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:
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 (F.J. Murray &amp; J. von Neumann 1936, 1937, 1943; J. von Neumann 1938, 1940, 1943, 1949), 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 (on a Hilbert space) containing the identity. In this definition the weak (operator) 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 (the *-operation) 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 ||"aa*"||=||"a"|| ||"a*"||.
Terminology.
Some of the terminology in von Neumann algebra theory can be confusing, and the terms often have different meanings outside the subject.
By forgetting about the topology on a von Neumann algebra, we can consider it a (unital) *-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. (The von Neumann algebra itself is in general not von Neumann regular.)
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"∞("X") for some measure space ("X", μ) and conversely, for every σ-finite measure space "X", the *-algebra "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; 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". (These results are a consequence of the polar decomposition).
Comparison theory of projections.
The basic theory of projections was worked out by . Two subspaces belonging to "M" are called (Murray–von Neumann) equivalent if there is a partial isometry mapping the first isomorphically onto the other that is an element of the von Neumann algebra (informally, if "M" "knows" that the subspaces are isomorphic). 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 "E"1 ~ "F"1 and "E"2 ~ "F"2. If "E"1 ⊥ "E"2 and "F"1 ⊥ "F"2, then "E"1 + "E"2 ~ "F"1 + "F"2. 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 (or subspace belonging to "M") "E" is said to be a finite projection if there is no projection "F" ; "E" (meaning "F" ≤ "E" and "F" ≠ "E") that is equivalent to "E". For example, all finite-dimensional projections (or subspaces) 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. However it is possible for infinite dimensional subspaces to be finite.
Orthogonal projections are noncommutative analogues of indicator functions in "L"∞(R). "L"∞(R) 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:
Type I factors.
A factor is said to be of type I if there is a minimal projection "E ≠ 0", 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 no minimal projections but there are non-zero finite projections. This implies that every projection "E" can be "halved" in the sense that there are two projections "F" and "G" that are Murray–von Neumann equivalent and satisfy "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, found by . 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. proved the fundamental result that 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. For many years it was an open problem to find a type II factor whose fundamental group was not the group of positive reals, but Connes then showed that the von Neumann group algebra of a countable discrete group with Kazhdan's property (T) (the trivial representation is isolated in the dual space), such as SL(3,Z), has a countable fundamental group. Subsequently, Sorin Popa showed that the fundamental group can be trivial for certain groups, including the semidirect product of Z2 by SL(2,Z).
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.
 found an uncountable family of such groups with non-isomorphic von Neumann group algebras, thus showing the existence of uncountably many different separable type II1 factors.
Type III factors.
Lastly, type III factors are factors that do not contain any nonzero finite projections at all. In their first paper were unable to decide whether or not they existed; the first examples were later found by . 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 notation IIIλ, where λ is a real number in the interval [0,1]. More precisely, if the Connes spectrum (of its modular group) is 1 then the factor is of type III0, if the Connes spectrum is all integral powers of λ for 0  ; λ  ; 1, then the type is IIIλ, and if the Connes spectrum