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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. In mathematics, a *-algebra is an associative algebra over the field of complex numbers with an antilinear antiautomorphism * : A->A which is an involution. ...
In mathematics, the operator norm is a norm defined on the space of bounded operators between two Banach spaces. ...
In mathematical formulations of quantum mechanics, an operator is a linear transformation from a Hilbert space to itself. ...
In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ...
In functional analysis, the weak operator topology, often abbreviated WOT, is the weakest topology on the set of bounded operators on a Hilbert space such that the functional sending an operator T to the complex number is continuous for any vectors x and y in the Hilbert space. ...
In functional analysis, the strong operator topology, often abbreviated SOT, is the weakest topology on the set of bounded operators on a Hilbert space such that the evaluation map sending an operator T to the real number is continuous for each vector x in the Hilbert space. ...
C*-algebras are an important area of research in functional analysis. ...
John von Neumann in the 1940s. ...
In physics, particularly in quantum physics a system observable is a property of the system state that can be determined by some sequence of physical operations. ...
Fig. ...
The von Neumann bicommutant theorem gives another description of von Neumann algebras, using algebraic rather than topological properties. The von Neumann bicommutant theorem relates the closure of a set of bounded operators on a Hilbert space in certain topologies to the bicommutant of that set. ...
Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ...
Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ...
There are two basic examples of von Neumann algebras to keep in mind. Firstly, if X is a space with a σ -finite measure μ; and L2(X,μ) is the Hilbert space of complex-valued square-integrable functions on X, then the space B(L2(X,μ)) of bounded linear operators on this space is a (highly non-commutative) von Neumann algebra. Inside this algebra we have the sub-algebra of bounded multiplication operators
which in fact is the most general example of a commutative von Neumann algebra as is stated below.
Commutative von Neumann algebras
Main article: Abelian von Neumann algebra In functional analysis, an Abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all its elements commute. ...
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. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...
In mathematics, a measure is a function that assigns a number, e. ...
In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...
C*-algebras are an important area of research in functional analysis. ...
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ...
In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...
In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ...
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. C*-algebras are an important area of research in functional analysis. ...
The strictly C*-algebraic part of the noncommutative geometry program. ...
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. In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ...
In functional analysis a partial isometry is a linear map W between Hilbert spaces H, K such that there is a closed vector subspace H1 of H such that W restricted to H1 is an isometric map and W restricted to the orthogonal complement of H1 is zero. ...
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. Center (American English), centre (Commonwealth English), has a number of meanings. ...
In mathematics and functional analysis a direct integral is a generalization of the concept of direct sum. ...
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∞. In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ...
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 best understood factors of type II are the hyperfinite type II1 factor and the hyperfinite type II∞ factor; the wealth of other type II factors is the subject of intensive study. The title given to this article is incorrect due to technical limitations. ...
In mathematics, the hyperfinite type II∞ factor is a special von Neumann algebra. ...
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. Trefoil knot, the simplest non-trivial knot. ...
Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ...
In mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ...
Geometry (from the Greek words Geo = earth and metro = measure) is the branch of mathematics first popularized in ancient Greek culture by Thales (circa 624-547 BC) dealing with spatial relationships. ...
The word probability derives from the Latin probare (to prove, or to test). ...
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