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A '''von Neumann algebra''' or '''W*-algebra''' (named for [[John von Neumann]]) is a [[star-algebra|*-algebra]] of [[Bounded linear operator|bounded]] [[operator (physics)|operators]] on a [[Hilbert space]] that is closed in the [[weak operator topology]], and contains the identity operator. They were believed by [[John von Neumann]] to capture the concept of an algebra of [[observable]]s in [[quantum mechanics]]. Von Neumann algebras are [[C*-algebra]]s. The [[von Neumann bicommutant theorem]] gives another description of von Neumann algebras, using [[abstract algebra|algebraic]] rather than [[topology|topological]] properties.
The two basic examples of von Neumann algebras are as follows. The ring ''L''<sup>∞</sup>('''R''') of bounded measurable functions on the real line (modulo null functions) is a commutative von Neumann algebra under pointwise operations, which acts on the Hilbert space ''L''<sup>2</sup>('''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).
==Definitions==
There are three common ways to define von Neumann algebras.
The first and most common way is to define them as weakly closed * algebras of bounded operators (on a Hilbert space) containing the identity. In this definition the weak (operator) topology can be replaced by almost any other common topology other than the norm topology,
in particular by the strong or [[ultrastrong topology|ultrastrong topologies]]. (The * algebras of bounded operators that are closed in the norm topology are C<sup>*</sup> algebras, so in particular any von Neumann algebra is a
C<sup>*</sup> algebra.)
The second definition is that a von Neumann algebra is a subset of the bounded operators closed under * and equal to its double commutator, or equivalently the commutator of some subset closed under *. The
[[von Neumann bicommutant theorem]] says that the first two definitions are equivalent.
The first two definitions define a von Neumann algebras concretely as a set of operators acting on some given Hilbert space. Von Neumann algebras can also be defined abstractly as C<sup>*</sup> algebras that have a [[predual]]; in other words the von Neumann algebra, considered as a Banach space, is the dual of some other Banach space called the predual. The predual of a von Neumann algebra is unique up to isomorphism.
Some authors use "von Neumann algebra" for the algebras together with a Hilbert space action, and "W<sup>*</sup> algebra" for the abstract concept, so a von Neumann algebra is a W<sup>*</sup> algebra together with a Hilbert space and a suitable faithful unital action on the Hilbert space. The concrete and abstract definitions of a von Neumann algebra are similar to the concrete and abstract definitions of a C<sup>*</sup> algebra, which can be defined either as norm-closed * algebras of operators on a Hilbert space, or as [[B* algebra|Banach *-algebra]]s such that ||''a'' ''a<sup>*</sup>''||=||''a''|| ||''a<sup>*</sup>''||.
== Commutative von Neumann algebras ==
''Main article: [[Abelian von Neumann algebra]]''
The relationship between [[commutative]] von Neumann algebras and [[measure space]]s is analogous to that between [[commutative]] [[C*-algebra]]s and [[locally compact]] [[Hausdorff space]]s. Every commutative von Neumann algebra is isomorphic to [[Lp space|L<sup>∞</sup>]](''X'') for some measure space (''X'', μ) and for every σ-finite measure space ''X'', conversely, L<sup>∞</sup>(''X'') is a von Neumann algebra.
Due to this analogy, the theory of von Neumann algebras has been called noncommutative measure theory, while the theory of [[C*-algebra]]s is sometimes called [[noncommutative topology]].
== 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. Another way of stating this is that ''E'' is equivalent to ''F'' if ''E=aa<sup>*</sup>'' and ''F=a<sup>*</sup>a'' for some ''a''. 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.
For factors this is a total order, described in the section on traces below.
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 (algebra)|center]] consists only of multiples of the identity operator is called a '''factor'''. Every von Neumann algebra on a separable Hilbert space is isomorphic to a [[direct integral]] of factors. This decomposition is essentially unique. Thus, the problem of classifying isomorphism classes of von Neumann algebras on separable Hilbert spaces can be reduced to that of classifying isomorphism classes of factors.
Every factor has one of 3 types as described below.
The type classification can be extended to von Neumann algebras that are not factors, and a von Neumann algebra is 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 I<sub>1</sub>. Every von Neumann algebra can be written
uniquely as a sum of von Neumann algebras of types I, II, and III.
There are several other ways to divide factors into classes that are sometimes used:
* A factor is called '''discrete''' (or occasionally '''tame''') if it has type I, and '''continuous''' (or occasionally '''wild''') if it has type II or III.
* A factor is called '''semifinite''' if it has type I or II, and '''purely infinite''' if it has type III.
* A factor is called '''finite''' if the projection 1 is finite and '''properly infinite''' otherwise. Factors of types I and II may be either finite or properly infinite, but factors of type III are always properly infinite.
