability for Banach algebras, we obtain an analogous characterization of amenable actions. As a lemma, we also proved a fixed--point property for amenable actions that strengthens the theorem of Dong and Wang (2015).

math.FA↗