Choquet edge

The Choquet edge , named after Gustave Choquet , is a term from the mathematical theory of the Banach algebras . It is a boundary of a commutative Banach algebra , which is always contained in the Shilow boundary .

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 . ${\displaystyle X}$ ${\displaystyle C(X)}$ ${\displaystyle X\rightarrow \mathbb {C} }$ ${\displaystyle \|\cdot \|_{\infty }}$ ${\displaystyle X}$ ${\displaystyle A\subset C(X)}$ ${\displaystyle x,y\in X}$ ${\displaystyle f\in A}$ ${\displaystyle f(x)\not =f(y)}$

It is further a Algebrennorm on , is the corresponding dual space and finally ${\displaystyle \|\cdot \|}$ ${\displaystyle A}$ ${\displaystyle A^{*}}$

${\displaystyle A_{1}^{*}:=\{f\in A\mid \|f\|=f(1)=1\}}$

the so-called state space of , where 1 here also denotes the constant function 1, which by definition is contained in and there plays the role of a single element . This is a convex , weakly - * - compact set in and therefore has many extreme points according to the Kerin-Milman theorem . Let it be the set of these extreme points. ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle A^{*}}$ ${\displaystyle \mathrm {ext} (A_{1}^{*})}$

For each , the scoring is apparently one item off . We are now interested in those points for which there is even an extreme point of the state space: ${\displaystyle x\in X}$ ${\displaystyle \delta _{x}\colon A\rightarrow \mathbb {C} ,\,\delta _{x}(f):=f(x)}$ ${\displaystyle A_{1}^{*}}$ ${\displaystyle x\in X}$ ${\displaystyle \delta _{x}}$

${\displaystyle \chi A:=\{x\in X\mid \delta _{x}\in \mathrm {ext} (A_{1}^{*})\}}$     is called Choquet-Rand of . ${\displaystyle A}$

If there is any commutative Banach algebra with one element and if its Gelfand space is , then one defines the Gelfand transform as the Choquet boundary of the function algebra in . 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. ${\displaystyle A}$ ${\displaystyle X_{A}}$ ${\displaystyle \chi A}$ ${\displaystyle C(X_{A})}$ ${\displaystyle A}$ ${\displaystyle C(X)}$ ${\displaystyle X}$ ${\displaystyle A}$

The choquet border is a border

Is a compact Hausdorff space and a function algebra, then holds ${\displaystyle X}$ ${\displaystyle A\subset C(X)}$

• ${\displaystyle \chi A\not =\emptyset }$, the Choquet margin is not empty.
• ${\displaystyle \chi A}$ is an edge for ${\displaystyle A}$
• ${\displaystyle {\overline {\chi A}}=\partial A}$, that is, the Choquet margin lies close to the Schilow margin.

Relationship to the Bishop Edge

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 . ${\displaystyle X}$ ${\displaystyle A\subset C(X)}$

Examples

• If a compact Hausdorff space is, then is and therefore agrees with the Schilow margin. There are examples of rooms for which the bishop edge of is empty, e.g. B. . ${\displaystyle X}$ ${\displaystyle \chi C(X)=X}$ ${\displaystyle X}$ ${\displaystyle C(X)}$ ${\displaystyle X=[0,1]^{\mathbb {R} }}$
• The standard example and model for the development of the marginal term is disk algebra on the unit circle . Here, too, the Choquet edge and the Schilow edge coincide and are equal to the topological edge . ${\displaystyle A(\mathbb {D} )\subset C(\mathbb {D} )}$ ${\displaystyle \mathbb {D} :=\{z\in \mathbb {C} \mid |z|\leq 1\}}$ ${\displaystyle \partial \mathbb {D} :=\{z\in \mathbb {C} \mid |z|=1\}}$

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 Choquet boundary 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 ${\displaystyle \mathrm {int} (\mathbb {D} )}$ ${\displaystyle \partial A}$

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

In the article on the Bishop-Rand it was stated that this is the same

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

is. According to the above relationship between the Bishop edge and the Choquet edge, this is also the same as the Choquet edge, which in this example is really contained in the Schilow edge. As expected from the above sentence, here is . ${\displaystyle {\overline {\chi A}}=\partial A}$

Descriptive dimensions

The Choquet margin can be characterized by so-called descriptive measures , which suggests the connection to the Choquet theory . For a compact Hausdorff space is Banach space of the regular complex measurements on the total variation as standard. A measure is called a representative measure for an if ${\displaystyle X}$ ${\displaystyle M(X)}$ ${\displaystyle X}$ ${\displaystyle \mu \in M(X)}$ ${\displaystyle \varphi \in A^{*}}$

${\displaystyle \|\varphi \|=\|\mu \|}$   and     for everyone   . ${\displaystyle \varphi (f)=\int _{X}f(x)\mathrm {d} \mu (x)}$ ${\displaystyle f\in A}$

For example , the one-point measure is a descriptive measure because ${\displaystyle \varphi =\delta _{x}}$ ${\displaystyle \varepsilon _{x}}$

${\displaystyle \|\delta _{x}\|=1=\|\varepsilon _{x}\|}$   and   . ${\displaystyle \delta _{x}(f)=f(x)=\int _{X}f(y)\mathrm {d} \varepsilon _{x}(y)}$

