Dissertation Defense: "Operator noncommutative function theory and partial matrix and operator convexity"
Abstract: This dissertation is composed of two pillars. The first details a purely operatorial notion of noncommutative function theory. We explore inversion theory in this context. In particular, we prove inverse and implicit function theorems. The second pillar concerns a general framework for partial noncommutative convexity, called $\Gamma$-convexity. Many well-studied examples of partial convexity fall under the framework of $\Gamma$-convexity. Several results along the lines of the Hahn-Banach Effros-Winkler separation theorem are established in this general setting. The two pillars of this dissertation are connected by a construction that yields an operator noncommutative version of the Heine-Borel theorem.
Host: John McCarthy
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