From Operator Algebras to Complexity Theory and Back

Abstract

Quantum mechanics and the theory of operator algebras have been intertwined since their origin. In the 1930s [20] von Neumann laid the foundations for the theory of (what are now known as) von Neumann algebras with the explicit goal of establishing Heisenberg’s matrix mechanics on a rigorous footing (quoting from the preface, in the translation by Beyer: “The object of this book is to present the new quantum mechanics in a unified representation which, so far as it is possible and useful, is mathematically rigorous”). Following the initial explorations of Murray and von Neumann, the new theory took on a life of its own, eventually leading to multiple applications unrelated to quantum mechanics, such as to free probability or noncommutative geometry.