Abelian von Neumann algebra

Abelian von Neumann algebras in the mathematical branch of functional analysis looked at von Neumann algebras whose multiplication is commutative.

Examples

• The algebra of diagonal matrices in the finite-dimensional Hilbert space is an abelian von Neumann algebra, which is obviously isomorphic to the algebra with componentwise multiplication. The subalgebra of constant multiples of the identity matrix is also an abelian von Neumann algebra.
• The sequence space with the component-wise multiplication is the infinite-dimensional generalization of the first example. This Abelian von Neumann algebra operates on the Hilbert space.
• If the Lebesgue measure on the unit interval [0,1], so every function defined by the formula a continuous linear operator. The algebra is an abelian von Neumann algebra that you simply denoted by.

Abelian von Neumann algebras as L ∞ - algebras

The example above is up to isomorphism already the most general case. The following applies:

Is an abelian von Neumann algebra on a Hilbert space, so there is a locally compact Hausdorff space and a positive measure on with support to so that isomorphic. Isomorphism in this context means isometric * - isomorphism. If the Hilbert space is separable, then one can choose as a compact metric space.

Conversely, if a measure space with lokalkompaktem, any function defined by the formula a continuous linear operator. The algebra is an abelian von Neumann algebra, which is isomorphic to. is maximum among all abelian von Neumann algebras.

Abelian von Neumann algebras on separable Hilbert spaces

The Isomorphisklassen of abelian von Neumann algebras on a separable Hilbert space can overlook completely; restrict ourselves to the maximum von Neumann algebras, we can even replace isomorphism by unitary equivalence.

Let to and isomorphic to the von Neumann algebras from the above examples. Every maximal abelian von Neumann algebra on a separable Hilbert space is unitarily equivalent to exactly one of the algebras

In this case, the names of two von Neumann algebras over and over unitarily equivalent if there is a unitary operator, so that is an isomorphism.

Abelian von Neumann algebras as C *-algebras

Abelian von Neumann algebras are particular commutative C *-algebras and as such according to the set of Gelfand - Neumark isomorphic to an algebra of continuous functions on a compact Hausdorff space. is an extremal of disconnected space. The converse is not true, ie there is extremal disconnected, compact Hausdorff spaces such that the algebra is isomorphic not to a von Neumann algebra.

Spectral

Is a self-adjoint, bounded linear operator on the Hilbert space, the generated by von Neumann algebra is abelian and contains all the spectral projections of. Abelian von Neumann algebras are therefore a natural framework for the development of spectral theory, which can be extended to unbounded self-adjoint operators. This program is executed consistently.