Approximate indicators for closed subgroups of locally compact groups with applications to weakly amenable groups

We generalize the notion of an approximate indicator for a closed subgroup $H$ of a locally compact group $G$ introduced by Aristov, Runde, and Spronk and extend their characterization of the existence of such nets in terms of the approximability of $\chi_{H}$ in an appropriate ${weak}^{*}$ topology. We find that this equivalent condition is satisfied whenever $H$ is weakly amenable and $\chi_{H}$, considered as acting on $\ell^{1}(G)$ by multiplication, extends to a bounded map on $VN(G)$. This occurs in particular when a natural projection $VN(G)arrow I(A(G),H)^{\perp}$ exists. Applications are obtained to the existence (and non-existence) of natural and invariant projections onto $I(A(G),H)^{\perp}$ and $I(A_{cb}(G),H)^{\perp}$ and to the existence of ($\Delta$-weak) bounded approximate identities in ideals of $A(G)$ and $A_{cb}(G)$. In particular, we exhibit a locally compact group without the invariant complementation property.