nLab
von Neumann algebra

Context

Algebra

Functional analysis

Measure and probability theory

AQFT

Contents

Idea

A von Neumann algebra or W *W^*-algebra is an important and special kind of operator algebra, relevant in particular to measure theory and quantum mechanics/quantum field theory in its algebraic formulation as AQFT. Specifically, (non-commutative) von Neumann algebras can be understood as the formal duals of (non-commutative) localisable measurable spaces (or measurable locales); see the section Relation to measurable spaces below.

History and terminology

Since terminology varies in the literature, we will say something about this first. There are no precise definitions here; see below for those.

(Of course, W *W^*-algebras should not be confused with W-algebras? in (logarithmic) conformal field theory.)

John von Neumann originally studied certain operator algebras (back then they were called rings of operators), defined as unital **-subalgebras of the algebra B(H)B(H) of bounded operators on some Hilbert space HH that are closed in any of the several operator topologies on B(H)B(H) (except for the norm topology, which gives C *C^*-algebras); the ultraweak topology is most convenient for our purposes.

One disadvantage of such a definition is that it makes it difficult to separate properties of von Neumann algebras from properties of their representations on Hilbert spaces. For example, all faithful representations induce the same ultraweak topology on a von Neumann algebra, but different representations induce different weak topologies. Furthermore, not all von Neumann algebras come automatically equipped with a representation on a Hilbert space, such as the coproduct of two von Neumann algebras (although such a representation can always be constructed). Finally, this definition unnecessarily confuses two very distinct notions: algebras and modules (or representations).

Therefore, we may use the modern abstract terminology in which a von Neumann algebra is defined as an algebra with certain structures and properties. It then becomes a theorem that every von Neumann algebra has a free representation on a Hilbert space (such as Haagerup's standard form), so we may study von Neumann algebras in the historical concrete sense if we wish; but now we think of these as particular representations of algebras.

In the old terminology, morphisms of representations of von Neumann algebras (von Neumann algebras in the historical concrete sense) are sometimes called spatial morphisms of von Neumann algebras (as opposed to the abstract morphisms that we will define below). Similarly, the concrete von Neumann algebras themselves are sometimes called von Neumann algebras, whereas the abstract von Neumann algebras are called W *W^*-algebras. Compare the historic definitions of C *C^*-algebras, as well as other examples of concrete and abstract structures such as manifolds.

The nPOV dictates that a clear distinction between the categories of algebras and modules must be maintained, in particular, modules should not be confused with algebras. Hence we stick to the modern terminology, which also seems to be preferred in new papers on von Neumann algebras, see for example arXiv:1110.5671v1.

Definitions

For completeness, we give both the modern abstract and historical concrete definitions.

Abstract

We build on the concepts of Banach space and (abstract) C *C^*-algebra. In this definition, a Banach space is a complex Banach space and a morphism of Banach spaces is a short linear map (a complex-linear map of norm at most 11); a C *C^*-algebra is a complex unital C *C^*-algebra, and a morphism of C *C^*-algebras is a unital **-homomorphism (which is necessarily also a short linear map). Note in particular that an isomorphism of either must be an isometry.

Given a Banach space AA, a predual of AA is a Banach space VV whose dual Banach space V *V^* is isomorphic to AA:

V *iA V^* \overset{i}\to A

(or more properly, equipped with such an isomorphism ii). Similarly, given a morphism f:ABf\colon A \to B (properly, with AA and BB so equipped), a predual of ff is a morphism t:WVt\colon W \to V whose dual is isomorphic to ff:

V * i A t * f W * j B. \array { V^* & \overset{i}\to & A \\ \mathllap{t^*}\downarrow & & \downarrow\mathrlap{f} \\ W^* & \underset{j}\to & B } .

With these preliminaries, a W *W^*-algebra is a C *C^*-algebra that admits a predual (or more properly, equipped with one), and a W *W^*-homomorphism is a C *C^*-homomorphism that admits a predual. In this way, the category of W *W^*-algebras becomes a subcategory of the category of C *C^*-algebras.

It is a theorem (see below) that the predual of a W *W^*-algebra or W *W^*-homomorphism is essentially unique; we speak of the? predual of AA, write it A *A_*, and identify AA with (A *) *(A_*)^* (and similarly for morphisms). (So in fact we don't need the word ‘equipped’; being a W *W^*-algebra is an extra property, not an extra structure, on a C *C^*-algebra.)

