crossed product
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In mathematics, and more specifically in the theory of
von Neumann algebra In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra. Von Neumann algebra ...
s, a crossed product is a basic method of constructing a new von Neumann algebra from a von Neumann algebra acted on by a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
. It is related to the semidirect product construction for groups. (Roughly speaking, ''crossed product'' is the expected structure for a
group ring In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the giv ...
of a semidirect product group. Therefore crossed products have a
ring theory In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their r ...
aspect also. This article concentrates on an important case, where they appear in
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...
.)


Motivation

Recall that if we have two finite groups G and ''N'' with an action of ''G'' on ''N'' we can form the semidirect product N \rtimes G. This contains ''N'' as a normal subgroup, and the action of ''G'' on ''N'' is given by
conjugation Conjugation or conjugate may refer to: Linguistics * Grammatical conjugation, the modification of a verb from its basic form * Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics * Complex conjugation, the chang ...
in the semidirect product. We can replace ''N'' by its complex group algebra ''C'' 'N'' and again form a product C \rtimes G in a similar way; this algebra is a sum of subspaces ''gC'' 'N''as ''g'' runs through the elements of ''G'', and is the group algebra of N \rtimes G. We can generalize this construction further by replacing ''C'' 'N''by any algebra ''A'' acted on by ''G'' to get a crossed product A \rtimes G, 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 In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
, 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 In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra. 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 A \rtimes G 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 In mathematics, a locally compact group is a topological group ''G'' for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are loc ...
''G'' acting on any von Neumann algebra ''A''. When A is an
abelian von Neumann algebra In functional analysis, an abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all elements commute. The prototypical example of an abelian von Neumann algebra is the algebra ''L''∞(''X'', μ) for μ a ...
, this is the original group-measure space construction of Murray and
von Neumann Von Neumann may refer to: * John von Neumann (1903–1957), a Hungarian American mathematician * Von Neumann family * Von Neumann (surname), a German surname * Von Neumann (crater), a lunar impact crater See also * Von Neumann algebra * Von Ne ...
.


Properties

We let ''G'' be an infinite countable discrete group acting on the abelian von Neumann algebra ''A''. The action is called free if ''A'' has no non-zero projections ''p'' such that some nontrivial ''g'' fixes all elements of ''pAp''. The action is called ergodic if the only invariant projections are 0 and 1. Usually ''A'' can be identified as the abelian von Neumann algebra L^\infty(X) of essentially bounded functions on a measure space ''X'' acted on by ''G'', and then the action of ''G'' on ''X'' is ergodic (for any measurable invariant subset, either the subset or its complement has measure 0) if and only if the action of ''G'' on ''A'' is ergodic. If the action of ''G'' on ''A'' is free and ergodic then the crossed product A \rtimes G is a factor. Moreover: * The factor is of type I if ''A'' has a minimal projection such that 1 is the sum of the ''G'' conjugates of this projection. This corresponds to the action of ''G'' on ''X'' being transitive. Example: ''X'' is the integers, and ''G'' is the group of integers acting by translations. *The factor has type II1 if ''A'' has a faithful finite normal ''G''-invariant trace. This corresponds to ''X'' having a finite ''G'' invariant measure, absolutely continuous with respect to the measure on ''X''. Example: ''X'' is the unit circle in the complex plane, and ''G'' is the group of all roots of unity. *The factor has type II if it is not of types I or II1 and has a faithful semifinite normal ''G''-invariant trace. This corresponds to ''X'' having an infinite ''G'' invariant measure without atoms, absolutely continuous with respect to the measure on ''X''. Example: ''X'' is the real line, and ''G'' is the group of rationals acting by translations. *The factor has type III if ''A'' has no faithful semifinite normal ''G''-invariant trace. This corresponds to ''X'' having no non-zero absolutely continuous ''G''-invariant measure. Example: ''X'' is the real line, and ''G'' is the group of all transformations ''ax''+''b'' for ''a'' and ''b'' rational, ''a'' non-zero. In particular one can construct examples of all the different types of factors as crossed products.


