Inmathematics, avon Neumann algebra orW*-algebra is a*-algebra ofbounded operators on aHilbert space that isclosed in theweak operator topology and contains theidentity operator. It is a special type ofC*-algebra.

Von Neumann algebras were originally introduced byJohn von Neumann, motivated by his study ofsingle operators,group representations,ergodic theory andquantum mechanics. Hisdouble commutant theorem shows that theanalytic definition is equivalent to a purelyalgebraic definition as an algebra of symmetries.

Two basic examples of von Neumann algebras are as follows:

• The ring${\displaystyle L^{\mathrm{\infty }}(\mathbb{R})}$ ofessentially boundedmeasurable functions on the real line is a commutative von Neumann algebra, whose elements act asmultiplication operators by pointwise multiplication on theHilbert space${\displaystyle L^2(\mathbb{R})}$ ofsquare-integrable functions.
• The algebra${\displaystyle \mathcal{B}(\mathcal{H})}$ of allbounded operators on a Hilbert space${\displaystyle \mathcal{H}}$ is a von Neumann algebra, non-commutative if the Hilbert space has dimension at least${\displaystyle 2}$.

Von Neumann algebras were first studied byvon Neumann (1930) in 1929; he andFrancis Murray developed the basic theory, under the original name ofrings 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 ofvon Neumann (1961).

Introductory accounts of von Neumann algebras are given in the online notes ofJones (2003) andWassermann (1991) and the books byDixmier (1981),Schwartz (1967),Blackadar (2005) andSakai (1971). The three volume work byTakesaki (1979) gives an encyclopedic account of the theory. The book byConnes (1994) discusses more advanced topic.

There are three common ways to define von Neumann algebras.

The first and most common way is to define them asweakly closed*-algebras of bounded operators (on a Hilbert space) containing the identity. In this definition the weak (operator) topology can be replaced by many othercommon topologies including thestrong,ultrastrong orultraweak operator topologies. The *-algebras of bounded operators that are closed in thenorm topology areC*-algebras, so in particular any von Neumann algebra is a C*-algebra.

The second definition is that a von Neumann algebra is a subalgebra of the bounded operators closed underinvolution (the *-operation) and equal to its doublecommutant, or equivalently thecommutant of some subalgebra closed under *. Thevon Neumann double commutant theorem (von Neumann 1930) says that the first two definitions are equivalent.

The first two definitions describe a von Neumann algebra concretely as a set of operators acting on some given Hilbert space.Sakai (1971) showed that von Neumann algebras can also be defined abstractly as C*-algebras that have apredual; in other words the von Neumann algebra, considered as aBanach space, is thedual of some other Banach space called the predual. The predual of a von Neumann algebra is in fact unique up to isomorphism. Some authors use "von Neumann algebra" for the algebras together with a Hilbert space action, and "W*-algebra" for the abstract concept, so a von Neumann algebra is a W*-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*-algebra, which can be defined either as norm-closed *-algebras of operators on a Hilbert space, or asBanach *-algebras such that${\displaystyle ||a{a}^{\ast }||=||a||||a^{\ast }||}$.

Some of the terminology in von Neumann algebra theory can be confusing, and the terms often have different meanings outside the subject.

• Afactor is a von Neumann algebra with trivial center, i.e. a center consisting only of scalar operators.
• Afinite von Neumann algebra is one which is thedirect integral of finite factors (meaning the von Neumann algebra has a faithful normal tracial state${\displaystyle \tau :M\to \mathbb{C}}$^[1]). 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 calledseparable. Note that such algebras are rarelyseparable in the norm topology.
• The von Neumann algebragenerated by a set of bounded operators on a Hilbert space is the smallest von Neumann algebra containing all those operators.
• Thetensor 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.

Byforgetting about the topology on a von Neumann algebra, we can consider it a (unital)*-algebra, or just a ring. Von Neumann algebras aresemihereditary: every finitely generated submodule of aprojective module is itself projective. There have been several attempts to axiomatize the underlying rings of von Neumann algebras, includingBaer *-rings andAW*-algebras. The*-algebra ofaffiliated operators of a finite von Neumann algebra is avon Neumann regular ring. (The von Neumann algebra itself is in general not von Neumann regular.)

