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In mathematics, and more specifically in the theory of von Neumann algebras, a crossed product is a basic method of constructing a new von Neumann algebra from a von Neumann algebra acted on by a group. It is related to the semidirect product construction for groups. (Roughly speaking, ''crossed product'' is the expected structure for a group ring of a semidirect product group. Therefore crossed products have a ring theory aspect also. This article concentrates on an important case, where they appear in functional analysis.) ==Motivation== Recall that if we have two finite groups and ''N'' with an action of ''G'' on ''N'' we can form the semidirect product . This contains ''N'' as a normal subgroup, and the action of ''G'' on ''N'' is given by conjugation in the semidirect product. We can replace ''N'' by its complex group algebra ''C''(), and again form a product in a similar way; this algebra is a sum of subspaces ''gC''() as ''g'' runs through the elements of ''G'', and is the group algebra of . We can generalize this construction further by replacing ''C''() by any algebra ''A'' acted on by ''G'' to get a crossed product , which is the sum of subspaces ''gA'' and where the action of ''G'' on ''A'' is given by conjugation in the crossed product. The crossed product of a von Neumann algebra by a group ''G'' acting on it is similar except that we have to be more careful about topologies, and need to construct a Hilbert space acted on by the crossed product. (Note that the von Neumann algebra crossed product is usually larger than the algebraic crossed product discussed above; in fact it is some sort of completion of the algebraic crossed product.) In physics, this structure appears in presence of the so called gauge group of the first kind. ''G'' is the gauge group, and ''N'' the "field" algebra. The observables are then defined as the fixed points of ''N'' under the action of ''G''. A result by Doplicher, Haag and Roberts says that under some assumptions the crossed product can be recovered from the algebra of observables. ==Construction== Suppose that ''A'' is a von Neumann algebra of operators acting on a Hilbert space ''H'' and ''G'' is a discrete group acting on ''A''. We let ''K'' be the Hilbert space of all square summable ''H''-valued functions on ''G''. There is an action of ''A'' on ''K'' given by *a(k)(g) = g−1(a)k(g) for ''k'' in ''K'', ''g'', ''h'' in ''G'', and ''a'' in ''A'', and there is an action of ''G'' on ''K'' given by *g(k)(h) = k(g−1h). The crossed product is the von Neumann algebra acting on ''K'' generated by the actions of ''A'' and ''G'' on ''K''. It does not depend (up to isomorphism) on the choice of the Hilbert space ''H''. This construction can be extended to work for any locally compact group ''G'' acting on any von Neumann algebra ''A''. When is an abelian von Neumann algebra, this is the original group-measure space construction of Murray and von Neumann. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Crossed product」の詳細全文を読む スポンサード リンク
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