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 algebras were originally introduced by John von Neumann, motivated by his study of single operators, group representations, ergodic theory and quantum mechanics. His double commutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries.
Two basic examples of von Neumann algebras are as follows:

The ring of essentially bounded measurable functions on the real line is a commutative von Neumann algebra, whose elements act as multiplication operators by pointwise multiplication on the Hilbert space of square-integrable functions.
The algebra of all bounded operators on a Hilbert space is a von Neumann algebra, non-commutative if the Hilbert space has dimension at least.

Von Neumann algebras were first studied by in 1929; he and Francis Murray developed the basic theory, under the original name of rings of operators, in a series of papers written in the 1930s and 1940s, reprinted in the collected works of.
Introductory accounts of von Neumann algebras are given in the online notes of and and the books by,, and. The three volume work by gives an encyclopedic account of the theory. The book by discusses more advanced topics.

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 containing the identity. In this definition the weak topology can be replaced by many other common topologies including the strong, ultrastrong or ultraweak operator 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 subalgebra of the bounded operators closed under involution and equal to its double commutant, or equivalently the commutant of some subalgebra closed under *. The von Neumann double commutant theorem says that the first two definitions are equivalent.
The first two definitions describe a von Neumann algebra concretely as a set of operators acting on some given Hilbert space. showed that 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 in fact 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 .

Terminology

Some of the terminology in von Neumann algebra theory can be confusing, and the terms often have different meanings outside the subject.

A factor is a von Neumann algebra with trivial center, i.e. a center consisting only of scalar operators.
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.

By forgetting about the topology on a von Neumann algebra, we can consider it a *-algebra, or just a ring. Von Neumann algebras are semihereditary: every finitely generated submodule of a projective module is itself projective. There have been several attempts to axiomatize the underlying rings of von Neumann algebras, including Baer *-rings and AW*-algebras. The *-algebra of affiliated operators of a finite von Neumann algebra is a von Neumann regular ring.

Commutative von Neumann algebras

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^∞ for some measure space and conversely, for every σ-finite measure space X, the *-algebra L^∞ 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; they are exactly the operators which give an orthogonal projection of H onto some closed subspace. A subspace of the Hilbert space H is said to belong to the von Neumann algebra M if it is the image of some projection in M. This establishes a 1:1 correspondence between projections of M and subspaces that belong to M. Informally these are the closed subspaces that can be described using elements of M, or that M "knows" about.
It can be shown that the closure of the image of any operator in M and the kernel of any operator in M belongs to M. Also, the closure of the image under an operator of M of any subspace belonging to M also belongs to M. .

Comparison theory of projections

The basic theory of projections was worked out by. Two subspaces belonging to M are called equivalent if there is a partial isometry mapping the first isomorphically onto the other that is an element of the von Neumann algebra. This induces a natural equivalence relation on projections by defining E to be equivalent to F if the corresponding subspaces are equivalent, or in other words 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=uu* and F=u*u for some partial isometry u in M.
The equivalence relation ~ thus defined is additive in the following sense: Suppose E₁ ~ F₁ and E₂ ~ F₂. If E₁ ⊥ E₂ and F₁ ⊥ F₂, then E₁ + E₂ ~ F₁ + F₂. Additivity would not generally hold if one were to require unitary equivalence in the definition of ~, i.e. if we say E is equivalent to F if u*Eu = F for some unitary u. The Schröder–Bernstein theorems for operator algebras gives a sufficient condition for Murray-von Neumann equivalence.
The subspaces belonging to M are partially ordered by inclusion, and this induces a partial order ≤ of projections. There is also a natural partial order on the set of equivalence classes of projections, induced by the partial order ≤ of projections. If M is a factor, ≤ is a total order on equivalence classes of projections, described in the section on traces below.
A projection E is said to be a finite projection if there is no projection F < E that is equivalent to E. For example, all finite-dimensional projections are finite, 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. However it is possible for infinite dimensional subspaces to be finite.
Orthogonal projections are noncommutative analogues of indicator functions in L^∞. L^∞ is the ||·||_∞-closure of the subspace generated by the indicator functions. Similarly, a von Neumann algebra is generated by its projections; this is a consequence of the spectral theorem for self-adjoint operators.
The projections of a finite factor form a continuous geometry.

Factors

A von Neumann algebra N whose center consists only of multiples of the identity operator is called a factor. As showed, 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.
showed that 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 if it has type I, and continuous 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.