On von Neumann’s Examples of Types

The factorization properties of operator algebras in separable Hilbert spaces was developed by John von Neumann in collaboration with F.J. Murray in four outstanding papers published from 1936 to 1943. Unfortunately, probably because of conceptual difficulties with physical interpretations, some relevant results presented in those papers, in particular the remarkable examples of factors, have not been adequately considered so far. The paper here presented aims to introduce the subject in a new although maybe unusual form pursuing three main goals: speculating about the physical reasons and motivations that are likely to have been at the origin of von Neumann’s investigation; describing the examples of factors provided by von Neumann and Murray with the purpose of clarifying the general concepts standing at the base of the classification of algebraic factors into three general types; outlining the perspective of extending the theory to non–separable Hilbert spaces with the purpose of suggesting a novel approach to the representation of infinite systems controlled by external gauge fields.