symmetric monoidal (∞,1)-category of spectra
AQFT and operator algebra
A von Neumann algebra or $W^*$-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) localizable measurable spaces (or measurable locales); see the section Relation to measurable spaces below.
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, $W^*$-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 $B(H)$ of bounded operators on some Hilbert space $H$ that are closed in any of the several operator topologies on $B(H)$ (except for the norm topology, which gives $C^*$-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 $W^*$-algebras. Compare the historic definitions of $C^*$-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.
For completeness, we give both the modern abstract and historical concrete definitions.
We build on the concepts of Banach space and (abstract) $C^*$-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 $1$); a $C^*$-algebra is a complex unital $C^*$-algebra, and a morphism of $C^*$-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 $A$, a predual of $A$ is a Banach space $V$ whose dual Banach space $V^*$ is isomorphic to $A$:
(or more properly, equipped with such an isomorphism $i$). Similarly, given a morphism $f\colon A \to B$ (properly, with $A$ and $B$ so equipped), a predual of $f$ is a morphism $t\colon W \to V$ whose dual is isomorphic to $f$:
With these preliminaries, a $W^*$-algebra or (“abstract”) von Neumann algebra is a $C^*$-algebra that admits a predual (or more properly, equipped with one), and a $W^*$-homomorphism is a $C^*$-homomorphism that admits a predual. In this way, the category of $W^*$-algebras becomes a subcategory of the category of $C^*$-algebras.
It is a theorem (see below) that the predual of a $W^*$-algebra or $W^*$-homomorphism is essentially unique; we speak of the predual of $A$, write it $A_*$, and identify $A$ with $(A_*)^*$ (and similarly for morphisms). (So in fact we don't need the word ‘equipped’; being a $W^*$-algebra is an extra property, not an extra structure, on a $C^*$-algebra.)
Fix a complex Hilbert space $H$ and consider the algebra $B(H)$ of bounded operators on $H$. A (“concrete”) von Neumann algebra on $H$ is a unital $*$-subalgebra of $B(H)$ that is closed in the weak operator topology, or equivalently in the ultraweak topology? or in the strong topology. As such is automatically closed in the norm topology, the von Neumann algebras form a (particularly nice) class of concrete $C^*$-algebras on $H$, where the latter are defined as unital $*$-subalgebras of $B(H)$ closed under the norm topology.
We equip a von Neumann algebra with the topology induced by its inclusion into $B(H)$ 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 $B(H)$ and treating it as an algebra on its own.
Alternatively, we can define a von Neumann algebra $A$ as a unital $*$-algebra that admits an injective morphism into $B(H)$ for some Hilbert space $H$ such that the image of the inclusion is closed in the ultraweak topology on $B(H)$. One can then prove that the topology induced on $A$ by the ultraweak topology on $B(H)$ does not depend on the choice of $H$ or the particular inclusion of $A$ into $B(H)$. 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) $W^*$-algebras and $W^*$-homomorphisms. Similarly, we get the category of representations of $W^*$-algebras on Hilbert spaces using instead the spatial morphisms of concrete von Neumann algebras.
Sakai’s theorem states that preduals considered in the abstract definition are necessarily unique. More precisely, given a von Neumann algebra $A$ we consider the category whose objects are pairs $(V,f)$, where $V$ is a Banach space and $f\colon V^*\to A$ is an isomorphism of Banach spaces. A morphism from $(V,f)$ to $(W,g)$ is a morphism $h\colon V\to W$ of Banach spaces such that $f h^* = g$.
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 $V$ of a von Neumann algebra $A$ equipped with the ultraweak topology. The topological vector space $V$ canonically embeds into the dual of $A$ as a Banach space, the embedding map being induced by the canonical continuous map from $A$ equipped with the norm topology to $A$ equipped with the ultraweak topology. Thus $V$ is also a Banach space. There is a canonical morphism of Banach spaces from $A$ to $V^*$ given by evaluating an element of $V$ on the given element of $A$. This morphism is in fact an isomorphism, hence $V$ is the predual of $A$. 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.
The easiest example of a von Neumann algebra is given by the $C^*$-algebra $B(H)$ of bounded operators on a complex Hilbert space $H$. The predual can be canonically identified with the Banach space of trace class operators.
Any $C^*$-subalgebra of $B(H)$ closed in the ultraweak topology is again a von Neumann algebra.
Another example is $L^\infty(X)$ under pointwise almost everywhere multiplication, where $X$ is a measure space or a localizable measurable space. These are (up to isomorphism) all of the commutative von Neumann algebras, according to a specialized version of the Gelfand–Naimark theorem. In the case where $X$ is a finite measure space, where we have $L^\infty(X) \subset L^2(X)$, a concrete realization $L^\infty(X) \hookrightarrow B(L^2(X))$ is given by considering $L^2(X)$ as an $L^\infty(X)$-module given by pointwise almost everywhere multiplication.
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
and its update
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.
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 $A'$ 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).
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.
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 $\mathcal{M}'$.
A vector $x \in \mathcal{H}$ is a cyclic vector if $\mathcal{M}x$ is dense in $\mathcal{H}$.
A vector $x \in \mathcal{H}$ is a separating vector if $M(x) = 0$ implies $M = 0$ for all $M \in \mathcal{M}$.
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 $\mathcal{M}'$.
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
Murray–von Neumann classification of factors
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:
Then, by the spectral theorem every selfadjoint element A is represented by it’s spectral measure E via
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.
See Gleason's theorem.
To be done…