nLab enveloping von Neumann algebra

Enveloping von Neumann algebras



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Enveloping von Neumann algebras


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


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.


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'' .

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).


A C *C^*-algebra and its enveloping von Neumann algebra rarely have the same spectrum. For example, consider the C *C^*-algebra A=C 0()A=C_{0}(\mathbb{N}), where \mathbb{N} is given the discrete topology. Then, the von Neumann enveloping algebra of AA 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 *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.


Some abstract 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.


  • 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 May 16, 2024 at 11:10:46. See the history of this page for a list of all contributions to it.