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

TL;DR

This paper extends the concept of approximate indicators for closed subgroups in locally compact groups, linking their existence to weak amenability and bounded maps on von Neumann algebras, with applications to projections and approximate identities.

Contribution

It generalizes approximate indicators, characterizes their existence via weak amenability, and applies these results to projections and identities in harmonic analysis.

Findings

Existence of approximate indicators is linked to weak amenability.

Identifies conditions for invariant projections onto certain ideals.

Provides examples of groups lacking invariant complementation property.

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 ${weak}^{*}$ 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.