Common spectrum

The common spectrum of finitely many elements of a commutative - Banach algebra generalizes the concept of the spectrum of an element used in mathematics for the investigation of Banach algebras . ${\displaystyle \mathbb {C} }$

Motivation and Definition

Let be a -Banach algebra with one element 1. The spectrum of an element is the set of all complex numbers for which the element cannot be inverted . If one denotes the set of all - homomorphisms , then in the case of a commutative Banach algebra one has the relation ${\displaystyle A}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \sigma (a)}$ ${\displaystyle a\in A}$ ${\displaystyle \lambda }$ ${\displaystyle a-\lambda \cdot 1}$ ${\displaystyle X_{A}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle A\rightarrow \mathbb {C} }$

${\displaystyle \sigma (a)\,=\,\{\phi (a);\,\phi \in X_{A}\}\,\subset \,\mathbb {C} }$.

This relationship can also be extended to several elements of a Banach algebra. For a commutative -Banach algebra with unity and elements one sets ${\displaystyle \mathbb {C} }$ ${\displaystyle A}$ ${\displaystyle a_{1},\ldots a_{n}\in A}$

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

${\displaystyle \sigma (a_{1},\ldots ,a_{n})}$is called the common spectrum of the elements . If the Banach algebra has no one element, one adjoint one element and define the common spectrum there. ${\displaystyle a_{1},\ldots a_{n}}$

properties

Invertibility

The relationship between spectrum and invertibility is generalized to the situation of several elements as follows:

Is a commutative -Banachalgebra with 1 , so the following are equivalent: ${\displaystyle A}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle a_{1},\ldots ,a_{n}\in A}$ ${\displaystyle \lambda _{1},\ldots \lambda _{n}\in \mathbb {C} }$

• ${\displaystyle (\lambda _{1},\ldots \lambda _{n})\notin \sigma (a_{1},\ldots ,a_{n})}$
• There is with ${\displaystyle b_{1},\ldots ,b_{n}\in A}$ ${\displaystyle \sum _{k=1}^{n}(a_{k}-\lambda _{k})b_{k}\,=\,1}$

compactness

The common spectrum of finitely many elements of a commutative -Banach algebra is a compact subset of . According to the definition of the weak - * - topology , which is considered in the Gelfand space , the mapping is continuous. Since the Gelfand space of a Banach algebra is compact with 1, this results in the compactness of the common spectrum, because continuous images of compact sets are again compact. ${\displaystyle \sigma (a_{1},\ldots ,a_{n})}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {C} ^{n}}$ ${\displaystyle X_{A}\rightarrow \mathbb {C} ^{n},\,\phi \mapsto (\phi (a_{1}),\ldots ,\phi (a_{n}))}$ ${\displaystyle X_{A}}$

Polynomial convexity

A Banach algebra is by definition produced by elements when the smallest sub- Banach algebra is of that contains. ${\displaystyle A}$ ${\displaystyle a_{1},\ldots ,a_{n}\in A}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle a_{1},\ldots ,a_{n}}$

For a subset one can show that if and only if is compact and connected it holds for a commutative -Banach algebra with one element, which is generated by an element . ${\displaystyle K\subset \mathbb {C} }$ ${\displaystyle K=\sigma (a)}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle a\in A}$ ${\displaystyle K}$ ${\displaystyle \mathbb {C} \setminus K}$

A corresponding topological characterization of sets im , which appear as a common spectrum of generating elements of a commutative -Banach algebra with one element, does not succeed. Since a compact set is polynomial convex if and only if is connected, the following theorem represents a generalization of the above facts: ${\displaystyle \mathbb {C} ^{n}}$ ${\displaystyle a_{1},\ldots ,a_{n}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle K\subset \mathbb {C} }$ ${\displaystyle \mathbb {C} \setminus K}$

For a set , the following statements are equivalent: ${\displaystyle K\subset \mathbb {C} ^{n}}$

• There is a one- element commutative -Banach algebra that is generated by elements such that . ${\displaystyle \mathbb {C} }$ ${\displaystyle n}$ ${\displaystyle a_{1},\ldots ,a_{n}\in A}$ ${\displaystyle K=\sigma (a_{1},\ldots ,a_{n})}$
• ${\displaystyle K}$is compact and polynomial convex .

If one has a finite number of elements that do not produce the entire Banach algebra, their common spectrum is generally not polynomial convex.

literature

• FF Bonsall, J. Duncan: Complete Normed Algebras . Springer-Verlag 1973, ISBN 3540063862
• Lars Hörmander : An Introduction to Complex Analysis in Several Variables , North-Holland Mathematical Library 1973