Editing Von Neumann algebra (section)

{{Short description|*-algebra of bounded operators on a Hilbert space}}
{{redirect-distinguish|operator ring|ring operator|operator assistance}}
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In [[mathematics]], a '''von Neumann algebra''' or '''W*-algebra''' is a [[*-algebra]] of [[Bounded linear operator|bounded operators]] on a [[Hilbert space]] that is [[Closed set|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 [[operator theory|single operator]]s, [[group representation]]s, [[ergodic theory]] and [[quantum mechanics]]. His [[von Neumann double commutant theorem|double commutant theorem]] shows that the [[Mathematical analysis|analytic]] definition is equivalent to a purely [[abstract algebra|algebraic]] definition as an algebra of symmetries.

Two basic examples of von Neumann algebras are as follows:
*The ring <math>L^\infty(\mathbb R)</math> of [[essentially bounded]] [[measurable function]]s on the real line is a commutative von Neumann algebra, whose elements act as [[multiplication operator]]s by pointwise multiplication on the [[Hilbert space]] <math>L^2(\mathbb R)</math> of [[square-integrable function]]s.
*The algebra <math>\mathcal B(\mathcal H)</math> of all [[bounded operator]]s on a Hilbert space <math>\mathcal H</math> is a von Neumann algebra, non-commutative if the Hilbert space has dimension at least <math>2</math>.

Von Neumann algebras were first studied by {{harvtxt|von Neumann|1930}} in 1929; he and [[Francis Joseph Murray|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 ({{harvard citations|nb=yes|first=F.J.|last=Murray|first2=J. |last2=von Neumann |year=1936|year2=1937|year3=1943}}; {{harvard citations|nb=yes|first=J. |last=von Neumann |year1=1938|year2=1940|year3=1943|year4=1949}}), reprinted in the collected works of {{harvtxt|von Neumann|1961}}.

Introductory accounts of von Neumann algebras are given in the online notes of {{harvtxt|Jones|2003}} and {{harvtxt|Wassermann|1991}} and the books by {{harvtxt|Dixmier|1981}}, {{harvtxt|Schwartz|1967}}, {{harvtxt|Blackadar|2005}} and {{harvtxt|Sakai|1971}}. The three volume work by {{Harvtxt|Takesaki|1979}} gives an encyclopedic account of the theory. The book by {{harvtxt|Connes|1994}} discusses more advanced topics.