Projective representations on operator algebras

We establish new restrictions on the values of the lifting obstruction for projective unitary representations of second countable, locally compact Hausdorff groups on operator algebras. Using these, we show that every projective representation on the Jiang--Su algebra lifts to a genuine group representation. We also characterise the possible values of lifting obstructions for finite group projective representations on UHF-algebras of infinite type and Cuntz algebras. Finally we show that certain 2-cocycles of Property (T) groups cannot arise as lifting invariants of projective representations on the Hyperfinite II$_{1}$ factor.