Edge (Banach algebra)

A boundary of a Banach algebra is a set considered in the mathematical theory of Banach algebras . In Funktionenalgebren over a set is a subset of , so that each function already assumes its maximum on this subset. In the general case of commutative Banach algebras, a boundary is a corresponding subset of the Gelfand space . ${\displaystyle X}$ ${\displaystyle X}$

Motivating example

Let it be the Banach algebra of the continuous functions from the unit circle into the complex numbers with the supremum norm . Therein we consider the disk algebra , which is the sub-algebra those functions that in the interior of the unit circle holomorphic are. According to the maximum principle of function theory, a is already uniquely determined by its values ​​on the boundary , it applies ${\displaystyle C(\mathbb {D} )}$ ${\displaystyle \mathbb {D} :=\{z\in \mathbb {C} \mid |z|\leq 1\}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \|\cdot \|_{\infty }}$ ${\displaystyle A(\mathbb {D} )}$ ${\displaystyle C(\mathbb {D} )}$ ${\displaystyle f\in A(\mathbb {D} )}$ ${\displaystyle \partial \mathbb {D} }$

${\displaystyle \|f\|_{\infty }=\sup _{z\in \mathbb {D} }|f(z)|}$.

In this case it even applies

For each there is a with . ${\displaystyle f\in A(\mathbb {D} )}$ ${\displaystyle y\in \partial \mathbb {D} }$ ${\displaystyle \|f\|_{\infty }=|f(y)|}$

We take this fact as an opportunity for the following definitions.

Function algebras

Let it be a compact Hausdorff space and the Banach algebra of continuous functions . A uniform algebra over is a sub- algebra that contains the constant functions and separates the points, i.e. for two different points there is a with . ${\displaystyle X}$ ${\displaystyle C(X)}$ ${\displaystyle X\rightarrow \mathbb {C} }$ ${\displaystyle X}$ ${\displaystyle A\subset C(X)}$ ${\displaystyle x,y\in X}$ ${\displaystyle f\in A}$ ${\displaystyle f(x)\not =f(y)}$

A margin for is a subset such that ${\displaystyle A}$ ${\displaystyle Y\subset X}$

${\displaystyle \forall f\in A:\exists y\in Y:\,\|f\|_{\infty }=|f(y)|}$.

Functions are already determined by their values ​​on the edge, because are with , so is and with it . ${\displaystyle A}$ ${\displaystyle f,g\in A}$ ${\displaystyle f|_{Y}=g|_{Y}}$ ${\displaystyle \textstyle \|f-g\|_{\infty }=\sup _{y\in Y}|(f-g)(y)|=\sup _{y\in Y}|f(y)-g(y)|=0}$ ${\displaystyle f=g}$

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. ${\displaystyle C_{0}(X)}$ ${\displaystyle x\in X}$ ${\displaystyle f\in A}$

Commutative Banach algebras

If a commutative Banach algebra with the Gelfand space , then the Gelfand transformation is a homomorphism . Has a single element , it is compact. The image is based on an algebra of functions . Subsets of that are Rand for are also simply called Rand for . ${\displaystyle A}$ ${\displaystyle X_{A}}$ ${\displaystyle A\rightarrow C_{0}(X_{A}),\,a\mapsto {\hat {a}}}$ ${\displaystyle A}$ ${\displaystyle X_{A}}$ ${\displaystyle {\hat {A}}:=\{{\hat {a}}\mid a\in A\}}$ ${\displaystyle X_{A}}$ ${\displaystyle X_{A}}$ ${\displaystyle {\hat {A}}}$ ${\displaystyle A}$

A boundary for a commutative Banach algebra determines the elements from only up to one element of the Jacobson radical . Namely, are with , then follows and therefore , because the Jacobson radical is exactly the core of the Gelfand transformation. So if semi- simple , the Jacobson radical disappears and each element is uniquely determined by the values ​​of the Gelfand transformation on an edge. ${\displaystyle Y\subset X_{A}}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle a,b\in A}$ ${\displaystyle {\hat {a}}|_{Y}={\hat {b}}|_{Y}}$ ${\displaystyle {\widehat {a-b}}={\hat {a}}-{\hat {b}}=0}$ ${\displaystyle a-b\in \mathrm {Rad} (A)}$ ${\displaystyle A}$

Note: If a Banach algebra is also understood as a function algebra , then there may be conflicts between these two definitions, because it is not necessarily the Gelfand space of . ${\displaystyle A}$ ${\displaystyle A\subset C(X)}$ ${\displaystyle X}$ ${\displaystyle A}$

Remarks

For many considerations it is sufficient to consider Banach algebras with one element, because if necessary one could adjoint one . In the following we therefore only consider algebras with unity.

One is of course interested in the smallest possible margins. However, there are generally no smallest margins, but GJ Schilow has shown that there is always a smallest closed margin, which is therefore called the Schilow margin and is often identified with the symbol for the topological margin . ${\displaystyle \partial A}$

For a function algebra is called a peak point when one is with and for all . By definition, peak points are included in each edge. The set of all peak points according to E. Bishop is called the Bishop edge and denotes it with , although this is generally not an edge, in extreme cases it can even be empty. For arbitrary commutative Banach algebras one sets ${\displaystyle A\subset C(X)}$ ${\displaystyle x\in X}$ ${\displaystyle f\in A}$ ${\displaystyle |f(x)|=\|f\|_{\infty }=1}$ ${\displaystyle |f(y)|<1}$ ${\displaystyle y\in X\setminus \{x\}}$ ${\displaystyle \rho A}$ ${\displaystyle \rho A}$ ${\displaystyle A}$ ${\displaystyle \rho A:=\rho {\hat {A}}\subset X_{A}}$

The situation is different for the so-called Choquet margin , the definition of which goes back to representational dimensions , as examined by G. Choquet within the framework of what is known today as the Choquet theory . This set is always an edge. Again one defines for arbitrary commutative Banach algebras . It applies ${\displaystyle \chi A}$ ${\displaystyle \chi A:=\chi {\hat {A}}\subset X_{A}}$

${\displaystyle \rho A\subset \chi A\subset \partial A}$

and in general the inclusions are real. In the standard example of disk algebra above, all three edges match.

literature

Ronald Larsen: Banach Algebras , Marcel Dekker (1973), ISBN 0-8247-6078-6

Individual evidence

1. ^ Eberhard Kaniuth: A Course in Commutative Banach Algebras , Springer-Verlag (2008), ISBN 978-0-387-72475-1 , definition 3.3.1