Abelian von Neumann algebra
Encyclopedia
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
for μ a σ-finite measure on X realized as an algebra of operators on the Hilbert space 
as follows: Each 
is identified with the multiplication operator
Of particular importance are the abelian von Neumann algebras on separable Hilbert spaces, particularly since they are completely classifiable by simple invariants.
Though there is a theory for von Neumann algebras on non-separable Hilbert spaces (and indeed much of the general theory still holds in that case) the theory is considerably simpler for algebras on separable spaces and most applications to other areas of mathematics or physics only use separable Hilbert spaces. Note that if the measure spaces (X, μ) is a standard measure space (that is X − N is a standard Borel space for some null set N and μ is a σ-finite measure) then L2μ(X) is separable.
s. Every commutative von Neumann algebra on a separable Hilbert space is isomorphic to L∞
(X) for some standard measure space (X, μ) and conversely, for every standard measure space X, L∞(X) is a von Neumann algebra. This isomorphism as stated is an algebraic isomorphism.
In fact we can state this more precisely as follows:
Theorem. Any abelian von Neumann algebra of operators on a separable Hilbert space is *-isomorphic to exactly one of the following
The isomorphism can be chosen to preserve the weak operator topology.
In the above list, the interval [0,1] has Lebesgue measure and the sets {1, 2, ..., n} and N have counting measure. The unions are disjoint unions. This classification is essentially a variant of Maharam's classification theorem for separable measure algebras. The version of Maharam's classification theorem that is most useful involves a point realization of the equivalence, and is somewhat of a folk theorem.
Notice that in the above result, it is necessary to clip away sets of measure zero to make the result work.
In the above theorem, the isomorphism is required to preserve the weak operator topology. As it turns out (and follows easily from the definitions), for algebras L∞μ(X), the following topologies agree on norm bounded sets:
However, for an abelian von Neumann algebra A the realization of A as an algebra of operators on a separable Hilbert space is highly non-unique. The complete classification of the operator algebra realizations of A is given by spectral multiplicity theory and requires the use of direct integral
s.
L2μ(X) are all maximal abelian. This means that they cannot be extended to properly larger abelian algebras. They are also referred to as Maximal abelian self-adjoint algebras (or M.A.S.A.s). Another phrase used to describe them is abelian von Neumann algebras of uniform multiplicity 1; this description makes sense only in relation to multiplicity theory described below.
Von Neumann algebras A on H, B on K are spatially isomorphic (or unitarily isomorphic) if and only if there is a unitary operator U: H → K such that
In particular spatially isomorphic von Neumann algebras are algebraically isomorphic.
To describe the most general abelian von Neumann algebra on a separable Hilbert space H up to spatial isomorphism, we need to refer the direct integral decomposition of H. The details of this decomposition are discussed in decomposition of abelian von Neumann algebras. In particular:
Theorem Any abelian von Neumann algebra on a separable Hilbert space H is spatially isomorphic to L∞μ(X) acting on
for some measurable family of Hilbert spaces {Hx}x ∈ X.
Note that for abelian von Neumann algebras acting on such direct integral spaces, the equivalence of the weak operator topology, the ultraweak topology and the weak* topology on norm bounded sets still hold.
reduce to problems about automorphisms of abelian von Neumann algebras. In that regard, the following results are useful:
Theorem. Suppose μ, ν are standard measures on X, Y respectively. Then any involutive isomorphism
which is weak*-bicontinuous corresponds to a point transformation in the following sense: There are Borel null subsets M of X and N of Y and a Borel isomorphism
such that
Note that in general we cannot expect η to carry μ into ν.
The next result concerns unitary transformations which induce a weak*-bicontinuous isomorphism between abelian von Neumann algebras.
Theorem. Suppose μ, ν are standard measures on X, Y and
for measurable families of Hilbert spaces {Hx}x ∈ X, {Ky}y ∈ Y. If U:H → K is a unitary such that
then there is an almost everywhere defined Borel point transformation η X → Y as in the previous theorem and a measurable family {Ux}x ∈ X of unitary operators
such that
where the expression in square root sign is the Radon–Nikodym derivative
of μ η -1 with respect to ν. The statement follows combining the theorem on point realization of automorphisms stated above with the theorem characterizing the algebra of diagonalizable operators stated in the article on direct integral
s.
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 and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, an abelian von Neumann algebra is a von Neumann algebra
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. They were originally introduced by John von Neumann, motivated by his study of single operators, group...
of operators on a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
in which all elements commute.
The prototypical example of an abelian von Neumann algebra is
the algebra
for μ a σ-finite measure on X realized as an algebra of operators on the Hilbert space as follows: Each is identified with the multiplication operatorOf particular importance are the abelian von Neumann algebras on separable Hilbert spaces, particularly since they are completely classifiable by simple invariants.
Though there is a theory for von Neumann algebras on non-separable Hilbert spaces (and indeed much of the general theory still holds in that case) the theory is considerably simpler for algebras on separable spaces and most applications to other areas of mathematics or physics only use separable Hilbert spaces. Note that if the measure spaces (X, μ) is a standard measure space (that is X − N is a standard Borel space for some null set N and μ is a σ-finite measure) then L2μ(X) is separable.
