Holomorphic functional calculus of several variables

The holomorphic functional calculus of several variables is used in mathematics to investigate commutative - Banach algebras . This functional calculus allows the application of a holomorphic function of several variables to a tuple consisting of elements of the Banach algebra. This generalizes the holomorphic functional calculus , which refers to the holomorphic functions of a variable. ${\displaystyle \mathbb {C} }$

motivation

The simplest holomorphic functions of several variables are polynomials with . ${\displaystyle p=p(z_{1},\ldots ,z_{n})=\sum _{j_{1},\ldots ,j_{n}=0}^{N}\beta _{j_{1},\ldots ,j_{n}}z_{1}^{j_{1}}\ldots z_{n}^{j_{n}}}$ ${\displaystyle \beta _{j_{1},\ldots ,j_{n}}\in \mathbb {C} }$

Substituting elements a - algebra in such a polynomial results . ${\displaystyle a_{1},\ldots ,a_{n}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle p(a_{1},\ldots ,a_{n})=\sum _{j_{1},\ldots ,j_{n}=0}^{N}\beta _{j_{1},\ldots ,j_{n}}a_{1}^{j_{1}}\ldots a_{n}^{j_{n}}}$

In order to define meaningful, one first needs an identity element 1 in the Banach algebra. If the set of polynomials is variable, a mapping is obtained . In order for this mapping to become a homomorphism, the elements must commute with one another , because it is a commutative ring and therefore must ${\displaystyle a_{j}^{0}}$ ${\displaystyle P_{n}}$ ${\displaystyle n}$ ${\displaystyle P_{n}\rightarrow A,\,p\mapsto p(a_{1},\ldots ,a_{n})}$ ${\displaystyle a_{1},\ldots ,a_{n}}$ ${\displaystyle P_{n}}$

${\displaystyle a_{j}a_{k}=z_{j}(a_{1},\ldots ,a_{n})z_{k}(a_{1},\ldots ,a_{n})=(z_{j}z_{k})(a_{1},\ldots ,a_{n})}$

${\displaystyle =(z_{k}z_{j})(a_{1},\ldots ,a_{n})=z_{k}(a_{1},\ldots ,a_{n})z_{j}(a_{1},\ldots ,a_{n})=a_{k}a_{j}}$

be. Therefore one has to limit oneself to commutative -Banach algebras with one element. If one has no one element, one can adjoint one . ${\displaystyle \mathbb {C} }$

The calculation

The holomorphic functional calculus of a variable for an element deals with holomorphic functions that are defined in a neighborhood of the spectrum . In the situation considered here, there are elements of a commutative Banach algebra with 1 and holomorphic functions are considered in variables that are defined in a neighborhood of the common spectrum . If the Gelfand space is from , then is ${\displaystyle a\in A}$ ${\displaystyle \sigma (a)}$ ${\displaystyle n}$ ${\displaystyle a_{1},\ldots ,a_{n}}$ ${\displaystyle n}$ ${\displaystyle \sigma (a_{1},\ldots ,a_{n})}$ ${\displaystyle X_{A}}$ ${\displaystyle A}$

${\displaystyle \sigma (a_{1},\ldots ,a_{n})\,=\,\{(\varphi (a_{1}),\ldots ,\varphi (a_{n}));\,\varphi \in X_{A}\}\,\subset \,\mathbb {C} ^{n}}$

a compact subset of the . With methods of function theory one shows ${\displaystyle \mathbb {C} ^{n}}$

• Let be a commutative -Banach algebra with 1 and be a holomorphic function defined in a neighborhood of . Then there is an element with ${\displaystyle A}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle a_{1},\ldots ,a_{n}\in A}$ ${\displaystyle f}$ ${\displaystyle \sigma (a_{1},\ldots ,a_{n})}$ ${\displaystyle a\in A}$

${\displaystyle \varphi (a)=f(\varphi (a_{1}),\ldots ,\varphi (a_{n}))}$ for all ${\displaystyle \varphi \in X_{A}}$

