We consider questions related to quantizing complex valued functions defined on a locally compact topological group. In the case of bounded functions, we generalize R. Werner's approach to prove the characterization of the associated normal covariant quantization maps.
Let X be a Banach space. Then there is a locally convex topology for X, the "Right topology," such that a linear map T , from X into a Banach space Y , is weakly compact, precisely when T is a continuous map from X, equipped with the "Right" topology, into Y equipped with the norm topology. When T is only sequentially continuous with respect to the Right topology, it is said to be pseudo weakly compact. This notion is related to Pelczynski's Property (V ).
Positive operator measures (with values in the space of bounded operators on a Hilbert space) and their generalizations, mainly positive sesquilinear form measures, are considered with the aim of providing a framework for their generalized eigenvalue type expansions. Though there are formal similarities with earlier approaches to special cases of the problem, the paper differs e.g. from standard rigged Hilbert space constructions and avoids the introduction of nuclear spaces. The techniques are predominantly measure theoretic and hence the Hilbert spaces involved are separable. The results range from a Naimark type dilation result to direct integral representations and a fairly concrete generalized eigenvalue expansion for unbounded normal operators.
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