# Von Neumann Algebra

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. They 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 L∞(R) of essentially bounded measurable functions on the real line is a commutative von Neumann algebra, which acts by pointwise multiplication on the Hilbert space L2(R) of square integrable functions. The algebra B(H) of all bounded operators on a Hilbert space H is a von Neumann algebra, non-commutative if the Hilbert space has dimension at least 2.

Von Neumann algebras were first studied by von Neumann (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.

### Other articles related to "von neumann algebra, algebra, von neumann, von neumann algebras, algebras":

Direct Integral - Decomposable Operators - Decomposition of Abelian Von Neumann Algebras
... For any Abelian von Neumann algebra A on a separable Hilbert space H, there is a standard Borel space X and a measure μ on X such that it is unitarily equivalent as an operator algebra to L∞μ(X ... this asserts more than just the algebraic equivalence of A with the algebra of diagonal operators ... If the Abelian von Neumann algebra A is unitarily equivalent to both L∞μ(X) and L∞ν(Y) acting on the direct integral spaces and μ, ν are standard measures ...
Timeline Of Computing Hardware 2400 BC–1949 - 1940–1949
1945 John von Neumann drafted a report describing the future computer eventually built as the EDVAC (Electronic Discrete Variable Automatic Computer) ... first published description of the design of a stored-program computer, giving rise to the term von Neumann architecture ... Based on ideas from John von Neumann about stored program computers, the EDSAC was the first complete, fully functional von Neumann architecture computer ...
Von Neumann Algebra - Applications
... Von Neumann algebras have found applications in diverse areas of mathematics like knot theory, statistical mechanics, Quantum field theory, Local quantum physics, Free probability, Noncommutative geometry ...
Finite Dimensional Von Neumann Algebra
... In mathematics, von Neumann algebras are self-adjoint operator algebras that are closed under a chosen operator topology ... When the underlying Hilbert space is finite dimensional, the von Neumann algebra is said to be a finite dimensional von Neumann algebra ... The finite dimensional case differs from the general von Neumann algebras in that topology plays no role and they can be characterized using Wedderburn's theory of semisimple algebras ...
Central Carrier - Related Results
... Suppose E and F are projections in a von Neumann algebra M ... Corollary Two projections E and F in a von Neumann algebra M contain two nonzero subprojections that are Murray-von Neumann equivalent if C(E)C(F) ≠ 0 ... Using this fact and a maximality argument, it can be deduced that the Murray-von Neumann partial order « on the family of projections in M becomes a total order if M is a factor ...

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