Model theory and operator algebras

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Model theory is a branch of mathematical logic which has had success in providing novel attacks on problems in other branches of mathematics. Generally speaking, to study a mathematical object model theoretically, one considers not the object in question but the theory of the object in a suitably chosen language. By studying all the models of this theory, it is sometimes possible to draw conclusions about the original structure. Recently, with the development of continuous model theory, applications to certain branches of analysis which were previously out of reach for model theory have become more tractable. In this proposal, I suggest a systematic model theoretic study of operator algebras based on recent success of work of myself and co-authors in settling questions related to ultrapowers of separable algebras. There seems to be the intriguing possibility that model theory can provide assistance with problems related to questions about particular tracial von Neumann algebras such as II$_1$ factors, and a critical classes of C*-algebras such as nuclear, separable C*-algebras. The short term objective is to widen the understanding of model theoretic methods in application areas but in the long run, with a better understanding of the techniques from both fields, it will be possible to prove significant results on the borders of two of Canada's strong research communities.