Weak property (T₋㵵) for discrete groups

We show that, for a countable discrete group, property (T₋㵵) of Bader, Furman, Gelander and Monod is equivalent to the property that, whenever an Lᵖ-representation of admits a net of almost invariant unit vectors, it has a non-zero invariant vector. Central in the proof is to show that the closure of the group of T-valued 1-coboundaries is a sufficient criteria for strong ergodicity of ergodic p. m. p. actions.