The relationship betweencommutative von Neumann algebras andmeasure spaces is analogous to that between commutativeC*-algebras andlocally compactHausdorff spaces. Every commutative von Neumann algebra is isomorphic toL^∞(X) for some measure space (X, μ) and conversely, for every σ-finite measure spaceX, the *-algebraL^∞(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 ofC*-algebras is sometimes callednoncommutative topology (Connes 1994).

OperatorsE in a von Neumann algebra for whichE =EE =E* are calledprojections; they are exactly the operators which give an orthogonal projection ofH onto some closed subspace. A subspace of the Hilbert spaceH is said tobelong to the von Neumann algebraM if it is the image of some projection inM. This establishes a 1:1 correspondence between projections ofM and subspaces that belong toM. Informally these are the closed subspaces that can be described using elements ofM, or thatM "knows" about.

It can be shown that the closure of the image of any operator inM and the kernel of any operator inM belongs toM. Also, the closure of the image under an operator ofM of any subspace belonging toM also belongs toM. (These results are a consequence of thepolar decomposition).

The basic theory of projections was worked out byMurray & von Neumann (1936). Two subspaces belonging toM are called (Murray–von Neumann)equivalent if there is a partial isometry mapping the first isomorphically onto the other that is an element of the von Neumann algebra (informally, ifM "knows" that the subspaces are isomorphic). This induces a naturalequivalence relation on projections by definingE to be equivalent toF if the corresponding subspaces are equivalent, or in other words if there is apartial isometry ofH that maps the image ofE isometrically to the image ofF and is an element of the von Neumann algebra. Another way of stating this is thatE is equivalent toF ifE=uu* andF=u*u for some partial isometryu inM.

The equivalence relation ~ thus defined is additive in the following sense: SupposeE₁ ~F₁ andE₂ ~F₂. IfE₁ ⊥E₂ andF₁ ⊥F₂, thenE₁ +E₂ ~F₁ +F₂. Additivity wouldnot generally hold if one were to require unitary equivalence in the definition of ~, i.e. if we sayE is equivalent toF ifu*Eu =F for some unitaryu. TheSchröder–Bernstein theorems for operator algebras gives a sufficient condition for Murray-von Neumann equivalence.

The subspaces belonging toM are partially ordered by inclusion, and this induces a partial order ≤ of projections. There is also a natural partial order on the set ofequivalence classes of projections, induced by the partial order ≤ of projections. IfM is a factor, ≤ is a total order on equivalence classes of projections, described in the section on traces below.

A projection (or subspace belonging toM)E is said to be afinite projection if there is no projectionF <E (meaningF ≤E andF ≠E) that is equivalent toE. For example, all finite-dimensional projections (or subspaces) 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. However it is possible for infinite dimensional subspaces to be finite.

Orthogonal projections are noncommutative analogues of indicator functions inL^∞(R).L^∞(R) is the ||·||_∞-closure of the subspace generated by the indicator functions. Similarly, a von Neumann algebra is generated by its projections; this is a consequence of thespectral theorem for self-adjoint operators.

The projections of a finite factor form acontinuous geometry.

A von Neumann algebraN whosecenter consists only of multiples of the identity operator is called afactor. Asvon Neumann (1949) showed, every von Neumann algebra on a separable Hilbert space is isomorphic to adirect 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.

Murray & von Neumann (1936) showed that 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 I₁. 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 calleddiscrete (or occasionallytame) if it has type I, andcontinuous (or occasionallywild) if it has type II or III.
• A factor is calledsemifinite if it has type I or II, andpurely infinite if it has type III.
• A factor is calledfinite if the projection 1 is finite andproperly infinite otherwise. Factors of types I and II may be either finite or properly infinite, but factors of type III are always properly infinite.

