Noncommutative Shift-Invariant Spaces

TL;DR

This paper develops a general framework for noncommutative shift-invariant spaces using Hilbert modules with operator-valued inner products, extending classical results to nonabelian groups.

Contribution

It introduces a novel theory of reproducing systems in noncommutative modular structures, generalizing shift-invariant space characterizations beyond abelian groups.

Findings

Characterization of Riesz and frame sequences in noncommutative settings

Extension of classical shift-invariant space results to nonabelian groups

Framework encompassing fundamental properties of group-invariant subspaces

Abstract

The structure of subspaces of a Hilbert space that are invariant under unitary representations of a discrete group is related to a notion of Hilbert modules endowed with inner products taking values in spaces of unbounded operators. A theory of reproducing systems in such modular structures is developed, providing a general framework that includes fundamental results of shift-invariant spaces. In particular, general characterizations of Riesz and frame sequences associated to group representations are provided, extending previous results for abelian groups and for cyclic subspaces of unitary representations of noncommutative discrete groups.