Von Neumann algebra

A von Neumann algebra (named in honour of John von Neumann) is a *-algebra of bounded operators on a Hilbert space which is closed in the weak operator topology, or equivalently, in the strong operator topology (under pointwise convergence) and contains the identity operator. They were believed by John von Neumann to capture abstractly the concept of an algebra of observables in quantum mechanics. Von Neumann algebras are automatically C*-algebras. The von Neumann bicommutant theorem gives another description of von Neumann algebras, using algebraic rather than topological properties.

Von Neumann algebras are also called W*-algebras. The more common name was suggested by Jacques Dixmier.

There are two basic examples of von Neumann algebras to keep in mind. Firstly, if X is a space with a $\sigma$ -finite^[1] measure $\mu$ and $L^{2}(X,\mu )$ is the Hilbert space of complex-valued square-integrable functions on X, then the space $B(L^{2}(X,\mu ))$ of bounded linear operators on this space is a von Neumann algebra. Inside this algebra we have the sub-algebra $L^{\infty }(X,\mu )$ of bounded multiplication operators

\psi \mapsto f\psi ,\quad \psi \in L_{\mu }^{2}(X)

which in fact is the most general example of a commutative von Neumann algebra as is stated below.

Definitions

There are three common ways to define von Neumann algebras.

The first and most common way is to define them as weakly closed * algebras of bounded operators (on a Hilbert space) containing the identity. In this definition the weak (operator) topology can be replaced by almost any other common topology other than the norm topology, in particular by the strong or ultrastrong topologies. (The * algebras of bounded operators that are closed in the norm topology are C^* algebras, so in particular any von Neumann algebra is a C^* algebra.)

The second definition is that a von Neumann algebra is a subset of the bounded operators closed under * and equal to its double commutator, or equivalently the commutator of some subset closed under *. The von Neumann bicommutant theorem says that the first two definitions are equivalent.

The first two definitions define a von Neumann algebras concretely as a set of operators acting on some given Hilbert space. Von Neumann algebras can also be defined abstractly as C^* algebras that have a predual; in other words the von Neumann algebra, considered as a Banach space, is the dual of some other Banach space called the predual. The predual of a von Neumann algebra is unique up to isomorphism. Some authors use "von Neumann algebra" for the algebras together with a Hilbert space action, and "W^* algebra" for the abstract concept, so a von Neumann algebra is a W^* algebra together with a Hilbert space and a suitable faithful unital action on the Hilbert space. The concrete and abstract definitions of a von Neumann algebra are similar to the concrete and abstract definitions of a C^* algebra, which can be defined either as norm-closed * algebras of operators on a Hilbert space, or as Banach *-algebras such that ||a a^*||=||a|| ||a^*||.

Commutative von Neumann algebras

Main article: Abelian von Neumann algebra

The relationship between commutative von Neumann algebras and measure spaces is analogous to that between commutative C*-algebras and locally compact Hausdorff spaces. Every commutative von Neumann algebra is isomorphic to L^∞(X) for some measure space (X, μ) and for every σ-finite measure space X, conversely, L^∞(X) is a von Neumann algebra.

Due to this analogy, the theory of von Neumann algebras has been called noncommutative measure theory, while the theory of C*-algebras is sometimes called noncommutative topology.

Projections

Operators E in a von Neumann algebra for which E = EE = E* are called projections. There is a natural equivalence relation on projections by defining E to be equivalent to F if there is a partial isometry of H that maps the image of E isometrically to the image of F and is an element of the von Neumann algebra. Another way of stating this is that E is equivalent to F if E=aa^* and F=a^*a for some a. There is also a natural partial order on the set of isomorphism classes of projections, induced by the partial order of the von Neumann algebra. For factors this is a total order, described in the section on traces below.

A projection E is said to be finite if there is no projection F < E that is equivalent to E. For example, all finite-dimensional projections are finite (since isometries between Hilbert spaces leave the dimension fixed), but the identity operator on an infinite-dimensional Hilbert space is not finite in the von Neumann algebra of all bounded operators on it, since it is isometrically isomorphic to a proper subset of itself.

Factors

A von Neumann algebra N whose center consists only of multiples of the identity operator is called a factor. Every von Neumann algebra on a separable Hilbert space is isomorphic to a direct integral of factors. This decomposition is essentially unique. Thus, the problem of classifying isomorphism classes of von Neumann algebras on separable Hilbert spaces can be reduced to that of classifying isomorphism classes of factors.

Every factor has one of 3 types as described below. The type classification can be extended to von Neumann algebras that are not factors, and a von Neumann algebra is of type X if it can be decomposed as a direct integral of type X factors; for example, every commutative von Neumann algebra has type I₁. Every von Neumann algebra can be written uniquely as a sum of von Neumann algebras of types I, II, and III.

There are several other ways to divide factors into classes that are sometimes used:

A factor is called discrete (or occasionally tame) if it has type I, and continuous (or occasionally wild) if it has type II or III.