The element from the above sentence is generally not clearly defined, because there can be different elements in with for everyone . But then applies to everyone . Since the kernels of the homomorphisms from are exactly the maximum ideals of (see article Banachalgebra ), lies in the average of all maximum ideals, i.e. in the Jacobson radical of . So if the Jacobson radical is (that is, if the Banach algebra is semi-simple ) then one can infer. In this case that is clearly determined from the above sentence. The following sentence is then obtained: ${\displaystyle a}$ ${\displaystyle a,b}$ ${\displaystyle A}$ ${\displaystyle \varphi (a)=\varphi (b)}$ ${\displaystyle \varphi \in X_{A}}$ ${\displaystyle a-b\in {\mbox{ker}}(\varphi )}$ ${\displaystyle \varphi \in X_{A}}$ ${\displaystyle X_{A}}$ ${\displaystyle A}$ ${\displaystyle a-b}$ ${\displaystyle A}$ ${\displaystyle \{0\}}$ ${\displaystyle a=b}$ ${\displaystyle a}$

• Let be a semi-simple, commutative -Banach algebra with 1, and let be an open neighborhood of the common spectrum . Is the set of all in defined holomorphic functions, so there is every exactly one element with ${\displaystyle A}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle a_{1},\ldots ,a_{n}\in A}$ ${\displaystyle U}$ ${\displaystyle \sigma (a_{1},\ldots ,a_{n})}$ ${\displaystyle H(U)}$ ${\displaystyle U}$ ${\displaystyle f\in H(U)}$ ${\displaystyle a\in A}$

${\displaystyle \varphi (a)=f(\varphi (a_{1}),\ldots ,\varphi (a_{n}))}$for everyone . ${\displaystyle \varphi \in X_{A}}$

This clearly defined element is called . In the situation of the above sentence then still applies ${\displaystyle f(a_{1},\ldots ,a_{n})}$

• The mapping is a homomorphism that continues the establishment . ${\displaystyle H(U)\rightarrow A,\,f\mapsto f(a_{1},\ldots ,a_{n})}$ ${\displaystyle P_{n}\rightarrow A}$

In this sense, one can insert elements of semi-simple, commutative -Banach algebras with 1 into holomorphic functions that are defined in a neighborhood of the common spectrum. ${\displaystyle \mathbb {C} }$

These theorems were proved by Schilow under the additional assumption that the Banach algebra is finitely generated . The general case was then shown by Arens and Calderón ; further versions can be found in the Bourbaki volume mentioned below .

Schilow's Idempotent Theorem

The best-known application of these methods goes back to Schilow himself. Schilow's idempotent theorem makes a statement about the existence of idempotent elements in commutative Banach algebras with 1:

• Let be a commutative -Banach algebra with one element and let the Gelfand space be a disjoint union of non-empty compact subsets and . Then there is an idempotent element with for all and for all ${\displaystyle A}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle X_{A}=K_{0}\cup K_{1}}$ ${\displaystyle K_{0}}$ ${\displaystyle K_{1}}$ ${\displaystyle e\in A}$ ${\displaystyle \varphi (e)=0}$ ${\displaystyle \varphi \in K_{0}}$ ${\displaystyle \varphi (e)=1}$ ${\displaystyle \varphi \in K_{1}}$

To the proof sketch of Schilow's idempotent theorem

For the proof, which can only be roughly indicated here, one obtains suitable elements so that their common spectrum is also a disjoint union of compact sets and . Then there are disjoint open environments and from and . The function that is equal to 0 and equal to 1 is holomorphic in a neighborhood of the common spectrum. Is also semi-simple, the element you are looking for is. In a further step of the proof, one frees oneself from the additional requirement of semi-simplicity. ${\displaystyle a_{1},\ldots ,a_{n}\in A}$ ${\displaystyle L_{0}}$ ${\displaystyle L_{1}}$ ${\displaystyle U_{0}}$ ${\displaystyle U_{1}}$ ${\displaystyle L_{0}}$ ${\displaystyle L_{1}}$ ${\displaystyle f}$ ${\displaystyle U_{0}}$ ${\displaystyle U_{1}}$ ${\displaystyle A}$ ${\displaystyle e=f(a_{1},\ldots ,a_{n})}$

Another important application is

• A semi-simple, commutative -Banach algebra has a unit if and only if the Gelfand space is compact. ${\displaystyle \mathbb {C} }$ ${\displaystyle A}$ ${\displaystyle X_{A}}$

literature

• FF Bonsall, J. Duncan: Complete Normed Algebras . Springer-Verlag 1973, ISBN 3540063862
• Bourbaki: Élements de mathématique, XXXII, Theories spectrales , Paris: Hermann 1967
• Gunning - Rossi : Analytic functions of several complex variables . Prentice Hall 1965
• Lars Hörmander : An Introduction to Complex Analysis in Several Variables , North-Holland Mathematical Library 1973