Weak separation properties for closed subgroups of locally compact groups
Volume 238 / 2017
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 $\mathrm {VN}(G)\rightarrow \mathrm {VN}_{H}(G)$, and (3) the approximability of the characteristic function $\chi _{H}$ by functions in $M_{cb}A(G)$ with respect to the weak$^{*}$ topology of $M_{cb}A(G_{d})$. 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 the fact that $G_{d}$ satisfies the approximation property, in combination with the existence of a natural projection (in the sense of Lau and Ülger), are shown to suffice to deduce (3). Several consequences of (2) involving the cb-multiplier completion of $A(G)$ 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 $H$.
