Operator Algebras, Operator Spaces, Frames and Applications

点击翻译按钮获取中文摘要

ABSTRACTPaulsen/Blecher/PapadakisThe Principal Investigators propose several main lines of research ontopics related to the theory and applications of operator algebras,operator spaces, and frames. Blecher will be studying the generaltheory of operator algebras and modules over operator algebras,Hilbert C*-modules, and questions relating to noncommutative Choquettheory. Paulsen will continue to study injective operator spaces andmodules, the weak expectation property, function theoretic operatortheory, interpolation theory from an operator algebra point of view.With Papadakis he will also be studying and frames and reconstructionswith a view to applying reproducing kernel Hilbert space methods andsymmetric orthogonalization results.The study of operator algebras originally grew out of quantum mechanics.It is often important to see how formulas involving numerical variablesbehave when these variables are allowed to be operator variables. It isout of such a process that the theory of operator spaces and completelybounded maps emerged. Blecher and Paulsen's research focuses mainly onquestions of how various theories behave under this `quantization'.On the other hand, interpolation theory started as a purely mathematicalexercise, and only in the past 20 years has it been found to haveimportant applications in engineering. For example, in electrical circuitdesign, one starts with a desired frequency response, for a few givenfrequencies, and wishes to design the simplest circuit with that givenresponse. Mathematically, this problem becomes one of finding thesimplest function of a given type that achieves certain given values atgiven points. This last problem is what we call an interpolation problem.Already the demands of electrical engineering take us beyond the knowninterpolation theories. Surprisingly, interpolation theory and the studyof operator algebras is interwoven, and this interplay has lead to somenew interpolation results. We have found that a better understanding ofthe "quantized", i.e., matrix-valued, interpolation is what is neededto answer many ordinary interpolation questions.Frame theory can be applied to the study of how we extract informationout of streams of data, and how we reconstruct the original data fromthe derived information. A typical example of a situation where thisarises is the CAT scan, where from a large quantity of data, one istrying to reconstruct a picture of the inside of a body. Our work isnot focused on particular examples, but on how one analyzes how "good"is a particular frame.