==Type I factors==
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 I<sub>n</sub>, and the bounded operators on a separable infinite-dimensional Hilbert space, a factor of type I<sub>∞</sub>.
==Type II factors==
A factor is said to be of '''type II''' if there are non-zero 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 II<sub>1</sub>; otherwise, it is said to be of type II<sub>∞</sub>. The best understood factors of type II are the [[hyperfinite type II-1 factor|hyperfinite type II<sub>1</sub> factor]] and the [[hyperfinite type II-infinity factor|hyperfinite type II<sub>∞</sub> factor]].
These are the unique hyperfinite factors of types II<sub>1</sub> and II<sub>∞</sub>; there are an uncountable number of other
factors of these types that are the subject of intensive study.
A factor of type II<sub>1</sub> has a unique finite tracial<!-- word looks suspicious. I am a stupid bot and I might be wrong --> state,
and the set of traces of projections is [0,1].
A factor of type II<sub>∞</sub> has a semifinite trace, unique up to
rescaling, and the set of traces of projections is [0,∞]. The set of real numbers λ such that there is an automorphism rescaling the trace by a factor of λ is called the '''fundamental group''' of the type II<sub>∞</sub> factor.
The tensor product of a factor of type II<sub>1</sub> and an infinite
type I factor has type II<sub>∞</sub>, and conversely any factor of
type II<sub>∞</sub> can be constructed like this. The '''fundamental group''' of a type
II<sub>1</sub> factor is defined to be the fundamental group of its tensor product with the infinite (separable) factor of type I.
An example of a type II<sub>1</sub> factor is the von Neumann group algebra
of a countable infinite discrete group such that every non-trivial conjugacy class is infinite.
==Type III factors==
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<sub>∞</sub> in the past, but recently that notation has been superseded by the introduction of a family of type III factors called type III<sub>λ</sub>, where λ is a real number in the interval [0,1]. The only trace on these factors takes value ∞ on all non-zero positive elements, and any two non-zero projections
are equivalent. At one time type III factors were considered to be wild, intractable objects; [[Tomita-Takesaki theory]]
has led to a good structure theory. In particular, any type III factor can be written in a canonical way as the [[crossed product]] of a type II<sub>∞</sub> factor and the real numbers.
==Terminology==
*A '''finite''' von Neumann algebra is one which is the direct integral of finite factors. Similarly, '''properly infinite''' von Neumann algebras are the direct integral of properly infinite factors.
*A von Neumann algebra that acts on a separable Hilbert space is called '''separable'''. Note that such algebras are rarely [[separable space|separable]] in the norm topology.
*The von Neumann algebra '''generated''' by a set of bounded operators on a Hilbert space is the smallest von Neumann algebra containing all those operators.
*The '''tensor product''' of two von Neumann algebras acting on two Hilbert spaces is defined to be the von Neumann algebra generated by their algebraic tensor product, considered as operators on the Hilbert space tensor product of the Hilbert spaces.
==The predual==
Any von Neumann algebra ''M'' has a '''predual''' ''M''<sub>*</sub>, which is the Banach space of all σ-weakly continuous linear functionals on ''M''. As the name suggests, ''M'' is (as a Banach space) the dual of its predual. The predual is unique in the sense that any other Banach space whose dual is ''M'' is canonically isomorphic to ''M''<sub>*</sub>. As mentioned above, the existence of a predual characterizes von Neumann algebras among C* algebras.
Examples:
#The predual of the von Neumann algebra ''L''<sup>∞</sup>('''R''') of essentially bounded functions on '''R''' is the Banach space ''L''<sup>1</sup>('''R''') of integrable functions.
#The predual of the von Neumann algebra ''B''(''H'') of bounded operators on a Hilbert space ''H'' is the Banach space of all [[trace class]] operators with the trace norm ||''A''||= Tr(|''A''|). The Banach space of trace class operators is itself the dual of the C*-algebra of compact operators (which is not a von Neumann algebra).
==Weights, states, and traces.==
*A '''weight''' ω on a von Neumann algebra is a linear map from the set of positive elements (those of the form ''aa<sup>*</sup>'') to [0,∞].
*A '''positive linear functional''' is a weight with ω(1) finite (or rather the extension of ω to the whole algebra by linearity).
*A '''state''' is a weight with ω(1)=1.
*A '''trace''' is a weight with ω(''aa<sup>*</sup>'')=ω(''a<sup>*</sup>a'') for all ''a''.
*A '''tracial state''' is a trace with ω(1)=1.
Any factor has a trace such that the trace of a non-zero projection is non-zero
and the trace of a projection is infinite if and only if the projection is infinite.
