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enveloping von Neumann algebra

Enveloping von Neumann algebras

Context

Algebra

Functional analysis

Operator algebra

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Enveloping von Neumann algebras

Definitions

Let AA be a C *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)E(A) of AA is the free von Neumann algebra on AA. That is, we have an adjunction

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

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

Definition defines the functor EE 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 AA; this is a Hilbert space HH and representation π:AB(H)\pi : A \to B(H), the a universal representation? of AA. The image π(A)\pi(A) is a subspace of B(H)B(H); consider its double commutant? (or equivalently its closure in the weak operator topology) AA''. Ignoring the representation of AA'' on HH, AA'' is a von Neumann algebra, the enveloping von Neumann algebra of AA.

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

Aπ(A)Cl wk *(π(A))=A. A \to \pi(A) \to Cl_{wk^*}(\pi(A)) = A'' .
Definition

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

Here, the unit of the adjunction is simply ii.

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 *C^*-algebra and its enveloping von Neumann algebra have the same spectrum. The functional calculus on a C *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 *C^*-algebra to obtain an element of its enveloping von Neumann algebra.

Applications

Some astract treatments of quantum mechanics use C *C^*-algebras, while others use von Neumann algebras. If a physical system is described by a C *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

Last revised on April 30, 2018 at 19:53:05. See the history of this page for a list of all contributions to it.