Von Neumann algebra
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A von Neumann algebra 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. Von Neumann algebras are also called W*-algebras. Von Neumann algebras are automatically C*-algebras. They are named for John von Neumann, a name suggested by Jacques Dixmier. They were believed by von Neumann to abstractly capture the concept of an algebra of observables in quantum mechanics.
The von Neumann bicommutant theorem gives another description of von Neumann algebras, using algebraic rather than topological properties.
There are two basic examples of von Neumann algebras to keep in mind. Firstly, if X is a space with a <math> \sigma <math> -finite measure <math> \mu <math>; and <math> L^2(X,\mu) <math> is the Hilbert space of complex-valued square-integrable functions on X, then the space <math> B(L^2(X,\mu)) <math> of bounded linear operators on this space is a (highly non-commutative) von Neumann algebra. Inside this algebra we have the sub-algebra <math> L^\infty (X,\mu) <math> of bounded multiplication operators
- <math> \psi \mapsto f \psi, \quad \psi \in L^2_\mu(X) <math>
which in fact is the most general example of a commutative von Neumann algebra as is stated below.
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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 locally compact 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. 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.
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 is isomorphic to a direct integral of factors; thus, the problem of classifying isomorphism classes of von Neumann algebras can be reduced to that of classifying isomorphism classes of 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 In, and the bounded operators on a separable infinite-dimensional Hilbert space, a factor of type I∞.
A factor is said to be of type II if there are 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 most important factors of type II are the hyperfinite type II1 factor and the hyperfinite type II∞ factor, but there are others.
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 IIIλ, where λ is a real number in the interval [0,1].
The type classification can be extended to von Neumann algebras that are not factors by defining a von Neumann algebra to be 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.
Applications
Von Neumann algebras have found applications in diverse areas of mathematics like knot theory, statistical mechanics, representation theory, geometry and probability.