von-Neumann-Algebra

A von Neumann algebra or W *-algebra is a mathematical structure in the functional analysis. Historically, the theory of von Neumann algebras with the fundamental 1936-1943 published works by Francis J. Murray and John von Neumann, On rings of operators starts. The name von Neumann algebra for such algebras goes back to a proposal by Jean Dieudonné.

Definition

A von Neumann algebra (named after John von Neumann ) or ( now obsolete ) a ring of operators is a * - subalgebra with identity of the algebra of bounded linear operators of a Hilbert space, the one ( and hence all ) of the following three equivalent conditions met:

• .
• Is complete in the strong operator topology.

Here is the commutant of the commutant and accordingly of.

The equivalence of the three statements above is called the von Neumann Doppelkommutantensatz or Bikommutantensatz. This statement can be strengthened as follows:

This formulation, which establishes an equivalence between the purely algebraic Kommutanten formation and purely topological density relationship or degree education is referred to as Bikommutantensatz. Thus, the Bikommutantensatz proves to be a leak rate. Together with the other leak rate of Kaplansky it represents the starting point of the theory of von Neumann algebras

A von Neumann algebra can be defined in the abstract without an underlying Hilbert space by a set of Shoichiro Sakai:

• A von Neumann algebra is a C * - algebra which is the topological dual space of a Banach space.

Factors

The von Neumann algebra is called a factor if it meets either of the following equivalent conditions:

• .
• Generated.

Since the set of operators is made ​​that commute with all the operators of the center is. Factors are therefore the von Neumann algebras with the smallest possible center. You can von Neumann algebras as a direct integral represent ( a generalization of the direct sum ) of factors, ie von Neumann algebras in this sense are composed of factors.

And examples of factors. With is also a factor; seems to be true and.

The factors 3 types, the Type I, Type II and Type III may be called, are distinguished.

Commutative von Neumann Algebras

Be a finite measure space -. Then L2 is a Hilbert space, and each essentially bounded function defined via multiplication an operator. The figure is a * - isomorphism of a commutative von Neumann algebra, one can show, that is the algebra wrong with their Kommutanten match. No proper superset algebra can therefore be commutative, is thus a maximum commutative von Neumann algebra.

Considering specifically the measure space ( unit interval with the Lebesgue measure ), then one can show that the Bikommutant of coincides with. The transition from the topological construct to measure theoretic construct corresponds to the transition of C * - algebras to von Neumann algebras. While speaking at C *-algebras for the set of Gelfand - Neumark of non- commutative topology, are the employees here viewing opportunity to see a von Neumann algebra as a noncommutative measure space, one is therefore also called non- commutative measure theory.

Properties

Each von Neumann algebra is a C *-algebra and thus a Banach.

As is clear from the limited Borel functional calculus, contain von Neumann algebras many orthogonal projections; each operator in the norm topology limit of linear combinations of orthogonal projections. This is an important difference to the C * - algebras, as the example of C ( [0,1] ) shows, in addition to 0 and 1, no further projections. One can construct a bandage from the set of projections; the structure of this association is used for type classification of von Neumann algebras.