On von Neumann's Examples of Types

TL;DR

This paper revisits von Neumann's examples of algebraic factors, exploring their physical motivations, clarifying their role in factor classification, and proposing extensions to non-separable Hilbert spaces for infinite systems.

Contribution

It offers a new perspective on von Neumann's factor examples, linking them to physical motivations and extending the theory to non-separable Hilbert spaces.

Findings

Clarifies the role of von Neumann's factors in algebraic classification

Proposes a novel approach to representing infinite systems with gauge fields

Suggests extensions of factor theory to non-separable Hilbert spaces

Abstract

The paper introduces in a new although maybe unusual form the examples of types provided by J. von Neumann and F.J. Murray in their outstanding papers on algebraic factorization (1936-1943)pursuing three main aims: 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 those authors with the purpose of clarifying the general concepts standing at the base of the classification of 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.