Introductory accounts of von Neumann algebras are given in the online notes of Jones (2003) and Wassermann (1991) and the books by Dixmier (1981), Schwartz (1967), Blackadar (2005) and Sakai (1971).
The first and most common way is to define them as weakly closed *-algebras of bounded operators (on a Hilbert space) containing the identity.
The first two definitions describe a von Neumann algebra concretely as a set of operators acting on some given Hilbert space.
Sakai (1971) 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.
Some of the terminology in von Neumann algebra theory can be confusing, and the terms often have different meanings outside the subject.
There have been several attempts to axiomatize the underlying rings of von Neumann algebras, including Baer *-rings and AW*-algebras.
The Schröder–Bernstein theorems for operator algebras gives a sufficient condition for Murray-von Neumann equivalence.
If M is a factor, ≤ is a total order on equivalence classes of projections, described in the section on traces below.
Similarly, a von Neumann algebra is generated by its projections; this is a consequence of the spectral theorem for self-adjoint operators.
A von Neumann algebra N whose center consists only of multiples of the identity operator is called a factor.
Murray & von Neumann (1937) 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].
The definition of the predual given above seems to depend on the choice of Hilbert space that M acts on, as this determines the ultraweak topology.
However the predual can also be defined without using the Hilbert space that M acts on, by defining it to be the space generated by all positive normal linear functionals on M. (Here "normal" means that it preserves suprema when applied to increasing nets of self adjoint operators; or equivalently to increasing sequences of projections.)
The predual M∗ is a closed subspace of the dual M* (which consists of all norm-continuous linear functionals on M) but is generally smaller.
The proof that M∗ is (usually) not the same as M* is nonconstructive and uses the axiom of choice in an essential way; it is very hard to exhibit explicit elements of M* that are not in M∗.
Examples: Weights and their special cases states and traces are discussed in detail in (Takesaki 1979).
The type of a factor can be read off from the possible values of this trace over the projections of the factor, as follows: If a von Neumann algebra acts on a Hilbert space containing a norm 1 vector v, then the functional a → (av,v) is a normal state.
The possible M-dimensions of modules are given as follows: Connes (1976) and others proved that the following conditions on a von Neumann algebra M on a separable Hilbert space H are all equivalent: There is no generally accepted term for the class of algebras above; Connes has suggested that amenable should be the standard term.
(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 flows.)
In fact they are precisely the factors arising as crossed products by free ergodic actions of Z or Z/nZ on abelian von Neumann algebras L∞(X).
Type II factors occur when X admits an equivalent finite (II1) or infinite (II∞) measure, invariant under an action of Z.
The commutation theorem for tensor products states that where M′ denotes the commutant of M. The tensor product of an infinite number of von Neumann algebras, if done naively, is usually a ridiculously large non-separable algebra.
A bimodule (or correspondence) is a Hilbert space H with module actions of two commuting von Neumann algebras.
The theory of subfactors, initiated by Vaughan Jones, reconciles these two seemingly different points of view.
For example, Connes and Jones gave a definition of an analogue of Kazhdan's property (T) for von Neumann algebras in this way.
Popa's work on fundamental groups of non-amenable factors represents another significant advance.