Borel functional calculus

The limited Borel functional calculus is a tool for the study of von Neumann algebras.

This functional calculus is an extension of the well known from the theory of C *-algebras continuous functional calculus for bounded Borel functions. This extension of the functional calculus is not possible in general C *-algebras, one has to restrict it to the smaller class of von Neumann algebras.

Construction

Considering a limited, monotonically increasing sequence of continuous real-valued function defined on the spectrum of a normal member of a von Neumann - algebra ( Hilbert space ), then the point-wise limit again is not continuous in general. In the consequence is being formed with the continuous functional calculus, a limited and monotonically increasing ( for location see Positive Operator) sequence of self-adjoint operators, from which one can show that it converges in the strong operator topology. Since von Neumann algebras are precisely the closed in strong operator topology sub -C *-algebras with unit element of, this limit is again.

Is a sequence of continuous real-valued functions other, the pointwise converges monotonically, as one can show that the limits correspond to and. Therefore, it is natural to denote this limit by.

Is the limit function even continuous, then by the theorem of Dini is uniform convergence before, and you realize that just whatever was set with the steady functional calculus is compatible. A continuation of this idea leads to the so-called bounded Borel functional calculus ( or just Borel calculus ).

The limited Borel calculus

Is a normal element of a von Neumann algebra and denotes the algebra of so applies to defined Borel functions:

• There is exactly one * - homomorphism with, and the following continuity property: real-valued functions converges the result pointwise monotonically in, so is the supremum of the von Neumann algebra.

It uses the suggestive notation. Following can be shown:

• Apply the formulas for all.
• For each.
• If and so true.
• For everyone.
• The restriction to the algebra of continuous functions is continuous functional calculus.

A spectral mapping theorem can not apply because the image of the spectrum is not compactly under a Borel function in general.

This functional calculus Borel limited functions is closely related to the spectral theorem. Is about self-adjoint, then the corresponding spectral function, wherein the characteristic function indicated.

Applications

As an application should only be noted that this functional calculus leads to the construction of many projections in von Neumann algebras. Is a Borel set and refers to the corresponding characteristic function, the following applies. Therefore, that is orthogonal to a.

Since continuous functions can be uniformly approximated by simple functions, one sees using the functional calculus that each element of a von Neumann algebra is a norm limit of linear combinations of orthogonal projections from. In this sense there is in von Neumann algebras are very many projections. Thus the von Neumann theory differs significantly from the theory of C * - algebras. The C * - algebra of continuous functions on the interval [ 0,1] is the only projections of the 0 - and 1- function, and so that as little as possible projections.

This richness of projections is one of the main starting points of the theory of von Neumann algebras, the factors are classified, for example, according to the structure of their projection associations.