A factor is called semifinite if it has type I or II, and purely infinite if it has type III.

A factor is called finite if the projection 1 is finite and properly infinite otherwise. Factors of types I and II may be either finite or properly infinite, but factors of type III are always properly infinite.

Type I factors

A factor is said to be of type I if there is a minimal projection, i.e. a projection E such that there is no other projection F with 0 < F < E. Any factor of type I is isomorphic to the von Neumann algebra of all bounded operators on some Hilbert space; since there is one Hilbert space for every cardinal number, isomorphism classes of factors of type I correspond exactly to the cardinal numbers. Since many authors consider von Neumann algebras only on separable Hilbert spaces, it is customary to call the bounded operators on a Hilbert space of finite dimension n a factor of type I_n, and the bounded operators on a separable infinite-dimensional Hilbert space, a factor of type I_∞.

Type II factors

A factor is said to be of type II if there are non-zero finite projections, but every projection E can be halved in the sense that there are equivalent projections F and G such that E = F + G. If the identity operator in a type II factor is finite, the factor is said to be of type II₁; otherwise, it is said to be of type II_∞. The best understood factors of type II are the hyperfinite type II₁ factor and the hyperfinite type II_∞ factor. These are the unique hyperfinite factors of types II₁ and II_∞; there are an uncountable number of other factors of these types that are the subject of intensive study. A factor of type II₁ has a unique finite tracial state, and the set of traces of projections is [0,1].

A factor of type II_∞ has a semifinite trace, unique up to rescaling, and the set of traces of projections is [0,∞]. The set of real numbers λ such that there is an automorphism rescaling the trace by a factor of λ is called the fundamental group of the type II_∞ factor.

The tensor product of a factor of type II₁ and an infinite type I factor has type II_∞, and conversely any factor of type II_∞ can be constructed like this. The fundamental group of a type II₁ factor is defined to be the fundamental group of its tensor product with the infinite (separable) factor of type I.

An example of a type II₁ factor is the von Neumann group algebra of a countable infinite discrete group such that every non-trivial conjugacy class is infinite.

Type III factors

Lastly, type III factors are factors that do not contain any nonzero finite projections at all. Since the identity operator is always infinite in those factors, they were sometimes called type III_∞ in the past, but recently that notation has been superseded by the introduction of a family of type III factors called type III_λ, where λ is a real number in the interval [0,1]. The only trace on these factors takes value ∞ on all non-zero positive elements, and any two non-zero projections are equivalent. At one time type III factors were considered to be wild, intractable objects; Tomita-Takesaki theory has led to a good structure theory. In particular, any type III factor can be written in a canonical way as the crossed product of a type II_∞ factor and the real numbers.

Type for C*-algebras

A C*-algebra A is of type I if and only if for all representations π of A such that the von Neumann algebra generated by π(A) (that is the bicommutant π(A)′′) is a factor, it is actually a type I factor. A locally compact group is said to be of type I if and only if its group algebra is type I.

However, if a C*-algebra has non-type I factor representations, then by results of James Glimm it also has representations of type II and type III. Thus for C*-algebras and locally compact groups, it is only meaningful to speak of type I and non type I properties.

Weights, states, and traces.

A weight ω on a von Neumann algebra is a linear map from the set of positive elements (those of the form aa^*) to [0,∞].

A positive linear functional is a weight with ω(1) finite (or rather the extension of ω to the whole algebra by linearity).

A state is a weight with ω(1)=1.

A trace is a weight with ω(aa^*)=ω(a^*a) for all a.

A tracial state is a trace with ω(1)=1.

Any factor has a trace such that the trace of a non-zero projection is non-zero and the trace of a projection is infinite if and only if the projection is infinite. Such a trace is unique up to rescaling. For factors that are separable or finite, two projections are equivalent if and only if they have the same trace. The type of a factor can be read off from the possible values of this trace as follows:

Type I_n: 0, x, 2x, ....,nx for some positive x (usually normalized to be 1/n or 1).
Type I_∞: 0, x, 2x, ....,∞ for some positive x (usually normalized to be 1).
Type II₁: [0,x] for some positive x (usually normalized to be 1).
Type II_∞: [0,∞].
Type III: 0,∞

If a von Neumann algebra acts on a Hilbert space containing a norm 1 vector v, then (av,v) gives a state. This construction can be reversed to give an action on a Hilbert space from a state: this is called the GNS construction.

Amenable von Neumann algebras

Connes and others proved that the following conditions on a von Neumann algebra N on a separable Hilbert space H are all equivalent:

N is hyperfinite or AFD or almost finite dimensional: this means the algebra contains an ascending sequence of finite dimensional subalgebras with dense union.

N is amenable: this means that the derivations of N with values in a normal dual Banach bimodule are all inner.

N has Schwartz's property P: for any bounded operator T on H the norm closed convex hull of the elements uTu^* contains an element commuting with N.

N is semidiscrete: this means the identity map from M to M is a weak pointwise limit of completely positive maps of finite rank.

