nLab enveloping von Neumann algebra

Definitions

Let $A$ be a $C^*$-algebra. We may define its enveloping von Neumann algebra in a few different but equivalent ways.

Definition

The enveloping von Neumann algebra $E(A)$ of $A$ is the free von Neumann algebra on $A$. That is, we have an adjunction

$$W^* Alg \overset{E}\underset{U}\leftrightharpoons C^* Alg ,$$

where $W^* Alg$ is the category of von Neumann algebras (which are $C^*$-algebras with preduals) and von Neumann algebra homomorphisms (which are $C^*$-algebra homomorphisms with preduals), $C^* Alg$ is the category of $C^*$-algebras and $C^*$-algebra homomorphisms, $U$ is the forgetful functor or inclusion functor, and $E$ is the functor that we wish to define.

Definition defines the functor $E$ up to unique natural isomorphism, if it exists. We may prove that it exists by the adjoint functor theorem or by proving that one of the explicit constructions below satisfies the relevant universal property.

Definition

Consider the direct sum of the the GNS representations of the positive? linear functionals on $A$; this is a Hilbert space $H$ and representation $\pi : A \to B(H)$, the a universal representation? of $A$. The image $\pi(A)$ is a subspace of $B(H)$; consider its double commutant? (or equivalently its closure in the weak operator topology) $A''$. Ignoring the representation of $A''$ on $H$, $A''$ is a von Neumann algebra, the enveloping von Neumann algebra of $A$.

To obtain an adjunction from Definition , we need also the unit of the adjunction, which is the map

$$A \to \pi(A) \to Cl_{wk^*}(\pi(A)) = A'' .$$

Definition

Think of $A$ as a Banach space, and consider its double dual $A^{**}$. We have (as with any Banach space) a short linear map $i\colon A \to A^{**}$, so that $i(A)$ has the structure of a $C^*$-algebra. Since $i(A)$ is weak-$*$-dense in $A^{**}$ and the $C^*$-algebraic operations are continuous, they extend to $A^{**}$. These extensions turn $A^{**}$ into a Banach algebra; the $C^*$ identity also extends, making $A^{**}$ into $C^*$-algebra. Since $A^{**}$ has $A^*$ as a predual, it is a von Neumann algebra, the enveloping von Neumann algebra of $A$.

Here, the unit of the adjunction is simply $i$. The counit of the adjunction is given by a similar procedure: for every $W^*$-algebra $M$ with predual $M_{*}$ we can combine the bidual of the canonical embedding $M_{*} \to (M_{*})^{**}$ with the isomorphism $(M_{*})^{*} \to M$ to provide a norm 1 projection $M^{**} \to M$. This can be shown to be a unital normal $*$-homomorphism.

The claim that the definitions above are all equivalent is the Sherman–Takeda theorem, due (naturally enough) to Sherman (1950) and Takeda (1954).

Properties

A $C^*$-algebra and its enveloping von Neumann algebra rarely have the same spectrum. For example, consider the $C^*$-algebra $A=C_{0}(\mathbb{N})$, where $\mathbb{N}$ is given the discrete topology. Then, the von Neumann enveloping algebra of $A$ is the set of all bounded sequences, whose spectrum is homeomorphic to the Stone-Čech compactification of $\mathbb{N}$.

The functional calculus on a $C^*$-algebra (which treats continuous functions) extends to the functional calculus on its enveloping von Neumann algebra (which treats measurable functions). In particular, we can apply a measurable function to an element of a $C^*$-algebra to obtain an element of its enveloping von Neumann algebra.

Applications

Some abstract treatments of quantum mechanics use $C^*$-algebras, while others use von Neumann algebras. If a physical system is described by a $C^*$-algebra in the first case, then it may described by its enveloping von Neumann algebra in the second case.

References

• S. Sherman (1950). The second adjoint of a C* algebra. Proceedings of the International Congress of Mathematicians 1950 (1): 470. American Mathematical Society.

• Zirô Takeda (1954). Conjugate spaces of operator algebras. Proceedings of the Japan Academy 30 (2): 90–95.

• Wikipedia (English): Enveloping von Neumann algebra, Sherman–Takeda theorem

category: operator algebras