A factor is said to be oftype I if there is a minimal projectionE ≠ 0, i.e. a projectionE such that there is no other projectionF with 0 <F <E. Any factor of type I is isomorphic to the von Neumann algebra ofall bounded operators on some Hilbert space; since there is one Hilbert space for everycardinal 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 dimensionn a factor of type I_n, and the bounded operators on a separable infinite-dimensional Hilbert space, a factor of type I_∞.

A factor is said to be oftype II if there are no minimal projections but there are non-zerofinite projections. This implies that every projectionE can be "halved" in the sense that there are two projectionsF andG that areMurray–von Neumann equivalent and satisfyE =F +G. If the identity operator in a type II factor is finite, the factor is said to be of type II₁; otherwise, it is said to be of type II_∞. The best understood factors of type II are thehyperfinite type II₁ factor and thehyperfinite type II_∞ factor, found byMurray & von Neumann (1936). These are the unique hyperfinite factors of types II₁ and II_∞; there are an uncountable number of other factors of these types that are the subject of intensive study.Murray & von Neumann (1937) proved the fundamental result that a factor of type II₁ has a unique finite tracial state, and the set of traces of projections is [0,1].

A factor of type II_∞ 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 thefundamental group of the type II_∞ factor.

The tensor product of a factor of type II₁ and an infinite type I factor has type II_∞, and conversely any factor of type II_∞ can be constructed like this. Thefundamental group of a type II₁ factor is defined to be the fundamental group of its tensor product with the infinite (separable) factor of type I. For many years it was an open problem to find a type II factor whose fundamental group was not the group ofpositive reals, butConnes then showed that the von Neumann group algebra of a countable discrete group withKazhdan's property (T) (the trivial representation is isolated in the dual space), such as SL(3,Z), has a countable fundamental group. Subsequently,Sorin Popa showed that the fundamental group can be trivial for certain groups, including thesemidirect product ofZ² by SL(2,Z).

An example of a type II₁ factor is the von Neumann group algebra of a countable infinite discrete group such that every non-trivial conjugacy class is infinite.McDuff (1969) found an uncountable family of such groups with non-isomorphic von Neumann group algebras, thus showing the existence of uncountably many different separable type II₁ factors.

Lastly,type III factors are factors that do not contain any nonzero finite projections at all. In their first paperMurray & von Neumann (1936) were unable to decide whether or not they existed; the first examples were later found byvon Neumann (1940). Since the identity operator is always infinite in those factors, they were sometimes called type III_∞ in the past, but recently that notation has been superseded by the notation III_λ, where λ is a real number in the interval [0,1]. More precisely, if the Connes spectrum (of its modular group) is 1 then the factor is of type III₀, if the Connes spectrum is all integral powers of λ for 0 < λ < 1, then the type is III_λ, and if the Connes spectrum is all positive reals then the type is III₁. (The Connes spectrum is a closed subgroup of the positive reals, so these are the only possibilities.) The only trace on type III 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 intractable objects, butTomita–Takesaki theory has led to a good structure theory. In particular, any type III factor can be written in a canonical way as thecrossed product of a type II_∞ factor and the real numbers.

Any von Neumann algebraM has apredualM_∗, which is the Banach space of all ultraweakly continuous linear functionals onM. 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 isM is canonically isomorphic toM_∗.Sakai (1971) showed that the existence of a predual characterizes von Neumann algebras among C* algebras.

The definition of the predual given above seems to depend on the choice of Hilbert space thatM acts on, as this determines the ultraweak topology. However the predual can also be defined without using the Hilbert space thatM acts on, by defining it to be the space generated by all positivenormal linear functionals onM. (Here "normal" means that it preserves suprema when applied to increasing nets of self adjoint operators; or equivalently to increasing sequences of projections.)

The predualM_∗ is a closed subspace of the dualM* (which consists of all norm-continuous linear functionals onM) but is generally smaller. The proof thatM_∗ is (usually) not the same asM* is nonconstructive and uses the axiom of choice in an essential way; it is very hard to exhibit explicit elements ofM* that are not inM_∗. For example, exotic positive linear forms on the von Neumann algebral^∞(Z) are given byfree ultrafilters; they correspond to exotic *-homomorphisms intoC and describe theStone–Čech compactification ofZ.

