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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:
The ring of essentially bounded measurable functions on the real line is a commutative von Neumann algebra, whose elements act as multiplication operators by pointwise multiplication on the Hilbert space of square-integrable functions.
The algebra of all bounded operators on a Hilbert space is a von Neumann algebra, non-commutative if the Hilbert space has dimension at least .
Von Neumann algebras were first studied by von Neumann (1930) 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 & J. von Neumann 1936, 1937, 1943; J. von Neumann 1938, 1940, 1943, 1949), reprinted in the collected works of von Neumann (1961).
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 three volume work by Takesaki (1979) gives an encyclopedic account of the theory. The book by Connes (1994) discusses more advanced topics.
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In mathematics, a vonNeumannalgebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology...
John vonNeumann (/vɒn ˈnɔɪmən/ von NOY-mən; Hungarian: Neumann János Lajos [ˈnɒjmɒn ˈjaːnoʃ ˈlɒjoʃ]; December 28, 1903 – February 8, 1957) was a Hungarian...
In functional analysis, an abelian vonNeumannalgebra is a vonNeumannalgebra of operators on a Hilbert space in which all elements commute. The prototypical...
the *-algebras closed in these topologies. If M is closed in the norm topology then it is a C*-algebra, but not necessarily a vonNeumannalgebra. One...
of vonNeumannalgebras. The concept was introduced in 1949 by John vonNeumann in one of the papers in the series On Rings of Operators. One of von Neumann's...
infinite and one finite. Murray and vonNeumann proved that up to isomorphism there is a unique vonNeumannalgebra that is a factor of type II1 and also...
closed under taking adjoints. These include C*-algebras, vonNeumannalgebras, and AW*-algebras. C*-algebras can be easily characterized abstractly by a...
constant for a one-dimensional algebra is 1. Nest algebras are hyper-reflexive with distance constant 1. Many vonNeumannalgebras are hyper-reflexive, but...
Typically the random variables lie in a unital algebra A such as a C*-algebra or a vonNeumannalgebra. The algebra comes equipped with a noncommutative expectation...
of B(H) is a vonNeumannalgebra if, and only if, M = M ′ ′ {\displaystyle M=M^{\prime \prime }} , and that if not, the vonNeumannalgebra it generates...
algebra Enveloping algebra, of an associative algebra: see Associative algebra § Enveloping algebra Enveloping vonNeumannalgebra, of a C*-algebra This...
defined). He proved that any such algebra is a Jordan algebra. Not every Jordan algebra is formally real, but Jordan, vonNeumann & Wigner (1934) classified...
specific vonNeumannalgebra acting on a Hilbert space in the presence of a trace. The first such result was proved by Francis Joseph Murray and John von Neumann...
canonical trace. In fact, corresponding to every tracial state on a vonNeumannalgebra there is a notion of Fuglede−Kadison determinant. For matrices over...
and are called JB* algebras or Jordan C* algebras. By analogy with the abstract characterisation of vonNeumannalgebras as C* algebras for which the underlying...
Gerard G. (1972), Algebraic methods in statistical mechanics and quantum field theory, Wiley-Interscience, ISBN 978-0-471-23900-0 vonNeumann, John (1927)...
representing commutative Banach algebras as algebras of continuous functions; the fact that for commutative C*-algebras, this representation is an isometric...
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