Operator-valued frames

TL;DR: In this paper, a generalization of vector-valued frame theory, called operator-valued frames, is presented, which can be viewed as the multiplicity-tiplicity-one case and extends to higher multiplicity their dilation approach.

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 mul- tiplicity one case and extends to higher multiplicity their dilation approach. We prove several results for operator-valued frames concerning duality, dis- jointeness, complementarity , and composition of operator valued frames and the relationship between the two types of similarity (left and right) of such frames. A key technical tool is the parametrization of Parseval operator val- ued frames in terms of a class of partial isometries in the Hilbert space of the analysis operator. We apply these notions to an analysis of multiframe gener- ators for the action of a discrete group G on a Hilbert space. One of the main results of the Han-Larson work was the parametrization of the Parseval frame generators in terms of the unitary operators in the von Neumann algebra gen- erated by the group representation, and the resulting norm path-connectedness of the set of frame generators due to the connectedness of the group of unitary operators of an arbitrary von Neumann algebra. In this paper we general- ize this multiplicity one result to operator-valued frames. However, both the parameterization and the proof of norm path-connectedness turn out to be necessarily more complicated, and this is at least in part the rationale for this paper. Our parameterization involves a class of partial isometries of a different von Neumann algebra. These partial isometries are not path-connected in the norm topology, but only in the strong operator topology. We prove that the set of operator frame generators is norm pathwise-connected precisely when the von Neumann algebra generated by the right representation of the group has no minimal projections. As in the multiplicity one theory there are analogous results for general (non-Parseval) frames.