ABSTRACT
A direct integral decomposition of a separable Hilbert space with respect to an abelian von Neumann algebra. In order to define these “direct integral” Hilbert spaces it will be necessary to be able to compare any separable Hilbert space of a given dimension with a standard, or distinguished, Hilbert space of that dimension. How these standard Hilbert spaces are defined is actually immaterial, although it will be convenient to have them nested; what is important is that they be fixed once and for all. The theory of direct integrals of Hilbert spaces and operators was first studied by von Neumann in the 1930s although his results were not published until 1949. Although coherences are certainly implicit in all treatments of direct integral theory, it was Effros who brought them to the forefront and made them part of the notation. Whether coherences are implicit or explicit does not affect how one thinks about or what one can do with direct integrals.