N has property E or the Hakeda-Tomiyama extension property: this means that there is a projection of norm 1 from bounded operators on H to M '.

N is injective: any completely positive linear map from any self adjoint closed subspace containing 1 of any unital C^*-algebra A to N can be extended to a completely positive map from A to N.

There is no generally accepted term for the class of algebras above; Connes has suggested that amenable should be the standard term.

The amenable factors have been classified: there is a unique one of each of the types I_n, I_∞, II₁, II_∞, III_λ, for 0<λ≤ 1, and the ones of type III₀ correspond to certain ergodic actions. (For type III₀ calling this a classification is a little misleading, as it is known that there is no easy way to classify the corresponding ergodic actions.) Von Neumann algebras of type I are always amenable, but for the other types there are an uncountable number of different non-amenable factors, which seem very hard to classify, or even distinguish from each other.

Algebraic properties

By forgetting about the topology on a von Neumann algebra, we can consider it a *-algebra (which is usually going to be noncommutative, but will always be unital), or just a ring.

Von Neumann algebras are semihereditary: every finitely generated submodule of a projective module is itself projective.

Despite the similarity in name, von Neumann algebras are not, in general, von Neumann regular rings; however, for finite von Neumann algebras, the algebra of affiliated operators, which makes all non-zero divisors invertible, can be constructed, and is von Neumann regular.

Terminology

A finite von Neumann algebra is one which is the direct integral of finite factors. Similarly, properly infinite von Neumann algebras are the direct integral of properly infinite factors.
A von Neumann algebra that acts on a separable Hilbert space is called separable. Note that such algebras are rarely separable in the norm topology.
The von Neumann algebra generated by a set of bounded operators on a Hilbert space is the smallest von Neumann algebra containing all those operators.
The tensor product of two von Neumann algebras acting on two Hilbert spaces is defined to be the von Neumann algebra generated by their algebraic tensor product, considered as operators on the Hilbert space tensor product of the Hilbert spaces.

Examples

The essentially bounded functions on a σ-finite measure space form a commutative (type I₁) von Neumann algebra acting on the L² functions^[1].

The bounded operators on any Hilbert space form a von Neumann algebra, indeed a factor, of type I.

The crossed product of a von Neumann algebra by a discrete (or more generally locally compact) group can be defined, and is a von Neumann algebra.

If we have any unitary representation of a group G on a Hilbert space H then the bounded operators commuting with G form a von Neumann algebra G′, whose projections correspond exactly to the closed subspaces of H invariant under G. The double commutator G′′ of G is also a von Neumann algebra.

The von Neumann group algebra of a discrete group G is the algebra of all bounded operators on $H=\ell ^{2}(G)$ commuting with the action of G on H through right multiplication. One can show that this is the von Neumann algebra generated by the operators corresponding to multiplication from the left with an element $g\in G$ .

One can define a tensor product of von Neumann algebras (a completion of the algebraic tensor product of the algebras considered as rings), which is again a von Neumann algebra. The tensor product of two finite algebras is finite, and the tensor product of an infinite algebra and a non-zero algebra is infinite. The tensor product of two von Neumann algebras of types X and Y (I, II, or III) has type equal to the maximum of X and Y.

The tensor product of an infinite number of von Neumann algebras, if done naively, is usually a ridiculously large non-separable algebra. Instead one usually chooses a state on each of the von Neumann algebras, uses this to define a state on the algebraic tensor product, which can be used to product a Hilbert space and a (reasonably small) von Neumann algebra. The factors of Powers and Araki-Woods were found like this. If all the factors are finite matrix algebras the factors are called ITPFI factors (ITPFI stands for "infinite tensor product of finite type I factors"). The type of the infinite tensor product can vary dramatically as the states are changed; for example, the infinite tensor product of an infinite number of type I₂ factors can have any type depending on the choice of states.

Krieger constructed some type III factors, known as Krieger's factors, from ergodic actions, which can be used to give all the hyperfinite type III₀ factors.

Applications

Von Neumann algebras have found applications in diverse areas of mathematics like knot theory, statistical mechanics, representation theory, geometry and probability.

References

Theory of Operator Algebras I, II, III by M. Takesaki. ISBN 3-540-42248-X ISBN 3-540-42914-X ISBN 3-540-42913-1
Non-commutative geometry by A. Connes, ISBN 0-12-185860-X, also available at ftp://ftp.alainconnes.org/book94bigpdf.pdf

^ ^a ^b For certain non-σ-finite measure spaces, usually considered pathological (mathematics), $L^{\infty }(X)$ is not a von Neumann algebra; for example, the σ-algebra of measurable sets might be the countable-cocountable algebra on an uncountable set.

[pathological_measure_spaces-1] For certain non-σ-finite measure spaces, usually considered pathological (mathematics), $L^{\infty }(X)$ is not a von Neumann algebra; for example, the σ-algebra of measurable sets might be the countable-cocountable algebra on an uncountable set.

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