But there could be other descriptive measures. For example, if the disk algebra, then for all of the Cauchy integral formula ${\displaystyle A=A(\mathbb {D} )\subset C(\mathbb {D} )}$ ${\displaystyle f\in A(\mathbb {D} )}$

${\displaystyle \delta _{0}(f)=f(0)={\frac {1}{2\pi i}}\int _{\partial \mathbb {D} }{\frac {f(z)}{z}}\mathrm {d} z=\int _{\mathbb {D} }f(z)\mathrm {d} \mu (z)}$

with a concentrated measure . In this case, the representing measure is not clear. A similar argument shows that the representing measure for none is unique. A uniqueness of performing measure is only functional with before. This situation also applies in the general case; more precisely, the following sentence applies: ${\displaystyle \partial \mathbb {D} }$ ${\displaystyle \mu }$ ${\displaystyle \delta _{z},\,|z|<1}$ ${\displaystyle \delta _{z}}$ ${\displaystyle |z|=1}$

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

• ${\displaystyle x\in \chi A}$
• The representing measure for is clearly determined. ${\displaystyle \delta _{x}}$

Completed function algebras

If one additionally demands that the function algebra is closed with regard to the supremum norm, then the following statements about an are equivalent: ${\displaystyle A\subset C(X)}$ ${\displaystyle x\in X}$

• ${\displaystyle x\in \chi A}$
• To and each open neighborhood of there is a with , and for all . ${\displaystyle 0<\alpha <\beta <1}$ ${\displaystyle U}$ ${\displaystyle x}$ ${\displaystyle f\in A}$ ${\displaystyle \|f\|_{\infty }=1}$ ${\displaystyle |f(x)|>\beta }$ ${\displaystyle |f(y)|<\alpha }$ ${\displaystyle y\in X\setminus U}$
• For every open environment of there is one with , and for all . ${\displaystyle U}$ ${\displaystyle x}$ ${\displaystyle f\in A}$ ${\displaystyle \|f\|_{\infty }=1}$ ${\displaystyle \textstyle |f(x)|>{\frac {3}{4}}}$ ${\displaystyle \textstyle |f(y)|<{\frac {1}{4}}}$ ${\displaystyle y\in X\setminus U}$
• For every open environment of there is a with and for all . ${\displaystyle U}$ ${\displaystyle x}$ ${\displaystyle f\in A}$ ${\displaystyle \|f\|_{\infty }=1=|f(x)|}$ ${\displaystyle \textstyle |f(y)|<1}$ ${\displaystyle y\in X\setminus U}$
• There's a family in with . ${\displaystyle (f_{\alpha })_{\alpha \in I}}$ ${\displaystyle A}$ ${\displaystyle \textstyle \{x\}=\bigcap _{\alpha \in I}\{y\in X\mid |f_{\alpha }(y)|=\|f\|_{\infty }\}}$

application

With the help of the Choquet margin one can prove the following theorem, which goes back to Robert Phelps :

Let it be a compact Hausdorff space and a function algebra. If a linear and surjective isometry is with , then it is multiplicative, that is, it applies to all . ${\displaystyle X}$ ${\displaystyle A\subset C(X)}$ ${\displaystyle T\colon A\rightarrow A}$ ${\displaystyle T(1)=1}$ ${\displaystyle T}$ ${\displaystyle T(fg)=T(f)T(g)}$ ${\displaystyle f,g\in A}$

The central argument in the proof is to use the multiplicativity of scores for points from the Choquet margin. This shows that and agree on all points of the Choquet edge and must therefore be the same, because the Choquet edge is an edge. This is set out in the textbook by R. Larsen mentioned below. ${\displaystyle \delta _{x}}$ ${\displaystyle x}$ ${\displaystyle T(fg)}$ ${\displaystyle T(f)T(g)}$

Individual evidence

1. ^ Ronald Larsen: Banach Algebras , Marcel Dekker (1973), ISBN 0-8247-6078-6 , definition 9.4.2
2. ^ Ronald Larsen: Banach Algebras , Marcel Dekker (1973), ISBN 0-8247-6078-6 , Corollary 9.4.1
3. ^ E. Bishop: A minimal boundary for function algebras , Pacific Journal of Mathematics (1959), Volume 9, pages 629-642
4. ^ Ronald Larsen: Banach Algebras , Marcel Dekker (1973), ISBN 0-8247-6078-6 , Theorem 9.7.2
5. ^ RR Phelps: Lectures on Choquet's Theorem , van Nostrand (1966), Corollary 8.2
6. ^ Ronald Larsen: Banach Algebras , Marcel Dekker (1973), ISBN 0-8247-6078-6 , example 9.3.5
7. ↑ Ronald Larsen: Banach Algebras , Marcel Dekker (1973), ISBN 0-8247-6078-6 , Theorem 9.6.7
8. ^ Ronald Larsen: Banach Algebras , Marcel Dekker (1973), ISBN 0-8247-6078-6 , Theorem 9.7.1
9. ↑ Ronald Larsen: Banach Algebras , Marcel Dekker (1973), ISBN 0-8247-6078-6 , Theorem 9.5.1