A note on the density theorem for projective unitary representations

It is well known that a Gabor representation on L 2 ( R d ) L^{2}(\mathbb {R}^{d}) admits a frame generator h ∈ L 2 ( R d ) h\in L^{2}(\mathbb {R}^{d}) if and only if the associated lattice satisfies the Beurling density condition, which in turn can be characterized as the “trace condition” for the associated von Neumann algebra. It happens that this trace condition is also necessary for any projective unitary representation of a countable group to admit a frame vector. However, it is no longer sufficient for general representations, and in particular not sufficient for Gabor representations when they are restricted to proper time-frequency invariant subspaces. In this short note we show that the condition is also sufficient for a large class of projective unitary representations, which implies that the Gabor density theorem is valid for subspace representations in the case of irrational types of lattices.