Vanishing-at-infinity criterion for maximal left ideals

Let $G$ be a Hermitian locally compact group, let $\widehat{G}$ denote its irreducible representations, and for $\pi\in\widehat{G}$ and a unit vector $\xi\in\mathcal{H}_\pi$ define

$$\mathcal{J}_{\pi,\xi}=\{\mu\in M(G):\pi(\mu)\xi=0\}.$$

A representation $\pi$ vanishes at infinity when its matrix coefficients vanish at infinity. Vanishing-at-infinity criterion. For every $\pi\in\widehat{G}$, there exists a unit vector $\xi\in\mathcal{H}_\pi$ for which $\mathcal{J}_{\pi,\xi}$ is a weak*-closed maximal left ideal of $M(G)$ if and only if $\pi$ vanishes at infinity. This is intended as an analogue of Barnes' theorem relating representation-theoretic properties to ideals in the measure algebra. The paper establishes related classification results for broad classes of groups, but the criterion remains open for general Hermitian locally compact groups.