Bounded Borel functional calculus

The bounded Borel functional calculus is a tool for studying Von Neumann algebras .

This functional calculus is an extension of the continuous functional calculus known from the theory of C * -algebras to limited Borel functions . This extension of the functional calculus is not possible in general C * -algebras, one has to restrict oneself to the smaller class of Von Neumann algebras.

construction

If one considers a bounded, monotonically growing sequence of continuous real-valued functions that are defined on the spectrum of a normal element of a Von Neumann algebra ( Hilbert space ), the pointwise limit is generally not continuous again. In is the sequence , where the continuous functional calculus is formed, a bounded and monotonically increasing (for the arrangement see positive operator ) sequence of self-adjoint operators , of which one can show that it converges in the strong operator topology . Since Von-Neumann algebras are exactly the sub-C * algebras of with one element that are closed in the strong operator topology, this limit value is again in . ${\displaystyle (f_{n})_{n}}$ ${\displaystyle a}$ ${\displaystyle A\subset L(H)}$ ${\displaystyle H}$ ${\displaystyle f}$ ${\displaystyle A}$ ${\displaystyle (f_{n}(a))_{n}}$ ${\displaystyle f_{n}(a)}$ ${\displaystyle L(H)}$ ${\displaystyle A}$

If there is another sequence of continuous real-valued functions that converges monotonically to point by point , then one can show that the limit values ​​of and agree. Therefore it makes sense to designate this limit value with . ${\displaystyle (g_{n})_{n}}$ ${\displaystyle \sigma (a)}$ ${\displaystyle f}$ ${\displaystyle (f_{n}(a))_{n}}$ ${\displaystyle (g_{n}(a))_{n}}$ ${\displaystyle f(a)}$

If the limit function is even continuous, then according to Dini's theorem , there is uniform convergence, and one recognizes that the definition just made is compatible with the continuous functional calculus. A continuation of these ideas leads to the so-called restricted Borel functional calculus (or Borel calculus for short). ${\displaystyle f}$

The limited Borel calculus

If a normal element of a Von Neumann algebra and denotes the algebra of the Borel functions defined on , then the following applies: ${\displaystyle a}$ ${\displaystyle A\subset L(H)}$ ${\displaystyle {\mathcal {B}}(\sigma (a))}$ ${\displaystyle \sigma (a)}$

• There is exactly one * -homomorphism with , and the following continuity property: If the sequence of real-valued functions converges monotonously to in point by point , then the supremum of is in the Von Neumann algebra . ${\displaystyle \Phi _{a}:{\mathcal {B}}(\sigma (a))\rightarrow A}$ ${\displaystyle \Phi _{a}(1)\,=\,\mathrm {id} _{H}}$ ${\displaystyle \Phi _{a}(\mathrm {id} _{\sigma (a)})\,=\,a}$ ${\displaystyle (f_{n})_{n}}$ ${\displaystyle f}$ ${\displaystyle {\mathcal {B}}(\sigma (a))}$ ${\displaystyle \Phi _{a}(f)}$ ${\displaystyle \{\Phi _{a}(f_{n});n\in {\mathbb {N} }\}}$ ${\displaystyle A}$

The suggestive spelling is used . The following can be shown: ${\displaystyle f(a)\,:=\,\Phi _{a}(f)}$

• There are formulas , for all . ${\displaystyle (f+g)(a)\,=\,f(a)+g(a)}$ ${\displaystyle (f\cdot g)(a)=f(a)\cdot g(a)}$ ${\displaystyle f,g\in {\mathcal {B}}(\sigma (a))}$
• Applies to everyone . ${\displaystyle f\in {\mathcal {B}}(\sigma (a))}$ ${\displaystyle f(a)^{*}={\overline {f}}(a)}$
• Is and so is true . ${\displaystyle f\in {\mathcal {B}}(\sigma (a))}$ ${\displaystyle g\in {\mathcal {B}}(\sigma (f(a)))}$ ${\displaystyle (g\circ f)(a)=g(f(a))}$
• ${\displaystyle \|f(a)\|\leq \sup\{|f(z)|;z\in \sigma (a)\}}$for everyone . ${\displaystyle f\in {\mathcal {B}}(\sigma (a))}$
• The restriction to the algebra of continuous functions is the continuous functional calculus .

A spectral mapping theorem cannot apply since the image of the spectrum under a Borel function is generally not compact again.

This functional calculus of bounded Borel functions is closely related to the spectral theorem . If, for example, self-adjoint, then the associated spectral family is, where the characteristic function denotes. ${\displaystyle a}$ ${\displaystyle (\chi _{(-\infty ,\lambda ]}|_{\sigma (a)})_{\lambda \in {\mathbb {R} }}}$ ${\displaystyle \chi _{(-\infty ,\lambda ]}}$

Applications

As an application, it should only be mentioned that this functional calculus leads to the construction of very many projections in Von Neumann algebras. If is a Borel set and denotes the associated characteristic function , then applies . Hence , that is, is an orthogonal projection in . ${\displaystyle B\subset \sigma (a)}$ ${\displaystyle \chi _{B}}$ ${\displaystyle \chi _{B}^{2}=\chi _{B}={\overline {\chi _{B}}}}$ ${\displaystyle \chi _{B}(a)^{2}\,=\,\chi _{B}(a)\,=\,\chi _{B}(a)^{*}}$ ${\displaystyle \chi _{B}(a)}$ ${\displaystyle A}$

Since continuous functions can be approximated uniformly by simple functions , one can see with the help of the functional calculus that every element of a Von Neumann algebra is a norm limit of linear combinations of orthogonal projections . In this sense, there are many projections in Von Neumann algebras. In this way, the Von Neumann theory differs considerably from the theory of C * algebras. The C * algebra of continuous functions on the interval [0,1] has the 0 and 1 function as the only projections, and thus as few projections as possible. ${\displaystyle A}$ ${\displaystyle C([0,1])}$

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

literature

• Richard V. Kadison , John R. Ringrose : Fundamentals of the Theory of Operator Algebras. Special Topics. Volume 1: Elementary Theory (= Pure and Applied Mathematics. Vol. 100, 1). Academic Press, Boston MA et al. 1983, ISBN 0-12-393301-3 .