Disc algebra

The disk algebra (sometimes Disc algebra) is in the mathematical areas of functional analysis and theory of functions considered algebra . Many functional analytical properties of disk algebra are direct consequences of theorems of function theory.

definition

If the disc denotes, then let the set of all continuous functions that are holomorphic inside . ${\displaystyle \mathbb {D} :=\{z\in \mathbb {C} ;\,|z|\leq 1\}}$ ${\displaystyle A(\mathbb {D} )}$ ${\displaystyle f:\mathbb {D} \rightarrow \mathbb {C} }$ ${\displaystyle \mathbb {D} ^{\circ }}$

The definitions

${\displaystyle {\begin{array}{rcl}(\lambda f)(z)&:=&\lambda f(z)\\(f+g)(z)&:=&f(z)+g(z)\\(fg)(z)&:=&f(z)g(z)\\(f^{*})(z)&:=&{\overline {f({\overline {z}})}}\\\end{array}}}$,

whereby , make a complex algebra with involution , the so-called disk algebra . ${\displaystyle \lambda \in \mathbb {C} ,z\in \mathbb {D} ,f,g\in A(\mathbb {D} )}$ ${\displaystyle A(\mathbb {D} )}$ ${\displaystyle *}$

Obviously it is a subalgebra of the function algebra of continuous functions . is concerning. the maximum norm that a Banach makes complete , because after the Weierstrass convergence theorem are uniform limits of analytic functions are also holomorphic. is therefore a Banach algebra itself, even with isometric involution , that is, it applies to all . Disk algebra is also sub- Banach algebra of , the Banach algebra of all holomorphic and limited functions with the supremum norm. ${\displaystyle A(\mathbb {D} )}$ ${\displaystyle C(\mathbb {D} )}$ ${\displaystyle \mathbb {D} \rightarrow \mathbb {C} }$ ${\displaystyle A(\mathbb {D} )}$ ${\displaystyle C(\mathbb {D} )}$ ${\displaystyle A(\mathbb {D} )}$ ${\displaystyle \|f^{*}\|=\|f\|}$ ${\displaystyle f\in A(\mathbb {D} )}$ ${\displaystyle H^{\infty }}$ ${\displaystyle \mathbb {D} ^{\circ }}$

An image is obtained by restricting to the edge of . According to the maximum principle for holomorphic functions, this mapping is an isometric homomorphism . In this sense one can also understand it as a sub-Banach algebra of , that is, the disk algebra becomes a uniform algebra over . is then the set of all continuous functions that can be holomorphically continued to. This would be an alternative definition of disk algebra. ${\displaystyle \partial \mathbb {D} }$ ${\displaystyle \mathbb {D} }$ ${\displaystyle A(\mathbb {D} )\rightarrow C(\partial \mathbb {D} ),\,f\mapsto f|_{\partial \mathbb {D} }}$ ${\displaystyle A(\mathbb {D} )}$ ${\displaystyle C(\partial \mathbb {D} )}$ ${\displaystyle {\partial \mathbb {D} }}$ ${\displaystyle A(\mathbb {D} )}$ ${\displaystyle \partial \mathbb {D} }$ ${\displaystyle \mathbb {D} ^{\circ }}$

The disk algebra is generated by, that is, the smallest sub-Banach algebra that contains this function is the disk algebra itself. ${\displaystyle \mathrm {id} _{\mathbb {D} }}$

The gel land space

For each , the point evaluation is a homomorphism and thus an element of the Gelfand space of disk algebra. One can show that with the all the homomorphisms of the disk algebra with values ​​in the complex numbers have already been found, and that the mapping is a homeomorphism , whereby the so-called Gelfandt topology is given by the relatively weak - * - topology . The gel-land space of disk algebra can therefore be identified with the circular disk. In this identification, the Gelfand transform is the identity on the disk algebra. ${\displaystyle z\in \mathbb {D} }$ ${\displaystyle \delta _{z}:A(\mathbb {D} )\rightarrow \mathbb {C} ,\,f\mapsto f(z)}$ ${\displaystyle X_{A(\mathbb {D} )}}$ ${\displaystyle \delta _{z}}$ ${\displaystyle \delta \colon \mathbb {D} \rightarrow X_{A(\mathbb {D} )},\,z\mapsto \delta _{z}}$ ${\displaystyle X_{a}\subset A^{\prime }}$

The non-regularity of disk algebra

On the gel-hand space of a commutative Banach algebra, one considers the so-called shell-core topology, which is created by the final operation ${\displaystyle X_{A}}$

${\displaystyle {\overline {E}}:=\{\delta \in X_{A};\,\ker(\delta )\supset \bigcap _{\phi \in E}\ker(\phi )\}}$

given is. If this coincides with the Gelfandt topology, the Banach algebra is called regular . Disk algebra is an example of a non-regular Banach algebra. In fact, the set is closed in the Gelfandt topology when it comes to identification. If now it follows for all , and it follows from the identity theorem for holomorphic functions . Therefore, and it follows with regard to the shell-core topology, the latter cannot therefore agree with the Gelfandt topology. ${\displaystyle X_{A(\mathbb {D} )}=\mathbb {D} }$ ${\displaystyle E:=\{0\}\cup \left\{{\frac {1}{n}};\,n\in \mathbb {N} \right\}}$ ${\displaystyle f\in \bigcap _{n\in \mathbb {N} }\ker(\delta _{\frac {1}{n}})}$ ${\displaystyle f\left({\frac {1}{n}}\right)=0}$ ${\displaystyle n}$ ${\displaystyle f=0}$ ${\displaystyle \bigcap _{\phi \in E}\ker(\phi )=\{0\}}$ ${\displaystyle {\overline {E}}=X_{A}}$

The Schilowrand

If one identifies with , then the topological boundary coincides with the Schilow boundary . It has to be shown that every function of disk algebra, which, because of the identification made, agrees with its Gelfand transform, assumes its absolute maximum on the edge of the circular disk, but that is exactly the statement of the maximum principle for holomorphic functions. ${\displaystyle X_{A(\mathbb {D} )}}$ ${\displaystyle \mathbb {D} }$ ${\displaystyle \partial \mathbb {D} =\{z\in \mathbb {C} ;\,|z|=1\}}$

Maximality

As mentioned above, one can understand by means of the constraint mapping as a sub-Banach algebra of . The maximality set of Wermer states that a maximum Unterbanachalgebra is. ${\displaystyle A(\mathbb {D} )}$ ${\displaystyle f\mapsto f|_{\partial \mathbb {D} }}$ ${\displaystyle C(\partial \mathbb {D} )}$ ${\displaystyle A(\mathbb {D} )\subset C(\partial \mathbb {D} )}$

Individual evidence

1. ^ FF Bonsall, J. Duncan: Complete Normed Algebras . Springer-Verlag 1973, ISBN 3-540-06386-2 , §1.16
2. ^ FF Bonsall, J. Duncan: Complete Normed Algebras . Springer-Verlag 1973, ISBN 3-540-06386-2 , §19.3
3. ^ FF Bonsall, J. Duncan: Complete Normed Algebras . Springer-Verlag 1973, ISBN 3-540-06386-2 , §23.9
4. ^ FF Bonsall, J. Duncan: Complete Normed Algebras . Springer-Verlag 1973, ISBN 3-540-06386-2 , §22.5 for n = 1