Operator Valued Measures Dilations And The Theory Of Frames

Operator-Valued Measures, Dilations, and the Theory of Frames

The authors develop elements of a general dilation theory for operator-valued measures. Hilbert space operator-valued measures are closely related to bounded linear maps on abelian von Neumann algebras, and some of their results include new dilation results for bounded linear maps that are not necessarily completely bounded, and from domain algebras that are not necessarily abelian. In the non-cb case the dilation space often needs to be a Banach space. They give applications to both the discrete and the continuous frame theory. There are natural associations between the theory of frames (including continuous frames and framings), the theory of operator-valued measures on sigma-algebras of sets, and the theory of continuous linear maps between -algebras. In this connection frame theory itself is identified with the special case in which the domain algebra for the maps is an abelian von Neumann algebra and the map is normal (i.e. ultraweakly, or weakly, or w*) continuous.

Operator-valued Measures, Dilations, and the Theory of Frames

Our methods extend to some cases where the domain algebra need not be commutative, leading to new dilation results for maps of general von Neumann algebras. This paper was motivated by some recent results in frame theory and the observation that there is a close connection between the analysis of dual pairs of frames (both the discrete and the continuous theory) and the theory of operator-valued measures.

Operator-Valued Measures, Dilations, and the Theory of Frames