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von Neumann algebra

Definition

A von Neumann algebra is a unital *-subalgebra of the algebra of bounded operators on a complex Hilbert space, which is closed in weak operator topology.

Clearly, they are automatically closed in norm topology, hence they form a (particularly nice) class of $C^{*}$-algebras.

Properties

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:

• Stefan Stolz and Peter Teichner, What is an elliptic object? (pdf)

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

Notice that the condition of $A$ 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 measure 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 $ℳ$ will always be a von Neumann algebra acting on a Hilbert space $\mathscr{H}$ with commutant $ℳ\prime$.

Vectors

Definition

A vector $x\in \mathscr{H}$ is a cyclic vector if $ℳx$ is dense in $\mathscr{H}$.

Definition

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

Theorem

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

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 $A$ of a von Neumann algebra can trivially be written as a linear combination of two selfadjoint elements:

$$A=\frac{1}{2}(A+A^{*})+i\frac{1}{2i}(A-A^{*})$$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

$$A={\int }_{-\parallel A\parallel }^{\parallel A\parallel }\lambda E(d\lambda )$$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…

Modular Theory

• modular theory

Miscellaneous

• split inclusion of von Neumann algebras

Remarks

• Von Neumann algebras may also be defined abstractly as (abstract) $C^{*}$-algebras with a predual.

• Von Neumann algebras are also sometimes called $W^{*}$-algebras; they should not be confused with $W$-algebras in (logarithmic) conformal field theory.

• Combining the previous two remarks, some authors use ‘$W^{*}$-algebra’ for the abstract concept and ‘von Neumann algebra’ for the concrete concept. Equivalently, then, a von Neumann algebra is a $W^{*}$-algebra equipped with a free action on a Hilbert space (and it's a theorem that any $W^{*}$-algebra may be so equipped).

References

• Jacob Lurie, von Neumann algebras, lecture series (2011) (web)