Editing Von Neumann algebra (section)

==Examples==
*The essentially bounded functions on a σ-finite measure space form a commutative (type I<sub>1</sub>) von Neumann algebra acting on the ''L''<sup>2</sup> functions. For certain non-σ-finite measure spaces, usually considered [[pathological (mathematics)|pathological]], ''L''<sup>∞</sup>(''X'') is not a von Neumann algebra;  for example, the σ-algebra of measurable sets might be the [[countable-cocountable algebra]] on an uncountable set. A fundamental approximation theorem can be represented by the [[Kaplansky density theorem]]. 
*The bounded operators on any Hilbert space form a von Neumann algebra, indeed a factor, of type I.
*If we have any [[unitary representation]] of a group ''G'' on a Hilbert space ''H'' then the bounded operators commuting with ''G'' form a von Neumann algebra ''{{prime|G}}'', whose projections correspond exactly to the closed subspaces of ''H'' invariant under ''G''. Equivalent subrepresentations correspond to equivalent projections in ''{{prime|G}}''. The double commutant ''{{prime|G}}''{{prime}} of ''G'' is also a von Neumann algebra.
* The '''von Neumann group algebra''' of a discrete group ''G'' is the algebra of all bounded operators on ''H'' = ''l''<sup>2</sup>(''G'') commuting with the action of ''G'' on ''H'' through right multiplication.  One can show that this is the von Neumann algebra generated by the operators corresponding to multiplication from the left with an element ''g'' ∈ ''G''. It is a factor (of type II<sub>1</sub>) if every non-trivial conjugacy class of ''G'' is infinite (for example, a non-abelian free group), and is the hyperfinite factor of type II<sub>1</sub> if in addition ''G'' is a union of finite subgroups (for example, the group of all permutations of the integers fixing all but a finite number of elements).
*The tensor product of two von Neumann algebras, or of a countable number with states, is a von Neumann algebra as described in the section above.
*The [[crossed product]] of a von Neumann algebra by a discrete (or more generally locally compact) group can be defined, and is a von Neumann algebra. Special cases are the '''group-measure space construction''' of Murray and [[John von Neumann|von Neumann]] and '''Krieger factors'''.
*The von Neumann algebras of a measurable [[equivalence relation]] and a measurable [[groupoid]] can be defined. These examples generalise von Neumann group algebras and the group-measure space construction.