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

Abstract

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 ${weak}^{\ast }$ 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 $VN(G)arrowI(A(G),H{)}^{\perp }$ 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.

A gap has been identified in the proof of Proposition 3.6 of [1], affecting the correctness of Proposition 5.2 of this article