$\mathcal{R}$ We Living in the Matrix?

Abstract

IntroductionIn the early years of the twentieth century, the foundations for quantum mechanics were laid out by Dirac, Heisenberg, Bohr, Schrödinger, and others. In his work on the foundations of quantum mechanics, John von Neumann postulated that physical phenomena should be modeled in terms of Hilbert spaces and operators, with observables corresponding to self-adjoint operators and states corresponding to unit vectors. Motivated by his interest in the theory of single operators, he would introduce the notion of what is now termed a von Neumann algebra. Von Neumann and Francis Murray subsequently published a series of fundamental papers, beginning with "On rings of operators" [13], that develop the basic properties of these algebras and establish operator algebras as an independent field of study.In the years after Murray and von Neumann's initial work, the field of operator algebras developed rapidly and split into subfields including * -algebras and von Neumann algebras. Moreover, operator algebraists began to examine generalizations of these objects, such as operator spaces and operator systems. The importance of operator algebras can be witnessed by its applications in Voiculescu's free probability theory, Popa's deformation/rigidity theory, and Jones' theory of subfactors. These areas give us insight into numerous fields, including random matrix theory, quantum field theory, ergodic theory, and knot theory.In a landmark paper unraveling the isomorphism classes of injective von Neumann algebras, Connes proves that it is possible to construct a sequence of approximate embeddings for a large class of von Neumann algebras into finite-dimensional matrix algebras; Connes somewhat casually remarks that this property should hold for all separable von Neumann algebras. Formally, Connes' embedding problem, as this assertion is now called, asks if every type II 1 factor acting on a separable Hilbert space is embeddable into an ultrapower of the hyperfinite II 1 factor via a nonprinciple ultrafilter.Our goal is to unravel the meaning behind Connes' embedding problem and to highlight its significance by providing equivalent formulations that have driven research in the field.