Examples:

1. The predual of the von Neumann algebraL^∞(R) of essentially bounded functions onR is the Banach spaceL¹(R) of integrable functions. The dual ofL^∞(R) is strictly larger thanL¹(R) For example, a functional onL^∞(R) that extends theDirac measure δ₀ on the closed subspace of bounded continuous functionsC⁰_b(R) cannot be represented as a function inL¹(R).
2. The predual of the von Neumann algebraB(H) of bounded operators on a Hilbert spaceH is the Banach space of alltrace 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 and their special cases states and traces are discussed in detail in (Takesaki 1979).

• Aweight ω on a von Neumann algebra is a linear map from the set ofpositive elements (those of the forma*a) to [0,∞].
• Apositive linear functional is a weight with ω(1) finite (or rather the extension of ω to the whole algebra by linearity).
• Astate is a weight with ω(1) = 1.
• Atrace is a weight with ω(aa*) = ω(a*a) for alla.
• Atracial 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 over the projections of the factor, as follows:

• Type I_n: 0,x, 2x, ....,nx for some positivex (usually normalized to be 1/n or 1).
• Type I_∞: 0,x, 2x, ....,∞ for some positivex (usually normalized to be 1).
• Type II₁: [0,x] for some positivex (usually normalized to be 1).
• Type II_∞: [0,∞].
• Type III: {0,∞}.

If a von Neumann algebra acts on a Hilbert space containing a norm 1 vectorv, then the functionala → (av,v) is a normal state. This construction can be reversed to give an action on a Hilbert space from a normal state: this is theGNS construction for normal states.

Given an abstract separable factor, one can ask for a classification of its modules, meaning the separable Hilbert spaces that it acts on. The answer is given as follows: every such moduleH can be given anM-dimension dim_M(H) (not its dimension as a complex vector space) such that modules are isomorphic if and only if they have the sameM-dimension. TheM-dimension is additive, and a module is isomorphic to a subspace of another module if and only if it has smaller or equalM-dimension.

A module is calledstandard if it has a cyclic separating vector. Each factor has a standard representation, which is unique up to isomorphism. The standard representation has an antilinear involutionJ such thatJMJ =M′. For finite factors the standard module is given by theGNS construction applied to the unique normal tracial state and theM-dimension is normalized so that the standard module hasM-dimension 1, while for infinite factors the standard module is the module withM-dimension equal to ∞.

The possibleM-dimensions of modules are given as follows:

• Type I_n (n finite): TheM-dimension can be any of 0/n, 1/n, 2/n, 3/n, ..., ∞. The standard module hasM-dimension 1 (and complex dimensionn².)
• Type I_∞ TheM-dimension can be any of 0, 1, 2, 3, ..., ∞. The standard representation ofB(H) isH⊗H; itsM-dimension is ∞.
• Type II₁: TheM-dimension can be anything in [0, ∞]. It is normalized so that the standard module hasM-dimension 1. TheM-dimension is also called thecoupling constant of the moduleH.
• Type II_∞: TheM-dimension can be anything in [0, ∞]. There is in general no canonical way to normalize it; the factor may have outer automorphisms multiplying theM-dimension by constants. The standard representation is the one withM-dimension ∞.
• Type III: TheM-dimension can be 0 or ∞. Any two non-zero modules are isomorphic, and all non-zero modules are standard.

