Majorization in von Neumann algebras, and local multipliers of C* algebras

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Operator algebras are the response by mathematicians (mostly John von Neumann, initially) to the development of quantum mechanics. Physicists found themselves using "mathematical" objects that somehow were natural but that did not make sense from the point of view of the then known mathematics. The algebras studied by von Neumann did not satisfy the initial goal of setting the framework for quantum mechanics, but surprisingly they achieved way more than that. During the last 50 years operator algebras have provided insight into areas as varied as quantum field theory, knot-theory, quantum computing, and logic, among many others.