Conjugacy co-amenability

Abstract

In this note we study a natural analytic property of inclusions of groups akin to co-amenability: the property of existence of a non-compactly supported invariant state for the conjugation action of a group $G$ on the von Neumann algebra generated by the characteristic functions $\{1_{gH{g}^{-1}}{\}}_{g\in G}$ viewed inside ${\ell }^{\infty }(G)$. Some interesting settings and examples of this phenomena are proved. We also comment on a consideration related to proper proximality, which motivated this property.

10 pages. Comments welcome