Weak separation properties for closed subgroups of locally compact groups

Abstract

Three separation properties for a closed subgroup $H$ of a locally compact group $G$ are studied: (1) the existence of a bounded approximate indicator for $H$, (2) the existence of a completely bounded invariant projection of $VN\left(G\right)$ onto $VN_{H}\left(G\right)$, and (3) the approximability of the characteristic function $\chi_{H}$ by functions in $M_{cb}A\left(G\right)$ with respect to the weak$^{*}$ topology of $M_{cb}A\left(G_{d}\right)$. We show that the $H$-separation property of Kaniuth and Lau is characterized by the existence of certain bounded approximate indicators for $H$ and that a discretized analogue of the $H$-separation property is equivalent to (3). Moreover, we give a related characterization of amenability of $H$ in terms of any group $G$ containing $H$ as a closed subgroup. The weak amenability of $G$ or that $G_{d}$ satisfies the approximation property, in combination with the existence of a natural projection (in the sense of Lau and Ulger), are shown to suffice to conclude (3). Several consequences of (2) involving the cb-multiplier completion of $A\left(G\right)$ are given. Finally, a convolution technique for averaging over the closed subgroup $H$ is developed and used to weaken a condition for the existence of a bounded approximate indicator for $H$.