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

We 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 our 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. We 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 C*-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 s-weakly, or w*) continuous. Some of the results for maps extend to the case where the domain algebra is non-commutative. It has been known for a long time that a necessary and sufficient condition for a bounded linear map from a unital C*-algebra into B(H) to have a Hilbert space dilation to a*-homomorphism is that the mapping needs to be completely bounded. Our theory shows that even if it is not completely bounded it still has a Banach space dilation to a homomorphism. For the special case when the domain algebra is an abelian von Neumann algebra and the map is normal, we show that the dilation can be taken to be normal with respect to the usual Banach space version of ultraweak topology on the range space. We view these results as generalizations of the known result of Cazzaza, Han and Larson that arbitrary framings have Banach dilations, and also the known result that completely bounded maps have Hilbertian dilations. 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.