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 .
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
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![{\ displaystyle \ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7)
![\ sigma (a)](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf9ea3f420f9f4df8439a86b1256e4157c632e9c)
![a \ in A](https://wikimedia.org/api/rest_v1/media/math/render/svg/a97387981adb5d65f74518e20b6785a284d7abd5)
![\ lambda](https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a)
![{\ displaystyle a- \ lambda \ cdot 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d80724bf9ae9229c98ed845d0a88c54dfc2ffacb)
![X_ {A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c3b750756d33b2e60efd90c59a059d0dae129cd)
![{\ displaystyle A \ rightarrow \ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90fd6ad9e59686ae2396b758e788f4d60b1024ec)
-
.
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![{\ displaystyle a_ {1}, \ ldots a_ {n} \ in A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5353578b3577f887674fc404726512499a472f)
-
.
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.
![a_ {1}, \ ldots a_ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36e5a85356a35cdefba758dc74d88be92a18b5c2)
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:
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![{\ displaystyle \ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7)
![a_ {1}, \ ldots, a_ {n} \ in A](https://wikimedia.org/api/rest_v1/media/math/render/svg/52c9160f76f73a4fa99182134bf78847d61bb473)
![{\ displaystyle \ lambda _ {1}, \ ldots \ lambda _ {n} \ in \ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/408943942d2db7cdc163da1ea86fc03d4074baf8)
![{\ displaystyle (\ lambda _ {1}, \ ldots \ lambda _ {n}) \ notin \ sigma (a_ {1}, \ ldots, a_ {n})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9670d6df4f36e9b6b96edb25e3b8dc055538dad)
- There is with
![{\ displaystyle b_ {1}, \ ldots, b_ {n} \ in A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/105f0ab89acc6574e9a9a0f8a6e3ab1b503d6dfa)
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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/660e7963fed5eb275b0e73db14cf50d712a57f84)
![{\ displaystyle \ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7)
![{\ displaystyle \ mathbb {C} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258)
![{\ displaystyle X_ {A} \ rightarrow \ mathbb {C} ^ {n}, \, \ phi \ mapsto (\ phi (a_ {1}), \ ldots, \ phi (a_ {n}))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30d3d41f8aa85e9f529a5a1f3b799ef2bd1cd370)
![X_ {A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c3b750756d33b2e60efd90c59a059d0dae129cd)
Polynomial convexity
A Banach algebra is by definition produced by elements when the smallest sub- Banach algebra is of that contains.
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![a_ {1}, \ ldots, a_ {n} \ in A](https://wikimedia.org/api/rest_v1/media/math/render/svg/52c9160f76f73a4fa99182134bf78847d61bb473)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![a_ {1}, \ ldots, a_ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/451345cc97e2ed923dd4656fcc400c3f37119cca)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03363ac0e26a2d473edbf8ffaebcd3f9fe089a8d)
![{\ displaystyle K = \ sigma (a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc96b3523e1b229228b652d6b75b9122253d412b)
![{\ displaystyle \ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7)
![a \ in A](https://wikimedia.org/api/rest_v1/media/math/render/svg/a97387981adb5d65f74518e20b6785a284d7abd5)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258)
![a_ {1}, \ ldots, a_ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/451345cc97e2ed923dd4656fcc400c3f37119cca)
![{\ displaystyle \ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7)
![{\ displaystyle K \ subset \ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03363ac0e26a2d473edbf8ffaebcd3f9fe089a8d)
![{\ displaystyle \ mathbb {C} \ setminus K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29515619c6fda22ba4132d75759e5aa863b25230)
For a set , the following statements are equivalent:
![{\ displaystyle K \ subset \ mathbb {C} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e384792633256704a3d2c8bbe259d25aa78446d)
- There is a one- element commutative -Banach algebra that is generated by elements such that .
![{\ displaystyle \ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![a_ {1}, \ ldots, a_ {n} \ in A](https://wikimedia.org/api/rest_v1/media/math/render/svg/52c9160f76f73a4fa99182134bf78847d61bb473)
![{\ displaystyle K = \ sigma (a_ {1}, \ ldots, a_ {n})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3f168c89dbe592a95af5d6e11a9fb5ed4431bab)
-
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
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Lars Hörmander : An Introduction to Complex Analysis in Several Variables , North-Holland Mathematical Library 1973