Shilov edge

The Schilow-Rand (after Georgi Schilow , after the English transcription also Shilov-Rand) is a mathematical concept from the theory of commutative - Banach algebras . A version of the maximum principle known from function theory is thus transferred to commutative Banach algebras. ${\displaystyle \mathbb {C} }$

motivation

For the sake of simplicity, we limit ourselves to commutative algebras with unity . Let it be a compact Hausdorff space and a subalgebra of the Banach algebra of continuous functions with the following properties: ${\displaystyle X}$ ${\displaystyle A\subset C(X)}$ ${\displaystyle C(X)}$ ${\displaystyle X\rightarrow \mathbb {C} }$

• ${\displaystyle 1\in A}$, that is, contains the constant function 1, ${\displaystyle A}$
• ${\displaystyle \forall x\not =y\in X:\exists a\in A:a(x)\not =a(y)}$, that is, separates the points from ${\displaystyle A}$ ${\displaystyle X}$

One then says briefly that there is a function algebra . ${\displaystyle A}$ ${\displaystyle X}$

A closed subset is maximizing (for ) if for all functions , the following applies: . ${\displaystyle E\subset X}$ ${\displaystyle A}$ ${\displaystyle a\in A}$ ${\displaystyle \sup\{|a(x)|;x\in X\}=\sup\{|a(x)|;x\in E\}}$

For example , the circular disk and the disk algebra , that is the algebra of all continuous functions which in the interior holomorphic are, it is because of the maximum principle of the theory of functions each closed subset that the border contains a maximized amount. In particular, is the smallest maximizing amount. ${\displaystyle X=\mathbb {D} :=\{z\in \mathbb {C} ;\,|z|\leq 1\}}$ ${\displaystyle A}$ ${\displaystyle \mathbb {D} }$ ${\displaystyle \mathbb {D} ^{\circ }}$ ${\displaystyle \partial \mathbb {D} }$ ${\displaystyle \partial \mathbb {D} }$

Schilow-Rand for function algebras

The example of disk algebra generalizes to the following theorem, which goes back to Schilow:

• If there is a compact Hausdorff space and an algebra of functions , then the intersection of all maximizing sets for not empty and again maximizing. ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle X}$ ${\displaystyle A}$

In particular, there is a smallest maximizing set. This is called the Schilow edge of function algebra , common names are or . Since maximizing sets are boundaries , the Shilov boundary is also a boundary. ${\displaystyle A}$ ${\displaystyle S(A),{\check {S}}(A)}$ ${\displaystyle \partial A}$

Schilow-Rand for commutative Banach algebras

Let be a commutative -Banach algebra with one element. The Gelfand-space is known to be a compact Hausdorff space and the Gelfand representation is a function algebra on from. The Schilow boundary of the function algebra is called the Schilow boundary of and is also denoted by or . ${\displaystyle A}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle X_{A}}$ ${\displaystyle A\rightarrow C(X_{A})}$ ${\displaystyle A}$ ${\displaystyle {\hat {A}}}$ ${\displaystyle X_{A}}$ ${\displaystyle {\hat {A}}}$ ${\displaystyle A}$ ${\displaystyle S(A),{\check {S}}(A)}$ ${\displaystyle \partial A}$

Examples

• Disk algebra's Gelfand space is the set of scores and the map is a homeomorphism . One identifies by means of this homeomorphism with , so and it is . ${\displaystyle A}$ ${\displaystyle \delta _{z}:A\rightarrow \mathbb {C} ,\,a\mapsto \delta _{z}(a):=a(z)}$ ${\displaystyle \mathbb {D} \rightarrow X_{A},\,z\mapsto \delta _{z}}$ ${\displaystyle \mathbb {D} }$ ${\displaystyle X_{A}}$ ${\displaystyle A={\hat {A}}}$ ${\displaystyle \partial A=\partial \mathbb {D} }$
• Let be the bicylinder with radius . be the sub-Banach algebra of generated by all polynomials in two variables . One can show that the Gelfand space of is the set of scores for and that is a homeomorphism. So you can identify with as above . Then you can show that . In this case the Schilow boundary is smaller than the topological boundary of in . ${\displaystyle X:=\{(z,w)\in \mathbb {C} ^{2};\,|z|\leq 1,|w|\leq 1\}}$ ${\displaystyle (1,1)}$ ${\displaystyle A}$ ${\displaystyle C(X)}$ ${\displaystyle A}$ ${\displaystyle \delta _{x}:A\rightarrow \mathbb {C} ,a\mapsto a(x)}$ ${\displaystyle x\in X}$ ${\displaystyle X\rightarrow X_{A},x\mapsto \delta _{x}}$ ${\displaystyle X}$ ${\displaystyle X_{A}}$ ${\displaystyle \partial A=\{(z,w)\in X;\,|z|=1=|w|\}}$ ${\displaystyle X}$ ${\displaystyle \mathbb {C} ^{2}}$
• Is a compact Hausdorff space and so is . ${\displaystyle X}$ ${\displaystyle A=C(X)}$ ${\displaystyle \partial A=X_{A}}$

Remarks

• If a commutative -Banach algebra with one element then applies to the Gelfand transform that . This follows directly from the definitions, because it is a maximizing set of function algebra . The Gelfand transforms thus fulfill a maximum principle with regard to the Schilow edge. In addition, the following local version of the maximum principle applies: ${\displaystyle A}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle {\hat {a}}:X_{A}\rightarrow \mathbb {C} }$ ${\displaystyle \sup\{|{\hat {a}}(\varphi )|;\,\varphi \in X_{A}\}\,=\,\sup\{|{\hat {a}}(\varphi )|;\,\varphi \in \partial A\}}$ ${\displaystyle \partial A}$ ${\displaystyle {\hat {A}}}$

Is     open, then applies to everyone and that . ${\displaystyle U\subset X_{A}\setminus \partial A}$ ${\displaystyle a\in A}$ ${\displaystyle \varphi \in U}$ ${\displaystyle \textstyle |{\hat {a}}(\varphi )|\leq \sup _{\psi \in \partial U}|{\hat {a}}(\psi )|}$

• The Choquet margin is always contained as a dense subset in the Schilow margin.

• As is well known, the formula applies to the spectrum of . The formula applies to the edges of the spectra . ${\displaystyle \sigma (a)}$ ${\displaystyle a\in A}$ ${\displaystyle \sigma (a)={\hat {a}}(X_{A})}$ ${\displaystyle \partial \sigma (a)\subset {\hat {a}}(\partial A)}$

Individual evidence

1. ^ FF Bonsall, J. Duncan: Complete Normed Algebras . Springer-Verlag 1973, ISBN 3-540-06386-2 , §22, definition 1
2. ^ FF Bonsall, J. Duncan: Complete Normed Algebras . Springer-Verlag 1973, ISBN 3-540-06386-2 , §22, Theorem 3
3. ^ FF Bonsall, J. Duncan: Complete Normed Algebras . Springer-Verlag 1973, ISBN 3-540-06386-2 , §22, definition 4
4. ^ FF Bonsall, J. Duncan: Complete Normed Algebras . Springer-Verlag 1973, ISBN 3-540-06386-2 , §22, Theorem 11
5. ^ Ronald Larsen: Banach Algebras , Marcel Dekker (1973), ISBN 0-8247-6078-6 , Corollary 9.4.1
6. ^ FF Bonsall, J. Duncan: Complete Normed Algebras . Springer-Verlag 1973, ISBN 3-540-06386-2 , §22, sentence 7