Spaces of von Neumann Type

Abstract

This chapter develops the mathematical foundations for the time-evolution of a physical systems as a three-dimensional motion picture (time-ordering). Our objective is to construct the mathematical version of a physical film on which space-time events can evolve. We first construct the film using infinite tensor products of Hilbert spaces, which is natural for physics. Although von Neumann [VN2] did not develop his theory for our purpose, it will be clear that it is natural for our approach. This film, as a Hilbert space, will be used as the ambient space in Chap. 7 for the Feynman (time-ordered) operator calculus. In order to make the theory available for applications beyond physics, we extend von Neumann’s method to construct infinite tensor products of Banach spaces. (This approach makes it easy to transfer the operator calculus to the Banach space setting.) We assume that the reader has read Sect. 1.4 of Chap. 1 This section provides a fairly complete introduction to the finite tensor product theory for both Hilbert and Banach spaces.