Bishop Edge

The Bishop margin , alternatively also called the minimal margin , is a term from the mathematical theory of Banach algebras . It goes back to Errett Bishop , who used this set to characterize the Choquet rim in certain cases. It is the minimum amount that is contained in each boundary of a commutative Banach algebra . One therefore speaks of the minimum margin . It should be noted that in general this is not a real border, in extreme cases it can even be empty, as demonstrated by an example below.

definition

Let it be a compact Hausdorff space and the Banach algebra of continuous functions with the supremum norm . A function algebra over is a subalgebra that contains the constant functions and separates the points, that is, for two different points there is a with . ${\displaystyle X}$ ${\displaystyle C(X)}$ ${\displaystyle X\rightarrow \mathbb {C} }$ ${\displaystyle \|\cdot \|_{\infty }}$ ${\displaystyle X}$ ${\displaystyle A\subset C(X)}$ ${\displaystyle x,y\in X}$ ${\displaystyle f\in A}$ ${\displaystyle f(x)\not =f(y)}$

A point is peak point to if there is a function there, so ${\displaystyle x\in X}$ ${\displaystyle A}$ ${\displaystyle f\in A}$

${\displaystyle \|f\|_{\infty }=|f(x)|=1}$   and     for everyone   . ${\displaystyle |f(y)|<1}$ ${\displaystyle y\in X\setminus \{x\}}$

The set of all peak points is called the Bishop margin or the minimum margin . ${\displaystyle \rho A}$

The definition can easily be generalized to locally compact spaces . We then consider the Banach algebra of those continuous functions that vanish at infinity , and replace the requirement to contain the constant functions by stating that there must not be a point in which each has the value 0. Finally, if there is any commutative Banach algebra with the Gelfand space , then one defines the Gelfand transform in as the Bishop boundary of the function algebra . The last definition can conflict with the first in constructed cases, because if a commutative Banach algebra is also implemented as a function algebra in an algebra , the Gelfand space does not necessarily have to be of. ${\displaystyle C_{0}(X)}$ ${\displaystyle x\in X}$ ${\displaystyle f\in A}$ ${\displaystyle A}$ ${\displaystyle X_{A}}$ ${\displaystyle \rho A}$ ${\displaystyle C_{0}(X_{A})}$ ${\displaystyle A}$ ${\displaystyle C(X)}$ ${\displaystyle X}$ ${\displaystyle A}$

Examples

• Is compact and metrizable , so is . In this case the minimum margin is equal to the maximum possible amount for it. ${\displaystyle X}$ ${\displaystyle \rho C(X)=X}$
• In the case of disk algebra on the unit circle , the Bishop boundary coincides with the topological boundary . ${\displaystyle A(\mathbb {D} )}$ ${\displaystyle \mathbb {D} }$ ${\displaystyle \partial \mathbb {D} =\{z\in \mathbb {C} \mid |z|=1\}}$

X is the union of the circular disk and the segment placed at the zero point

• We now give a function algebra for which the Bishop margin is not closed. In addition be

${\displaystyle X:=\mathbb {D} \cup [0,1]\cdot e_{3}\subset \mathbb {R} ^{3}}$with the relative topology .

${\displaystyle X}$is a compact space and contains the function algebra ${\displaystyle C(X)}$

${\displaystyle A:=\{f\in C(X)\mid f|_{\mathrm {int} (\mathbb {D} )}{\text{ ist holomorph }}\}}$,

where denotes the interior of the unit circle. For the Schilow edge one shows ${\displaystyle \mathrm {int} (\mathbb {D} )}$ ${\displaystyle \partial A}$

${\displaystyle \partial A=\partial \mathbb {D} \cup [0,1]\cdot e_{3}}$.

The bishop margin turns out to be one point smaller

${\displaystyle \rho A=\partial \mathbb {D} \cup (0,1]\cdot e_{3}=\partial A\setminus \{(0,0,0)\}}$.

We only show that there is no peak point and therefore does not belong to the Bishop edge. If 0 were a peak point, there would be a with and for all other points. But since it is holomorphic, this contradicts the maximum principle of function theory . ${\displaystyle 0=(0,0,0)}$ ${\displaystyle f\in A}$ ${\displaystyle |f(0)|=1}$ ${\displaystyle |f(x)|<1}$ ${\displaystyle f|_{\mathrm {int} (\mathbb {D} )}}$

• Is an uncountable set , so is the product chamber with the product topology according to the set of Tikhonov compact. One can show that in this case there are no peak points at all, that is, it is . So in this case the Bishop Edge is not a Edge. ${\displaystyle \Lambda }$ ${\displaystyle X=[0,1]^{\Lambda }}$ ${\displaystyle \rho C(X)=\emptyset }$

characterization

For closed function algebras, peak points can be characterized topologically differently, which then leads to a characterization of the Bishop boundary. To this end, a point is called a strong edge point if there is a function with and for all for every open environment of . With this definition the following sentence applies: ${\displaystyle A\subset C(X)}$ ${\displaystyle x\in X}$ ${\displaystyle U}$ ${\displaystyle x}$ ${\displaystyle f\in A}$ ${\displaystyle \|f\|_{\infty }=|f(x)|=1}$ ${\displaystyle |f(y)|<1}$ ${\displaystyle y\in X\setminus U}$

If a compact Hausdorff space and a closed function algebra, then the following statements about are equivalent: ${\displaystyle X}$ ${\displaystyle A\subset C(X)}$ ${\displaystyle x\in X}$

• ${\displaystyle x\in \rho A}$
• ${\displaystyle x}$is a strong boundary point and is a G _δ -set . ${\displaystyle \{x\}}$

The last example mentioned becomes clear, because in uncountable there is no one-point set a G _δ -set. ${\displaystyle X=[0,1]^{\Lambda }}$ ${\displaystyle \Lambda }$

Relationship to the Choquet edge

E. Bishop examined the space of the peak points to show the following theorem:

If a compact Hausdorff space and a closed function algebra, then the Bishop boundary agrees with the Choquet boundary and is a G _δ -set. ${\displaystyle X}$ ${\displaystyle A\subset C(X)}$

In particular, the bishop edge is an edge in this case.

Individual evidence

1. ^ Ronald Larsen: Banach Algebras , Marcel Dekker (1973), ISBN 0-8247-6078-6 , definition 9.1.3
2. ^ Ronald Larsen: Banach Algebras , Marcel Dekker (1973), ISBN 0-8247-6078-6 , example 9.3.1
3. ^ Ronald Larsen: Banach Algebras , Marcel Dekker (1973), ISBN 0-8247-6078-6 , example 9.3.1
4. ↑ Ronald Larsen: Banach Algebras , Marcel Dekker (1973), ISBN 0-8247-6078-6 , example 9.3.4
5. ^ Ronald Larsen: Banach Algebras , Marcel Dekker (1973), ISBN 0-8247-6078-6 , example 9.3.5
6. ↑ Ronald Larsen: Banach Algebras , Marcel Dekker (1973), ISBN 0-8247-6078-6 , Lemma 9.7.3
7. ^ E. Bishop: A minimal boundary for function algebras , Pacific Journal of Mathematics (1959), Volume 9, pages 629-642
8. ^ Ronald Larsen: Banach Algebras , Marcel Dekker (1973), ISBN 0-8247-6078-6 , Theorem 9.7.2