In all of the above, one may write ‘von Neumann’ in place of ‘W *W^*‑’.

Concrete

Fix a complex Hilbert space HH and consider the algebra B(H)B(H) of bounded operators on HH. A von Neumann algebra on HH is a unital **-subalgebra of B(H)B(H) that is closed in the weak operator topology, or equivalently in the ultraweak topology?. As such is automatically closed in the norm topology, the von Neumann algebras form a (particularly nice) class of concrete C *C^*-algebras.

We equip a von Neumann algebra with the topology induced by its inclusion into B(H)B(H) equipped with the ultraweak topology. An abstract morphism of von Neumann algebras can then be defined as a unital **-homomorphism that is continous in the ultraweak topology. Here we are disregarding the data of the inclusion of a von Neumann algebra into B(H)B(H) and treating it as an algebra on its own.

Alternatively, we can define a von Neumann algebra AA as a unital **-algebra that admits an injective morphism into B(H)B(H) for some Hilbert space HH such that the image of the inclusion is closed in the ultraweak topology on B(H)B(H). One can then prove that the topology induced on AA by the ultraweak topology on B(H)B(H) does not depend on the choice of HH or the particular inclusion of AA into B(H)B(H). Hence one can define an abstract morphism of von Neumann algebras as a unital morphism of **-algebras that is continuous in the ultraweak topology.

It is a theorem that the category of (concrete) von Neumann algebras and abstract morphisms is equivalent to the category of (abstract) W *W^*-algebras and W *W^*-homomorphisms. Similarly, we get the category of representations of W *W^*-algebras on Hilbert spaces using instead the spatial morphisms of concrete von Neumann algebras.

Sakai’s theorem and properties of preduals

Sakai’s theorem states that preduals considered in the above definition are necessarily unique. More precisely, given a von Neumann algebra AA we consider the category whose objects are pairs (V,f)(V,f), where VV is a Banach space and f:V *Af\colon V^*\to A is an isomorphism of Banach spaces. A morphism from (V,f)(V,f) to (W,g)(W,g) is a morphism h:VWh\colon V\to W of Banach spaces such that fh *=gf h^* = g.

Sakai’s theorem then states that in the above category there is exactly one morphism between any pair of objects, which is necessarily an isomorphism. In particular, the category of preduals is canonically isomorphic to the terminal category.

Sakai’s theorem can be extended to morphisms of von Neumann algebras. Thus preduals of von Neumann algebras and their morphisms are unique up to a unique isomorphism, in particular we can talk about the? predual of a von Neumann algebra and the? predual of a morphism of von Neumann algebras.

The weak topology induced on a von Neumann algebra by its predual is called the ultraweak topology. The role of the ultraweak topology for von Neumann algebras is analogous to the role of the norm topology for C*-algebras. In particular, morphisms of von Neumann algebras are precisely those morphisms of C*-algebras that are continuous in the ultraweak topology.

Consider the dual space VV of a von Neumann algebra AA equipped with the ultraweak topology. The topological vector space VV canonically embeds into the dual of AA as a Banach space, the embedding map being induced by the canonical continuous map from AA equipped with the norm topology to AA equipped with the ultraweak topology. Thus VV is also a Banach space. There is a canonical morphism of Banach spaces from AA to V *V^* given by evaluating an element of VV on the given element of AA. This morphism is in fact an isomorphism, hence VV is the predual of AA. In other words, the predual of a von Neumann algebra is canonically isomorphic to its dual in the ultraweak topology. Similarly, the predual of a morphism of von Neumann algebras is canonically isomorphic to its dual in the ultraweak topology.

Elementary examples

The easiest example of a von Neumann algebra is given by the C*-algebra B(H)B(H) of bounded operators on a complex Hilbert space HH. The predual can be canonically identified with the Banach space of trace class operators.

Any C*-subalgebra of B(H)B(H) closed in the ultraweak topology is again a von Neumann algebra.

Properties of morphisms of von Neumann algebras

W *W^*-categories

Modules over von Neumann algebras

Bimodules over von Neumann algebras and Connes fusion

Modular algebra and Tomita–Takesaki theory

Gelfand duality for commutative von Neumann algebras

Relevance

The Gel’fand–Naimark theorem states that there is a contravariant equivalence between the category of commutative von Neumann algebras and the category of localizable measurable spaces; that is, the opposite category of one is equivalent to the other. See Relation to Measurable Spaces below. General von Neumann algebras are seen then as a ‘noncommutative’ measurable spaces in a sense analogous to noncommutative geometry.

