The Principal Investigators propose several main lines of research on topics related to the theory and applications of operator algebras, operator spaces, and frames. Blecher will be studying the general theory of operator algebras and modules over operator algebras, Hilbert C*-modules, and questions relating to noncommutative Choquet theory. Paulsen will continue to study injective operator spaces and modules, the weak expectation property, function theoretic operator theory, interpolation theory from an operator algebra point of view. With Papadakis he will also be studying and frames and reconstructions with a view to applying reproducing kernel Hilbert space methods and symmetric orthogonalization results.
The study of operator algebras originally grew out of quantum mechanics. It is often important to see how formulas involving numerical variables behave when these variables are allowed to be operator variables. It is out of such a process that the theory of operator spaces and completely bounded maps emerged. Blecher and Paulsen's research focuses mainly on questions of how various theories behave under this `quantization'. On the other hand, interpolation theory started as a purely mathematical exercise, and only in the past 20 years has it been found to have important applications in engineering. For example, in electrical circuit design, one starts with a desired frequency response, for a few given frequencies, and wishes to design the simplest circuit with that given response. Mathematically, this problem becomes one of finding the simplest function of a given type that achieves certain given values at given points. This last problem is what we call an interpolation problem. Already the demands of electrical engineering take us beyond the known interpolation theories. Surprisingly, interpolation theory and the study of operator algebras is interwoven, and this interplay has lead to some new interpolation results. We have found that a better understanding of the "quantized", i.e., matrix-valued, interpolation is what is needed to answer many ordinary interpolation questions. Frame theory can be applied to the study of how we extract information out of streams of data, and how we reconstruct the original data from the derived information. A typical example of a situation where this arises is the CAT scan, where from a large quantity of data, one is trying to reconstruct a picture of the inside of a body. Our work is not focused on particular examples, but on how one analyzes how "good" is a particular frame.
