In mathematics and functional analysis a direct integral is a generalization of the concept of direct sum. The theory is most developed for direct integrals of Hilbert spaces and direct integrals of von Neumann algebras. 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*-algebras 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 systems of imprimitivity and his general theory of induced representations of locally compact separable groups.
Read more about Direct Integral: Direct Integrals of Hilbert Spaces, Decomposable Operators, Direct Integrals of Von Neumann Algebras, Central Decomposition, Measurable Families of Representations
Famous quotes containing the words direct and/or integral:
“One merit in Carlyle, let the subject be what it may, is the freedom of prospect he allows, the entire absence of cant and dogma. He removes many cartloads of rubbish, and leaves open a broad highway. His writings are all unfenced on the side of the future and the possible. Though he does but inadvertently direct our eyes to the open heavens, nevertheless he lets us wander broadly underneath, and shows them to us reflected in innumerable pools and lakes.”
—Henry David Thoreau (1817–1862)
“An island always pleases my imagination, even the smallest, as a small continent and integral portion of the globe. I have a fancy for building my hut on one. Even a bare, grassy isle, which I can see entirely over at a glance, has some undefined and mysterious charm for me.”
—Henry David Thoreau (1817–1862)