Classification
The relationship between commutative von Neumann algebras and measure spaces is analogous to that between commutative C*-algebras and locally compact Hausdorff spaceHausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
s. Every commutative von Neumann algebra on a separable Hilbert space is isomorphic to L∞
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
(X) for some standard measure space (X, μ) and conversely, for every standard measure space X, L∞(X) is a von Neumann algebra. This isomorphism as stated is an algebraic isomorphism.
In fact we can state this more precisely as follows:
Theorem. Any abelian von Neumann algebra of operators on a separable Hilbert space is *-isomorphic to exactly one of the following
The isomorphism can be chosen to preserve the weak operator topology.
In the above list, the interval [0,1] has Lebesgue measure and the sets {1, 2, ..., n} and N have counting measure. The unions are disjoint unions. This classification is essentially a variant of Maharam's classification theorem for separable measure algebras. The version of Maharam's classification theorem that is most useful involves a point realization of the equivalence, and is somewhat of a folk theorem.
- Let μ an ν be non-atomic probability measures on standard Borel spaces X and Y respectively. Then there is a μ null subset N of X, a ν null subset M of Y and a Borel isomorphism
- which carries μ into ν.
Notice that in the above result, it is necessary to clip away sets of measure zero to make the result work.
In the above theorem, the isomorphism is required to preserve the weak operator topology. As it turns out (and follows easily from the definitions), for algebras L∞μ(X), the following topologies agree on norm bounded sets:
- The weak operator topology on L∞μ(X);
- The ultraweak operator topology on L∞μ(X);
- The topology of weak* convergence on L∞μ(X) considered as the dual space of L1μ(X).
However, for an abelian von Neumann algebra A the realization of A as an algebra of operators on a separable Hilbert space is highly non-unique. The complete classification of the operator algebra realizations of A is given by spectral multiplicity theory and requires the use of direct integral
Direct integral
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...
s.
Spatial isomorphism
Using direct integral theory, it can be shown that the abelian von Neumann algebras of the form L∞μ(X) acting as operators onL2μ(X) are all maximal abelian. This means that they cannot be extended to properly larger abelian algebras. They are also referred to as Maximal abelian self-adjoint algebras (or M.A.S.A.s). Another phrase used to describe them is abelian von Neumann algebras of uniform multiplicity 1; this description makes sense only in relation to multiplicity theory described below.
Von Neumann algebras A on H, B on K are spatially isomorphic (or unitarily isomorphic) if and only if there is a unitary operator U: H → K such that
In particular spatially isomorphic von Neumann algebras are algebraically isomorphic.
To describe the most general abelian von Neumann algebra on a separable Hilbert space H up to spatial isomorphism, we need to refer the direct integral decomposition of H. The details of this decomposition are discussed in decomposition of abelian von Neumann algebras. In particular:
Theorem Any abelian von Neumann algebra on a separable Hilbert space H is spatially isomorphic to L∞μ(X) acting on
for some measurable family of Hilbert spaces {Hx}x ∈ X.
Note that for abelian von Neumann algebras acting on such direct integral spaces, the equivalence of the weak operator topology, the ultraweak topology and the weak* topology on norm bounded sets still hold.
Point and spatial realization of automorphisms
Many problems in ergodic theoryErgodic theory
Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
reduce to problems about automorphisms of abelian von Neumann algebras. In that regard, the following results are useful:
Theorem. Suppose μ, ν are standard measures on X, Y respectively. Then any involutive isomorphism
which is weak*-bicontinuous corresponds to a point transformation in the following sense: There are Borel null subsets M of X and N of Y and a Borel isomorphism
such that
- η carries the measure μ into a measure μ' on Y which is equivalent to ν in the sense that μ' and ν have the same sets of measure zero;
- η realizes the transformation Φ, that is
Note that in general we cannot expect η to carry μ into ν.
The next result concerns unitary transformations which induce a weak*-bicontinuous isomorphism between abelian von Neumann algebras.
Theorem. Suppose μ, ν are standard measures on X, Y and
for measurable families of Hilbert spaces {Hx}x ∈ X, {Ky}y ∈ Y. If U:H → K is a unitary such that
then there is an almost everywhere defined Borel point transformation η X → Y as in the previous theorem and a measurable family {Ux}x ∈ X of unitary operators
such that
where the expression in square root sign is the Radon–Nikodym derivative
Radon–Nikodym theorem
In mathematics, the Radon–Nikodym theorem is a result in measure theory that states that, given a measurable space , if a σ-finite measure ν on is absolutely continuous with respect to a σ-finite measure μ on , then there is a measurable function f on X and taking values in [0,∞), such that\nu =...
of μ η -1 with respect to ν. The statement follows combining the theorem on point realization of automorphisms stated above with the theorem characterizing the algebra of diagonalizable operators stated in the article on direct integral
Direct integral
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...
s.