Holomorphic functional calculus

The holomorphic functional calculus is a fundamental method from the mathematical theory of Banach algebras . Roughly speaking, in this functional calculus elements of a -Banach algebra are inserted into holomorphic functions that are defined in a neighborhood of the spectrum of the element, whereby the insertion into polynomials is generalized. ${\displaystyle \mathbb {C} }$

construction

Let it be a -Banach algebra with one element . If , then the spectrum is non-empty (see Gelfand-Mazur theorem ). Let further be a holomorphic function defined in an open neighborhood of . Whilst it can not directly use , but the Cauchy integral formula provides a representation of the function values of , in such a setting can still be carried out. ${\displaystyle A}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle e}$ ${\displaystyle a\in A}$ ${\displaystyle \sigma (a)}$ ${\displaystyle f:U\rightarrow {\mathbb {C} }}$ ${\displaystyle U}$ ${\displaystyle \sigma (a)}$ ${\displaystyle a}$ ${\displaystyle f}$ ${\displaystyle f}$

The cycle shown in red includes the spectrum shown in blue.

There is a cycle of simply closed paths that run entirely within and encompass the spectrum. The Cauchy's integral formula reads for points within , and the Banach algebra element can actually be used in it. One can show that the integral ${\displaystyle \Gamma =\gamma _{1}+\ldots +\gamma _{n}}$ ${\displaystyle U}$ ${\displaystyle \textstyle f(z)={\frac {1}{2\pi i}}\int _{\Gamma }{\frac {f(\zeta )}{\zeta -z}}d\zeta }$ ${\displaystyle z}$ ${\displaystyle \Gamma }$

${\displaystyle {\frac {1}{2\pi i}}\int _{\Gamma }f(\zeta )(\zeta e-a)^{-1}d\zeta }$

converges in the sense of the standard topology . There , the expression is defined in the integrand and is a continuous function . It can also be shown that this value does not depend on the specific choice of . This is why the value of this integral is denoted by in suggestive notation . ${\displaystyle \Gamma \cap \sigma (a)=\emptyset }$ ${\displaystyle (\zeta e-a)^{-1}}$ ${\displaystyle \zeta \mapsto f(\zeta )(\zeta e-a)^{-1}}$ ${\displaystyle \Gamma \to A}$ ${\displaystyle \Gamma }$ ${\displaystyle f(a)}$

For a compact set, let the set of holomorphic functions be defined in a neighborhood . Are and two such functions, we can and on the average of the domains of and explain. This becomes an algebra. With the above definitions we get a mapping . This mapping is called the holomorphic functional calculus of a . ${\displaystyle K}$ ${\displaystyle {\mathcal {H}}(K)}$ ${\displaystyle K}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle f\,+\,g}$ ${\displaystyle f\cdot g}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle {\mathcal {H}}(K)}$ ${\displaystyle {\mathbb {C} }}$ ${\displaystyle \Phi _{a}:{\mathcal {H}}(\sigma (a))\rightarrow A,\,\,f\mapsto f(a)}$

The requirement that a one element has is not an essential restriction, because if necessary one can adjoint a one element and apply the functional calculus in the enlarged Banach algebra. ${\displaystyle A}$

properties

The holomorphic functional calculus for an element has the following properties. ${\displaystyle \Phi _{a}}$ ${\displaystyle a\in A}$

• ${\displaystyle \Phi _{a}:{\mathcal {H}}(\sigma (a))\rightarrow A}$is a homomorphism, i. H. apply formulas , . ${\displaystyle (f+g)(a)\,=\,f(a)+g(a)}$ ${\displaystyle (f\cdot g)(a)=f(a)\cdot g(a)}$
• If there is a power series representation in a neighborhood of the spectrum , then in is considered to be an absolutely convergent series . ${\displaystyle f\in {\mathcal {H}}(\sigma (a))}$ ${\displaystyle f(z)=\sum _{n=0}^{\infty }\lambda _{n}z^{n}}$ ${\displaystyle f(a)=\sum _{n=0}^{\infty }\lambda _{n}a^{n}}$ ${\displaystyle A}$
• Is and so is true . ${\displaystyle f\in {\mathcal {H}}(\sigma (a))}$ ${\displaystyle g\in {\mathcal {H}}(\sigma (f(a)))}$ ${\displaystyle (g\circ f)(a)=g(f(a))}$
• The spectral mapping principle applies : for everyone . ${\displaystyle \sigma (f(a))\,=\,f(\sigma (a))}$ ${\displaystyle f\in {\mathcal {H}}(\sigma (a))}$

So one can imagine actually using the Banach algebra elements in holomorphic functions; the obvious algebraic operations behave as expected.

application

As a typical application of the holomorphic functional calculus we prove the following theorem:

For a -Banach algebra with one element are equivalent: ${\displaystyle \mathbb {C} }$ ${\displaystyle A}$ ${\displaystyle e}$

• ${\displaystyle A}$owns projections with . ${\displaystyle p}$ ${\displaystyle 0\not =p\not =e}$
• ${\displaystyle A}$has elements with a disjoint spectrum.

Since for a projection with is obviously incoherent, it only needs to be shown that there is a projection different from 0 and when has an incoherent spectrum. As is incoherent, there are open sets and in so , , and . The function , which is equal to 1 and equal to 0, is holomorphic as a locally constant function, i.e. an element from . Then according to the spectral mapping theorem, and therefore . There follows . Hence a projection of the kind we are looking for. ${\displaystyle \sigma (p)=\{0,1\}}$ ${\displaystyle p}$ ${\displaystyle 0\not =p\not =e}$ ${\displaystyle e}$ ${\displaystyle a\in A}$ ${\displaystyle \sigma (a)}$ ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle U\cap \sigma (a)\not =\emptyset }$ ${\displaystyle V\cap \sigma (a)\not =\emptyset }$ ${\displaystyle \sigma (a)\subset U\cup V}$ ${\displaystyle U\cap V=\emptyset }$ ${\displaystyle f}$ ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle {\mathcal {H}}(\sigma (a))}$ ${\displaystyle \sigma (f(a))=f(\sigma (a))=\{0,1\}}$ ${\displaystyle 0\not =f(a)\not =e}$ ${\displaystyle f=f\cdot f}$ ${\displaystyle f(a)=(f\cdot f)(a)=f(a)\cdot f(a)}$ ${\displaystyle f(a)}$

This statement can be tightened to Schilow's idempotent theorem , which requires the deeper holomorphic functional calculus of several variables .

literature

• FF Bonsall, J. Duncan: Complete Normed Algebras . Springer-Verlag 1973, ISBN 3-540-06386-2 , Ch. 1, §7: "A Functional Calculus for a Single Banach Algebra Element"
• J. Dixmier : Les C * -algèbres et leurs représentations , Gauthier-Villars, 1969
• RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras , 1983, ISBN 0-12-393301-3
• M. Takesaki, Theory of Operator Algebras I (Springer 1979, 2002)