John von Neumann and the Theory of Operator Algebras.pdf

John von Neumann andthe Theory of Operator Algebras *Denes Petz 1 and Miklos Redei 2After some earlier work on single operators, von Neumann turned to families ofoperators in [1?. He initiated the study of rings of operators whi h are ommonly alled von Neumann algebras today. The papers whi h onstitute the series \Ringsof operators opened a new eld in mathemati s and in uen ed resear h for half a entury (or even longer). In the standard theory of modern operator algebras, many on epts and ideas have their origin in von Neumanns work. Sin e its in eption,operator algebra theory has been in intimate relation to physi s. The mathemati alformalism of quantum theory was one of the motivations leading naturally to algebrasof Hilbert spa e operators. After de ades of relative isolation, again physi s fertil-ized the operator algebra theory by mathemati al questions of quantum statisti alme hani s and quantum eld theory.The obje tive of the present arti le is two-fold. On the one hand, to sket h theearly development of von Neumann algebras, to show how the fundamental lassi- ation of algebras emerged from the latti e of proje tions. These old ideas of vonNeumann and Murray revived mu h later in onne tion with Jordan operator al-gebras and the K-theory of C*-algebras. On the other hand, to review brie y somerelatively new developments su h as the lassi ation of hypernite fa tors, the indextheory of subfa tors and elements of Jordan algebras. These developments are on-ne ted to the programs initiated by von Neumann himself. The last part of the paperis devoted to those topi s of operator algebra theory whi h are losest to physi alappli ations.Our overview of the lega y of von Neumann in operator algebra theory is neitherentirely histori al nor is it omplete. It re e ts the s ienti taste and knowledge ofthe authors. The theory of operator algebras is a te hni al subje t and to present areadable a ount of the development of many years is a di? ult task. To fa