The importance of von Neumann algebras for (higher) category theory and topology lays in the evidence that von Neumann algebras are deeply connected with the low dimensional quantum field theory (2d CFT, TQFT in low dimensions, inclusions of factors, quantum groups and knot theory; elliptic cohomology: works of Wenzl, Vaughan Jones, Anthony Wasserman, Kerler, Kawahigashi, Ocneanu, Szlachanyi etc.).

The highlights of their structure theory include the results on classification of factors (Alain Connes, 1970s) and theory of inclusions of subfactors (V. Jones). (Hilbert) bimodules over von Neumann algebras have a remarkable tensor product due Connes (Connes fusion). Following Segal’s manifesto

  • Graeme Segal, Elliptic cohomology (after Landweber-Stong, Ochanine, Witten, and others). Séminaire Bourbaki, Vol. 1987/88. Astérisque No. 161-162 (1988), Exp. No. 695, 4, 187–201 (1989).

and its update

  • Graeme Segal, What is an elliptic object? Elliptic cohomology, 306–317, London Math. Soc. Lecture Note Ser., 342, Cambridge Univ. Press, Cambridge, 2007.

on hypothetical connections between CFT and elliptic cohomology, Stolz and Teichner have made a case for a role von Neumann algebras seem to play in a partial realization of that program:

See also the Wikipedia entry entry for more on von Neumann algebras and a list of references and links.

General

The bicommutant theorem (as known as the double commutant theorem , or von Neumann’s double commutant theorem ) is the following result.

Let AB(H)A \subseteq B(H) be a sub-star-algebra of the C-star algebra of bounded linear operators on a Hilbert space HH. Then AA is a von Neumann algebra on HH if and only if A=AA = A'', where AA' denotes the commutant of AA.

Notice that the condition of AA being a von Neumann algebra (being closed in the weak operator topology; “weak” here can be replaced by “strong”, “ultrastrong”, or “ultraweak” as described in operator topology), which is a topological condition, is by this result equivalent to an algebraic condition (being equal to its bicommutant).

Relation to measurable spaces

The Gel’fand–Naimark theorem states that the category of localizable measurable spaces is contravariantly equivalent to (that is equivalent to the opposite of) the category of commutative von Neumann algebras. As such, arbitrary von Neumann algebras may be interpreted as ‘noncommutative’ measurable spaces in a sense analogous to noncommutative geometry.

Topics of interest for the understanding of AQFT

This paragraph will collect some facts of interest for the aspects of AQFT.

In this paragraph \mathcal{M} will always be a von Neumann algebra acting on a Hilbert space \mathcal{H} with commutant \mathcal{M}'.

Vectors

Definition

A vector xx \in \mathcal{H} is a cyclic vector if x\mathcal{M}x is dense in \mathcal{H}.

Definition

A vector xx \in \mathcal{H} is a separating vector if M(x)=0M(x) = 0 implies M=0M = 0 for all MM \in \mathcal{M}.

Theorem

The notions of cyclic and separating are dual with respect to the commutant, that is a vector is cyclic for \mathcal{M} iff it is separating for \mathcal{M}'.

Projections in von Neumann algebras

One crucial feature of von Neumann algebras is that they contain “every projection one could wish for”: there are three points that make this statement precise:

  • the linear combinations of projections are norm dense in a von Neumann algebra

  • Gleason's theorem

  • Murray–von Neumann classification of factors

Projections are norm dense

First let us note that every element AA of a von Neumann algebra can trivially be written as a linear combination of two selfadjoint elements:

A=12(A+A *)+i12i(AA *) A = \frac{1}{2} (A + A^*) + i\frac{1}{2i} (A - A^*)

Then, by the spectral theorem every selfadjoint element A is represented by it’s spectral measure E via

A= A AλE(dλ) A = \integral_{-\|A\|}^{\|A\|} \lambda E(d\lambda)

The integral converges in norm to A and all spectral projections are elements of the von Neumann algebra. It immediatly follows that the set of finite sums of multiples of projections is norm dense in every von Neumann algebra.

Gleason’s theorem

See Gleason's theorem.

Murray–von Neumann classification of factors

To be done

Miscellaneous

References

Revised on October 12, 2013 21:42:03 by Toby Bartels (98.19.41.253)