Duality

If A is a
von Neumann algebra In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra. Von Neumann algebra ...
on which a locally compact Abelian G acts, then \Gamma, the dual group of
characters Character or Characters may refer to: Arts, entertainment, and media Literature * ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk * ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
\chi of G, acts by unitaries on K : * (\chi\cdot k)(h) =\chi(h) k(h) These unitaries normalise the crossed product, defining the dual action of \Gamma. Together with the crossed product, they generate A\otimes B(L^2(G)), which can be identified with the iterated crossed product by the dual action (A\rtimes G) \rtimes \Gamma . Under this identification, the double dual action of G (the dual group of \Gamma) corresponds to the tensor product of the original action on A and conjugation by the following unitaries on L^2(G) : * (g\cdot f)(h)=f(hg) The crossed product may be identified with the fixed point algebra of the double dual action. More generally A is the fixed point algebra of \Gamma in the crossed product. Similar statements hold when G is replaced by a non-Abelian locally compact group or more generally a locally compact quantum group, a class of Hopf algebra related to
von Neumann algebra In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra. Von Neumann algebra ...
s. An analogous theory has also been developed for actions on C* algebras and their crossed products. Duality first appeared for actions of the reals in the work of Connes and Takesaki on the classification of Type III factors. According to
Tomita–Takesaki theory In the theory of von Neumann algebras, a part of the mathematical field of functional analysis, Tomita–Takesaki theory is a method for constructing modular automorphisms of von Neumann algebras from the polar decomposition of a certain involution ...
, every vector which is cyclic for the factor and its commutant gives rise to a 1-parameter ''modular automorphism group''. The corresponding crossed product is a Type II_\infty
von Neumann algebra In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra. Von Neumann algebra ...
and the corresponding dual action restricts to an ergodic action of the reals on its centre, an
Abelian von Neumann algebra In functional analysis, an abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all elements commute. The prototypical example of an abelian von Neumann algebra is the algebra ''L''∞(''X'', μ) for μ a ...
. This
ergodic flow In mathematics, ergodic flows occur in geometry, through the geodesic and horocycle flows of closed hyperbolic surfaces. Both of these examples have been understood in terms of the theory of unitary representations of locally compact groups: if ...
is called the ''flow of weights''; it is independent of the choice of cyclic vector. The ''Connes spectrum'', a closed subgroup of the
positive reals In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used fo ...
+, is obtained by applying the exponential to the kernel of this flow. * When the kernel is the whole of R, the factor is type III_1. * When the kernel is (\log \lambda) Z for \lambda in (0,1), the factor is type III_\lambda. * When the kernel is trivial, the factor is type III_0. Connes and Haagerup proved that the Connes spectrum and the flow of weights are complete invariants of hyperfinite Type III factors. From this classification and results in ergodic theory, it is known that every infinite-dimensional hyperfinite factor has the form L^\infty(X)\rtimes Z for some free ergodic action of Z.


Examples

*If we take C to be the complex numbers, then the crossed product C \rtimes G is called the von Neumann group algebra of ''G''. *If G is an infinite discrete group such that every conjugacy class has infinite order then the von Neumann group algebra is a factor of type II1. Moreover if every finite set of elements of G generates a finite subgroup (or more generally if ''G'' is amenable) then the factor is the hyperfinite factor of type II1.


See also

* Crossed product algebra


References

* , (II), (III) * * {{Citation , last1=Pedersen , first1=Gert Kjaergard , title=C*-algebras and their automorphism groups , publisher=
Academic Press Academic Press (AP) is an academic book publisher founded in 1941. It was acquired by Harcourt, Brace & World in 1969. Reed Elsevier bought Harcourt in 2000, and Academic Press is now an imprint of Elsevier. Academic Press publishes referen ...
, location=Boston, MA , series=London Math. Soc. Monographs , isbn=978-0-12-549450-2 , year=1979 , volume=14 Operator theory Von Neumann algebras