Such a trace is unique up to rescaling. For factors that are separable or finite, two projections are equivalent if and only if they have the same trace. The type of a factor can be read off from the possible values of this trace as follows:
*Type I<sub>n</sub>: 0, ''x'', 2''x'', ....,''nx'' for some positive ''x'' (usually normalized to be 1/''n'' or 1).
*Type I<sub>∞</sub>: 0, ''x'', 2''x'', ....,''∞'' for some positive ''x'' (usually normalized to be 1).
*Type II<sub>1</sub>: [0,''x''] for some positive ''x'' (usually normalized to be 1).
*Type II<sub>∞</sub>: [0,∞].
*Type III: 0,∞
If a von Neumann algebra acts on a Hilbert space containing a norm 1 vector ''v'',
then (''av'',''v'') gives a state. This construction can be reversed to give an action on
a Hilbert space from a state: this is called the [[GNS construction]].
==Amenable von Neumann algebras==
Connes and others proved that the following conditions on a von Neumann algebra ''N''
on a separable Hilbert space ''H'' are all equivalent:
* ''N'' is '''hyperfinite''' or '''AFD''' or '''almost finite dimensional''': this means the algebra contains an ascending sequence of finite dimensional subalgebras with dense union.
*''N'' is '''amenable''': this means that the [[derivation (abstract algebra)|derivation]]s of ''N'' with values in a normal dual Banach bimodule are all inner.
*''N'' has Schwartz's '''property P''': for any bounded operator ''T'' on ''H'' the norm closed convex hull of the elements ''uTu<sup>*</sup>'' contains an element commuting with ''N''.
*''N'' is '''semidiscrete''': this means the identity map from ''M'' to ''M'' is a weak pointwise limit of completely positive maps of finite rank.
*''N'' has '''property E''' or the '''Hakeda-Tomiyama extension property''': this means that there is a projection of norm 1 from bounded operators on ''H'' to ''M'' '.
*''N'' is '''injective''': any completely positive linear map from any self adjoint closed subspace containing 1 of any unital C<sup>*</sup>-algebra ''A'' to ''N'' can be extended to a completely positive map from ''A'' to ''N''.
There is no generally accepted term for the class of algebras above; Connes has suggested that '''amenable''' should be the standard term.
The amenable factors have been classified: there is a unique one of each of the types I<sub>''n''</sub>, I<sub>∞</sub>, II<sub>1</sub>, II<sub>∞</sub>, III<sub>λ</sub>,
for 0<λ≤ 1, and the ones of type III<sub>0</sub> correspond to certain ergodic
actions. (For type III<sub>0</sub> calling this a classification is a little misleading, as it is known that there is no easy way to classify the corresponding ergodic actions.) Von Neumann algebras of type I are always amenable, but for the other types there are an uncountable number of different non-amenable factors, which seem very hard to classify, or even distinguish from each other.
==Algebraic properties==
By [[forgetting (mathematics)|forgetting]] about the topology on a von Neumann algebra, we can consider it a [[star-algebra|*-algebra]] (which is usually going to be noncommutative, but will always be unital), or just a ring.
Von Neumann algebras are [[semihereditary ring|semihereditary]]: every finitely generated submodule of a projective module is itself projective.
Despite the similarity in name, von Neumann algebras are not, in general, [[von Neumann regular ring]]s; however, for finite von Neumann algebras, the algebra of [[affiliated operator]]s, which makes all non-zero divisors invertible, can be constructed, and is von Neumann regular.
==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]], <math>L^\infty(X)</math> is not a von Neumann algebra; for example, the σ-algebra of measurable sets might be the [[countable-cocountable algebra]] on an uncountable set.
*The bounded operators on any Hilbert space form a von Neumann algebra, indeed a factor, of type I.
*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.
*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 ''G''′, whose projections correspond exactly to the closed subspaces of ''H'' invariant under ''G''. The double commutator ''G''′′ 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 <math>H = \ell^2(G)</math> 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 <math>g\in G</math>.
*One can define a tensor product of von Neumann algebras (a completion of the algebraic tensor product of the algebras considered as rings), which is again a von Neumann algebra. The tensor product of two finite algebras is finite, and the tensor product of an infinite algebra and a non-zero algebra is infinite. The tensor product of two von Neumann algebras of types X and Y (I, II, or III) has type equal to the maximum of X and Y.
*The tensor product of an infinite number of von Neumann algebras, if done naively, is usually a ridiculously large non-separable algebra. Instead one usually chooses a state on each of the von Neumann algebras, uses this to define a state on the algebraic tensor product, which can be used to product a Hilbert space and a (reasonably small) von Neumann algebra. The factors of '''Powers''' and '''Araki-Woods''' were found like this. If all the factors are finite matrix algebras the factors are called '''ITPFI factors''' (ITPFI stands for "infinite tensor product of finite type I factors"). The type of the infinite tensor product can vary dramatically as the states are changed; for example, the infinite tensor product of an infinite number of type I<sub>2</sub> factors can have any type depending on the choice of states.