Connes (1976) and others proved that the following conditions on a von Neumann algebraM on a separable Hilbert spaceH are allequivalent:

• M ishyperfinite orAFD orapproximately finite dimensional orapproximately finite: this means the algebra contains an ascending sequence of finite dimensional subalgebras with dense union. (Warning: some authors use "hyperfinite" to mean "AFD and finite".)
• M isamenable: this means that thederivations ofM with values in a normal dual Banach bimodule are all inner.^[2]
• M has Schwartz'sproperty P: for any bounded operatorT onH the weak operator closed convex hull of the elementsuTu* contains an element commuting withM.
• M issemidiscrete: this means the identity map fromM toM is a weak pointwise limit of completely positive maps of finite rank.
• M hasproperty E or theHakeda–Tomiyama extension property: this means that there is a projection of norm 1 from bounded operators onH toM '.
• M isinjective: any completely positive linear map from any self adjoint closed subspace containing 1 of any unital C*-algebraA toM can be extended to a completely positive map fromA toM.

There is no generally accepted term for the class of algebras above; Connes has suggested thatamenable should be the standard term.

The amenable factors have been classified: there is a unique one of each of the types I_n, I_∞, II₁, II_∞, III_λ, for 0 < λ ≤ 1, and the ones of type III₀ correspond to certain ergodic flows. (For type III₀ calling this a classification is a little misleading, as it is known that there is no easy way to classify the corresponding ergodic flows.) The ones of type I and II₁ were classified byMurray & von Neumann (1943), and the remaining ones were classified byConnes (1976), except for the type III₁ case which was completed by Haagerup.

All amenable factors can be constructed using thegroup-measure space construction ofMurray andvon Neumann for a singleergodic transformation. In fact they are precisely the factors arising ascrossed products by free ergodic actions ofZ orZ/nZ on abelian von Neumann algebrasL^∞(X). Type I factors occur when themeasure spaceX isatomic and the action transitive. WhenX is diffuse ornon-atomic, it isequivalent to [0,1] as ameasure space. Type II factors occur whenX admits anequivalent finite (II₁) or infinite (II_∞) measure, invariant under an action ofZ. Type III factors occur in the remaining cases where there is no invariant measure, but only aninvariant measure class: these factors are calledKrieger factors.

The Hilbert space tensor product of two Hilbert spaces is the completion of their algebraic tensor product. 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, and act on the tensor product of the corresponding Hilbert spaces. 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 type of the tensor product of two von Neumann algebras (I, II, or III) is the maximum of their types. Thecommutation theorem for tensor products states that

${\displaystyle (M\otimes N{)}^{\mathrm{\prime }}=M^{\mathrm{\prime }}\otimes N^{\mathrm{\prime }},}$

whereM′ denotes thecommutant ofM.

The tensor product of an infinite number of von Neumann algebras, if done naively, is usually a ridiculously large non-separable algebra. Insteadvon Neumann (1938) showed that one should choose a state on each of the von Neumann algebras, use this to define a state on the algebraic tensor product, which can be used to produce a Hilbert space and a (reasonably small) von Neumann algebra.Araki & Woods (1968) studied the case where all the factors are finite matrix algebras; these factors are calledAraki–Woods factors orITPFI 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 I₂ factors can have any type depending on the choice of states. In particularPowers (1967) found an uncountable family of non-isomorphic hyperfinite type III_λ factors for 0 < λ < 1, calledPowers factors, by taking an infinite tensor product of type I₂ factors, each with the state given by:

All hyperfinite von Neumann algebras not of type III₀ are isomorphic to Araki–Woods factors, but there are uncountably many of type III₀ that are not.

Abimodule (or correspondence) is a Hilbert spaceH with module actions of two commuting von Neumann algebras. Bimodules have a much richer structure than that of modules. Any bimodule over two factors always gives asubfactor since one of the factors is always contained in the commutant of the other. There is also a subtle relative tensor product operation due toConnes on bimodules. The theory of subfactors, initiated byVaughan Jones, reconciles these two seemingly different points of view.

Bimodules are also important for the von Neumann group algebraM of a discrete group Γ. Indeed, ifV is anyunitary representation of Γ, then, regarding Γ as the diagonal subgroup of Γ × Γ, the correspondinginduced representation onl²(Γ,V) is naturally a bimodule for two commuting copies ofM. Importantrepresentation theoretic properties of Γ can be formulated entirely in terms of bimodules and therefore make sense for the von Neumann algebra itself. For example, Connes and Jones gave a definition of an analogue ofKazhdan's property (T) for von Neumann algebras in this way.

