Crossed product - Citizendia

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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the It is related to the semidirect product construction for groups. In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can (Roughly speaking, crossed product is the expected structure for a group ring of a semidirect product group. In Mathematics, a group ring is a ring R constructed from a ring R and a group G (written multiplicatively Therefore crossed products have a ring theory aspect also. In Mathematics, ring theory is the study of rings, Algebraic structures in which addition and multiplication are defined and have similar properties to those This article concentrates on an important case, where they appear in functional analysis. For functional analysis as used in psychology see the Functional analysis (psychology article )

1 Motivation
2 Construction
3 Properties
4 Duality
5 Examples
6 References

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 in the semidirect product. We can replace N by its complex group algebra C[N], and again form a product $C[N] \rtimes G$ 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, 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. )

Construction

Suppose that A is an abelian von Neumann algebra of operators acting on a Hilbert space H and G is a discrete group acting on A. In Functional analysis, an Abelian von Neumann algebra is a Von Neumann algebra of operators on a Hilbert space in which all elements commute 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 H. 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 $A$ is an abelian von Neumann algebra, this is the original group-measure space construction of Murray and von Neumann. In Functional analysis, an Abelian von Neumann algebra is a Von Neumann algebra of operators on a Hilbert space in which all elements commute

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. In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with

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 on which a locally compact Abelian $G$ acts, then $Γ$ , the dual group of characters $χ$ 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 $Γ$ . In Mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the In Mathematics, a character is (most commonly a special kind of function from a group to a field (such as the Complex numbers) 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 $Γ$ ) 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 $Γ$ 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 algebras. Non-abelian may describe Non-abelian group, in mathematics a group that is not abelian (commutative Non-abelian gauge theory, in physics The locally compact (lc quantum group is a relatively new C*-algebraic formalism for Quantum groups generalizing the Kac algebra, Compact quantum In Mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a ( Unital associative algebra, a Coalgebra In Mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the An analogous theory has also been developed for actions on C* algebras and their crossed products. C*-algebras (pronounced "C-star" are an important area of research in Functional analysis, a branch of Mathematics.

Duality first appeared for actions of the reals in the work of Connes and Takesaki on the classification of Type III factors. In Mathematics, the real numbers may be described informally in several different ways Alain Connes (born 1 April 1947 is a French Mathematician, currently Professor at the College de France, IHÉS and Vanderbilt University In Mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the According to Tomita-Takesaki theory, every vector which is cyclic for the factor and its commutant gives rise to a 1-parameter modular automorphism group. 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 In Algebra, the commutant of a Subset S of a Semigroup (such as an algebra or a group) A is the subset The corresponding crossed product is a Type $II_\infty$ 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 Mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the In Mathematics and Physics, the adjective ergodic is used to imply that a system satisfies the Ergodic hypothesis of Thermodynamics or that In Mathematics, the real numbers may be described informally in several different ways In Functional analysis, an Abelian von Neumann algebra is a Von Neumann algebra of operators on a Hilbert space in which all elements commute This ergodic flow is called the flow of weights; it is independent of the choice of cyclic vector. Ergodic theory is a branch of Mathematics that studies Dynamical systems with an Invariant measure and related problems The Connes spectrum, a closed subgroup of the positive reals, is obtained by applying the exponential to the kernel of this flow. In Mathematics, the real numbers may be described informally in several different ways

When the kernel is the whole of $R$ , the factor is type $I I I 1$ .
When the kernel is $(logλ) Z$ for $λ$ in (0,1), the factor is type $I I I λ$ .
When the kernel is trivial, the factor is type $I I I 0$ .

Connes and Haagerup proved that the Connes spectrum and the flow of weights are complete invariants of hyperfinite Type III factors. Alain Connes (born 1 April 1947 is a French Mathematician, currently Professor at the College de France, IHÉS and Vanderbilt University In Mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the In Mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the 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 $Z$ . Ergodic theory is a branch of Mathematics that studies Dynamical systems with an Invariant measure and related problems

Examples

If we take the algebra A to be the complex numbers C, then the crossed product $A \rtimes G$ 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.

References

Takesaki, Masamichi, Theory of Operator Algebras I, II, III, Berlin, New York: Springer-Verlag, ISBN 978-3-540-42248-8 , ISBN 3-540-42914-X (II), ISBN 3-540-42913-1 (III)
Connes, Alain (1994), Non-commutative geometry, Boston, MA: Academic Press, ISBN 978-0-12-185860-5, <ftp://ftp.alainconnes.org/book94bigpdf.pdf>
Pedersen, Gert Kjaergard (1979), C*-algebras and their automorphism groups, vol. Springer Science+Business Media or Springer (ˈʃpʁɪŋɐ is a worldwide Publishing company based in Germany, which publishes textbooks academic Alain Connes (born 1 April 1947 is a French Mathematician, currently Professor at the College de France, IHÉS and Vanderbilt University Academic Press ( London, New York and San Diego) was an Academic Book Publisher that is now part of Elsevier. 14, London Math. Soc. Monographs, Boston, MA: Academic Press, ISBN 978-0-12-549450-2

Academic Press ( London, New York and San Diego) was an Academic Book Publisher that is now part of Elsevier.

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