Wermer's maximality theorem

The maximality set of Wermer , even Wermers maximality set called English Wermer's maximality theorem is a mathematical theorem that between complex analysis and functional analysis is based. The theorem goes back to the mathematician John Wermer and deals with maximality properties of a special Banach function algebra over the field of complex numbers .

Formulation of the sentence

Wermer's maximality theorem can be given as follows:

Let be the closed unit disk in the body of complex numbers , whose topological edge is the unit sphere . ${\displaystyle \mathbb {D} =\{z\in \mathbb {C} \colon |z|\leq 1\}\subseteq \mathbb {C} }$ ${\displaystyle S^{1}=\{z\in \mathbb {C} \colon |z|=1\}}$

Let the -Banach algebra of continuous complex-valued functions be provided with the usual point-by-point defined operations and the maximum norm . ${\displaystyle {\mathcal {C}}={\mathcal {C}}(S^{1}\;,\;\mathbb {C} )}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle f\colon S^{1}\to \mathbb {C} }$

Finally, let us be the subset of those functions which have a continuous continuation on such that this continuation function is even holomorphic on the open unit disk . ${\displaystyle {\mathcal {A}}}$ ${\displaystyle f\in {\mathcal {C}}}$ ${\displaystyle \mathbb {D} }$ ${\displaystyle \mathbb {E} ={\mathbb {D} }^{\circ }=\{z\in \mathbb {C} \colon |z|<1\}}$

Then:

${\displaystyle {\mathcal {A}}}$forms a true closed partial algebra of and is maximal as such . ${\displaystyle {\mathcal {C}}}$

That means:

${\displaystyle {\mathcal {A}}}$is a true closed partial algebra of and there is no other closed partial algebra of with . ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {B}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {A}}\subsetneqq {\mathcal {B}}\subsetneqq {\mathcal {C}}}$

Characterization of the partial algebra

With regard to the affiliation of a given function to the partial algebra , the following criterion applies  : ${\displaystyle f\in {\mathcal {C}}}$ ${\displaystyle {\mathcal {A}}}$

${\displaystyle f\in {\mathcal {A}}\iff }$     ${\displaystyle \forall k\in \mathbb {N} \colon \int _{0}^{2\pi }f(e^{{i}\phi })\;e^{ik\phi }\;d{\phi }=0}$

Generalization of the maximality theorem

Wermer's maximality theorem has the following generalization, from which it emerges, among other things, that there are also further maximal closed partial algebras in : ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {C}}}$

Be a closed partial algebra of which ${\displaystyle {\mathcal {D}}}$ ${\displaystyle {\mathcal {C}}}$

(1) contains the constant complex-valued functions

and

(2) a function whose restriction to the unit sphere is injective . ${\displaystyle g\in {\mathcal {C}}}$ ${\displaystyle g{\upharpoonright }{S^{1}}}$

Then forms a true closed partial algebra of , which as such is maximal, or it is . ${\displaystyle {\mathcal {D}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {D}}={\mathcal {C}}}$

See also

• Disc algebra

swell

• Paul J. Cohen : A note on constructive methods in Banach algebras . In: Proceedings of the American Mathematical Society . tape 12 , 1961, pp. 159-163 , doi : 10.2307 / 2034144 . MR0124515
• Edmund Landau , Dieter Gaier : Presentation and justification of some recent results of the function theory . 3rd, expanded edition. Springer-Verlag , Berlin (inter alia) 1986, ISBN 3-540-16886-9 . MR0869998
• G. Lumer : On Wermer's maximality theorem . In: Inventiones Mathematicae . tape 8 , 1969, p. 236-237 , doi : 10.1007 / BF01406075 . MR0251542
• Walter Rudin : Analyticity, and the maximum modulus principle . In: Duke Mathematical Journal . tape 20 , 1953, pp. 449-457 , doi : 10.1215 / S0012-7094-53-02045-6 . MR0056076
• John Wermer : On algebras of continuous functions . In: Proceedings of the American Mathematical Society . tape 4 , 1953, pp. 866-869 , doi : 10.2307 / 2031819 . MR0058877

Individual evidence

1. ↑ ^{a b c d} Edmund Landau, Dieter Gaier: Presentation and justification of some recent results of the theory of functions. 1986, pp. 174-181
2. ↑ is the complex amount function . ${\displaystyle z\mapsto |z|}$
3. So ↑ consists of the inner points of . ${\displaystyle \mathbb {E} }$ ${\displaystyle \mathbb {D} }$
4. ↑ is essentially to be equated with disk algebra . ${\displaystyle {\mathcal {A}}}$