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. Nevertheless,Voiculescu has shown that the class of non-amenable factors coming from the group-measure space construction isdisjoint from the class coming from group von Neumann algebras of free groups. LaterNarutaka Ozawa proved that group von Neumann algebras ofhyperbolic groups yieldprime type II₁ factors, i.e. ones that cannot be factored as tensor products of type II₁ factors, a result first proved by Leeming Ge for free group factors using Voiculescu'sfree entropy. Popa's work on fundamental groups of non-amenable factors represents another significant advance. The theory of factors "beyond the hyperfinite" is rapidly expanding at present, with many new and surprising results; it has close links withrigidity phenomena ingeometric group theory andergodic theory.

• The essentially bounded functions on a σ-finite measure space form a commutative (type I₁) von Neumann algebra acting on theL² functions. For certain non-σ-finite measure spaces, usually consideredpathological,L^∞(X) is not a von Neumann algebra; for example, the σ-algebra of measurable sets might be thecountable-cocountable algebra on an uncountable set. A fundamental approximation theorem can be represented by theKaplansky density theorem.
• The bounded operators on any Hilbert space form a von Neumann algebra, indeed a factor, of type I.
• If we have anyunitary representation of a groupG on a Hilbert spaceH then the bounded operators commuting withG form a von Neumann algebraG′, whose projections correspond exactly to the closed subspaces ofH invariant underG. Equivalent subrepresentations correspond to equivalent projections inG′. The double commutantG′′ ofG is also a von Neumann algebra.
• Thevon Neumann group algebra of a discrete groupG is the algebra of all bounded operators onH =l²(G) commuting with the action ofG onH 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 elementg ∈G. It is a factor (of type II₁) if every non-trivial conjugacy class ofG is infinite (for example, a non-abelian free group), and is the hyperfinite factor of type II₁ if in additionG 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.
• Thecrossed 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 thegroup-measure space construction of Murray andvon Neumann andKrieger factors.
• The von Neumann algebras of a measurableequivalence relation and a measurablegroupoid can be defined. These examples generalise von Neumann group algebras and the group-measure space construction.

Von Neumann algebras have found applications in diverse areas of mathematics likeknot theory,statistical mechanics,quantum field theory,local quantum physics,free probability,noncommutative geometry,representation theory,differential geometry, anddynamical systems.

For instance,C*-algebra provides an alternative axiomatization to probability theory. In this case the method goes by the name ofGelfand–Naimark–Segal construction. This is analogous to the two approaches to measure and integration, where one has the choice to construct measures of sets first and define integrals later, or construct integrals first and define set measures as integrals of characteristic functions.

• AW*-algebra
• Central carrier
• Tomita–Takesaki theory – Mathematical method in functional analysis

