Riesz and frame systems generated by unitary actions of discrete groups

Abstract

We characterize orthonormal bases, Riesz bases and frames which arise from the action of a countable discrete group Γ on a single element ψ of a given Hilbert space H. As Γ might not be abelian, this is done in terms of a bracket map taking values in the L1-space associated to the group von Neumann algebra of Γ. Our result generalizes recent work for LCA groups in [26]. In many cases, the bracket map can be computed in terms of a noncommutative form of the Zak transform