On Quasi-greedy Bases Associated with Unitary Representations of Countable Groups

We consider the natural generating system for a cyclic subspace of a Hilbert space generated by a dual integrable unitary representation of a countable abelian group. We prove, under mild hypothesis, that whenever the generating system is a quasi-greedy basis it must also be an unconditional Riesz basis. A number of applications to Gabor systems and to general Vilenkin systems are considered. In particular, we show that any Gabor Schauder basis that also forms a quasi-greedy system in L2(R) is in fact a Riesz basis, and therefore satisfies the classical Balian-Low theorem.