• Araki, H.; Woods, E. J. (1968), "A classification of factors",Publ. Res. Inst. Math. Sci. Ser. A,4 (1):51–130,doi:10.2977/prims/1195195263MR 0244773
• Blackadar, B. (2005),Operator algebras, Springer,ISBN 3-540-28486-9,corrected manuscript(PDF), 2013, archived fromthe original(PDF) on 2017-02-15, retrieved2015-12-15
• Connes, A. (1976), "Classification of Injective Factors",Annals of Mathematics, Second Series,104 (1):73–115,doi:10.2307/1971057,JSTOR 1971057
• Connes, A. (1994),Non-commutative geometry, Academic Press,ISBN 0-12-185860-X.
• Dixmier, J. (1981),Von Neumann algebras, 凡異出版社,ISBN 0-444-86308-7 (A translation ofDixmier, J. (1957),Les algèbres d'opérateurs dans l'espace hilbertien: algèbres de von Neumann, Gauthier-Villars, the first book about von Neumann algebras.)
• Jones, V.F.R. (2003),von Neumann algebras(PDF); incomplete notes from a course.
• Kostecki, R.P. (2013),W*-algebras and noncommutative integration,arXiv:1307.4818,Bibcode:2013arXiv1307.4818P.
• McDuff, Dusa (1969), "Uncountably many II₁ factors",Annals of Mathematics, Second Series,90 (2):372–377,doi:10.2307/1970730,JSTOR 1970730
• Murray, F. J. (2006), "The rings of operators papers",The legacy of John von Neumann (Hempstead, NY, 1988), Proc. Sympos. Pure Math., vol. 50, Providence, RI.: Amer. Math. Soc., pp. 57–60,ISBN 0-8218-4219-6 A historical account of the discovery of von Neumann algebras.
• Murray, F.J.; von Neumann, J. (1936), "On rings of operators",Annals of Mathematics, Second Series,37 (1):116–229,doi:10.2307/1968693,JSTOR 1968693. This paper gives their basic properties and the division into types I, II, and III, and in particular finds factors not of type I.
• Murray, F.J.; von Neumann, J. (1937), "On rings of operators II",Trans. Amer. Math. Soc.,41 (2), American Mathematical Society:208–248,doi:10.2307/1989620,JSTOR 1989620. This is a continuation of the previous paper, that studies properties of the trace of a factor.
• Murray, F.J.; von Neumann, J. (1943), "On rings of operators IV",Annals of Mathematics, Second Series,44 (4):716–808,doi:10.2307/1969107,JSTOR 1969107. This studies when factors are isomorphic, and in particular shows that all approximately finite factors of type II₁ are isomorphic.
• Powers, Robert T. (1967), "Representations of Uniformly Hyperfinite Algebras and Their Associated von Neumann Rings",Annals of Mathematics, Second Series,86 (1):138–171,doi:10.2307/1970364,JSTOR 1970364
• Sakai, S. (1971),C*-algebras and W*-algebras, Springer,ISBN 3-540-63633-1
• Schwartz, Jacob (1967),W-* Algebras, Gordon & Breach Publishing,ISBN 0-677-00670-5
• Shtern, A.I. (2001) [1994],"von Neumann algebra",Encyclopedia of Mathematics,EMS Press
• Takesaki, M. (1979),Theory of Operator Algebras I, II, III, Springer,ISBN 3-540-42248-X
• von Neumann, J. (1930), "Zur Algebra der Funktionaloperationen und Theorie der normalen Operatoren",Math. Ann.,102 (1):370–427,Bibcode:1930MatAn.102..685E,doi:10.1007/BF01782352,S2CID 121141866. The original paper on von Neumann algebras.
• von Neumann, J. (1936), "On a Certain Topology for Rings of Operators",Annals of Mathematics, Second Series,37 (1):111–115,doi:10.2307/1968692,JSTOR 1968692. This defines the ultrastrong topology.
• von Neumann, J. (1938),"On infinite direct products",Compos. Math.,6:1–77. This discusses infinite tensor products of Hilbert spaces and the algebras acting on them.
• von Neumann, J. (1940), "On rings of operators III",Annals of Mathematics, Second Series,41 (1):94–161,doi:10.2307/1968823,JSTOR 1968823. This shows the existence of factors of type III.
• von Neumann, J. (1943), "On Some Algebraical Properties of Operator Rings",Annals of Mathematics, Second Series,44 (4):709–715,doi:10.2307/1969106,JSTOR 1969106. This shows that some apparently topological properties in von Neumann algebras can be defined purely algebraically.
• von Neumann, J. (1949), "On Rings of Operators. Reduction Theory",Annals of Mathematics, Second Series,50 (2):401–485,doi:10.2307/1969463,JSTOR 1969463. This discusses how to write a von Neumann algebra as a sum or integral of factors.
• von Neumann, John (1961), Taub, A.H. (ed.),Collected Works, Volume III: Rings of Operators, NY: Pergamon Press. Reprints von Neumann's papers on von Neumann algebras.
• Wassermann, A. J. (1991),Operators on Hilbert space, archived fromthe original on 2007-02-16

• Operator theory
• Von Neumann algebras
• John von Neumann

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