*Krieger constructed some type III factors, known as '''Krieger's factors''', from ergodic actions, which can be used to give all the hyperfinite type III<sub>0</sub> factors.
==Applications==
Von Neumann algebras have found applications in diverse areas of mathematics like [[knot theory]], [[statistical mechanics]], [[representation theory]], [[geometry]] and [[probability]].
==See also==
*[[Quantum mechanics]]
*[[Quantum field theory]]
*[[Local quantum physics]]
*[[Free probability]]
*[[C*-algebra]]
*[[Noncommutative geometry]]
*[[Topologies on the set of operators on a Hilbert space]]
*[[Measure theory]]
==References==
*{{springer|id=V/v096900|title=von Neumann algebra|author=A.I. Shtern}}
*J. Dixmier, ''Les algèbres d'opérateurs dans l'espace hilbertien: algèbres de von Neumann'' , Gauthier-Villars (1957). ''von Neumann algebras'', ISBN 0444863087
*S. Sakai, ''C*-algebras and W*-algebras'' , Springer (1971) ISBN 3540636331
*M. Takesaki ''Theory of Operator Algebras I, II, III'' ISBN 3-540-42248-X ISBN 3-540-42914-X ISBN 3-540-42913-1
*A. Connes [ftp://ftp.alainconnes.org/book94bigpdf.pdf ''Non-commutative geometry''], ISBN 0-12-185860-X.
===von Neumann's papers===
* J. von Neumann [http://links.jstor.org/sici?sici=0003-486X%28193601%292%3A37%3A1%3C111%3AOACTFR%3E2.0.CO%3B2-S On a Certain Topology for Rings of Operators] The Annals of Mathematics 2nd Ser., Vol. 37, No. 1 (Jan., 1936), pp. 111-115. This defines the ultrastrong topology.
*F.J. Murray, J. von Neumann, [http://links.jstor.org/sici?sici=0003-486X%28193601%292%3A37%3A1%3C116%3AOROO%3E2.0.CO%3B2-Y ''On rings of operators''] Ann. of Math. (2) , 37 (1936) pp. 116–229. The original paper on von Neumann algebras, giving their basic properties and the division into types I, II, and III.
*F.J. Murray, J. von Neumann, [http://links.jstor.org/sici?sici=0002-9947%28193703%2941%3A2%3C208%3AOROOI%3E2.0.CO%3B2-9 ''On rings of operators II''] Trans. Amer. Math. Soc. , 41 (1937) pp. 208–248. This is a continuation of the previous paper, that studies properties of the trace of a factor.
*J. von Neumann, [http://links.jstor.org/sici?sici=0003-486X%28194001%292%3A41%3A1%3C94%3AOROOI%3E2.0.CO%3B2-U'' On rings of operators III''] Ann. of Math. (2) , 41 (1940) pp. 94–161. This shows the existence of factors of type III.
*F.J. Murray, J. von Neumann, [http://links.jstor.org/sici?sici=0003-486X%28194310%292%3A44%3A4%3C716%3AOROOI%3E2.0.CO%3B2-O ''On rings of operators IV''] Ann. of Math. (2) , 44 (1943) pp. 716–808. This studies when factors are isomorphic, and in particular shows that all almost finite factors of type II<sub>1</sub> are isomorphic.
*J. von Neumann, [http://www.numdam.org/item?id=CM_1939__6__1_0 ''On infinite direct products''] Compos. Math. , 6 (1938) pp. 1–77. This discusses infinite tensor products of Hilbert spaces and the algebras acting on them.
*J. von Neumann, [http://links.jstor.org/sici?sici=0003-486X%28194904%292%3A50%3A2%3C401%3AOROORT%3E2.0.CO%3B2-H On Rings of Operators. Reduction Theory] The Annals of Mathematics 2nd Ser., Vol. 50, No. 2 (Apr., 1949), pp. 401-485. This discusses how to write a von Neumann algebra as a sum or integral of factors.
*J. von Neumann, [http://links.jstor.org/sici?sici=0003-486X%28194310%292%3A44%3A4%3C709%3AOSAPOO%3E2.0.CO%3B2-L On Some Algebraical Properties of Operator Rings] The Annals of Mathematics 2nd Ser., Vol. 44, No. 4 (Oct., 1943), pp. 709-715. This shows that some apparently topological properties in von Neumann algebras can be defined purely algebraically.
[[Category:Operator theory]]
[[Category:Von Neumann algebras|*]]
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