Operator valued frames

TL;DR

This paper introduces operator-valued frame theory, extending vector-valued frames to higher multiplicities using operator-algebraic methods, and explores their properties and applications in group actions on Hilbert spaces.

Contribution

It generalizes vector-valued frame theory to operator-valued frames, including their parametrization, duality, and relationships, and applies these concepts to group actions and von Neumann algebras.

Findings

Multiframe generators form a norm pathwise-connected set when the von Neumann algebra has no minimal projections.

In the multiplicity one case, the class reduces to the unitary group, which is path-connected in norm.

In infinite multiplicity, the class is only path-connected in the strong operator topology.

Abstract

We develop a natural generalization of vector-valued frame theory, we term operator-valued frame theory, using operator-algebraic methods. This extends work of the second author and D. Han which can be viewed as the multiplicity one case and extends to higher multiplicity (e.g., multiframes) their dilation approach. We prove several results for operator-valued frames concerning their parametrization, duality, disjointeness, complementarity, and composition and the relationship between the two types of similarity (left and right) of such frames. We then apply these notions to prove that the collection of multiframe generators for the action of a discrete group on a Hilbert space is norm pathwise-connected precisely when the von Neumann algebra generated by the right representation of the group has no minimal projections. The proof is obtained by parametrizing this collection by a class of partial isometries in a larger von Neumann algebra. In the multiplicity one case this class reduces to the unitary class which is path-connected in norm, but in the infinite multiplicity case this class is path connected only in the strong operator topology and the proof depends on properties of tensor product slice maps.