Editing Direct integral (section)

{{Short description|Generalization of the concept of direct sum in mathematics}}
{{More footnotes needed|date=May 2025}}
In [[mathematics]] and [[functional analysis]], a '''direct integral''' or '''Hilbert integral''' is a generalization of the concept of [[direct sum]]. The theory is most developed for direct integrals of [[Hilbert space]]s and direct integrals of [[von Neumann algebra]]s. The concept was introduced in 1949 by [[John von Neumann]] in one of the papers in the series ''On Rings of Operators''. One of von Neumann's goals in this paper was to reduce the classification of (what are now called) von Neumann algebras on separable Hilbert spaces to the classification of so-called factors. Factors are analogous to full matrix algebras over a field, and von Neumann wanted to prove a continuous analogue of the [[Artin–Wedderburn theorem]] classifying semi-simple rings.

Results on direct integrals can be viewed as generalizations of results about finite-dimensional [[C*-algebra]]s of matrices; in this case the results are easy to prove directly. The infinite-dimensional case is complicated by measure-theoretic technicalities.

Direct integral theory was also used by [[George Mackey]] in his analysis of [[system of imprimitivity|systems of imprimitivity]] and his general theory of [[induced representation]]s of [[locally compact group|locally compact separable groups]].