Operator Algebras, Modules and Completely Bounded Maps

点击翻译按钮获取中文摘要

Abstract Paulsen/Blecher The Principal Investigators propose several main lines of research on topics related to the theory of completely bounded maps. Blecher will be studying the general theory of operator algebras and modules over operator algebras, Hilbert C*-modules, and questions relating to realizations of Banach spaces as operator spaces. Paulsen will continue to study the completely bounded Hochschild cohomology through its presentation as a completely bounded relative Yoneda cohomology, some problems concerning polynomially bounded operators, and questions in interpolation theory. The study of operator algebras originally grew out of quantum mechanics. The set of "observables" in a quantum mechanical system is described as an algebra of operators. For this reason, it is often important to see how formulas involving numerical variables behave when these variables are allowed to be operator variables. This process is often referred to as finding "quantized" versions of the old theories and it was out of this process that the theory of completely bounded maps grew. Blecher and Paulsen's research focuses mainly on questions of how various theories behave under this type of 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 some 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 ha ve found that a better understanding of the "quantized", i.e., matrix-valued, interpolation is what is needed to answer many ordinary interpolation questions.