Uniform algebra

Uniform algebras are investigated in the mathematical branch of functional analysis , more precisely in the theory of Banach algebras . These are closed sub- algebras of algebras of continuous functions on a compact form with regard to the supreme norm . Since you last the uniform standard calls, because it defines the topology of uniform convergence (Engl. Uniform convergence ), the common name in the German explained uniform algebra .

Definitions

For a compact Hausdorff space, let the -algebra of continuous functions . contains the constant functions and separates after Lemma Urysohn the points of , that is, two different points there is a function with . With the supremum norm there becomes a commutative Banach algebra. ${\displaystyle X}$ ${\displaystyle C(X)}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle C\rightarrow \mathbb {C} }$ ${\displaystyle C(X)}$ ${\displaystyle X}$ ${\displaystyle x_{1},x_{2}\in X}$ ${\displaystyle f\in C(X)}$ ${\displaystyle f(x_{1})\not =f(x_{2})}$ ${\displaystyle \textstyle \|f\|_{X}:=\sup _{x\in X}|f(x)|}$ ${\displaystyle C(X)}$

A uniform algebra on a compact Hausdorff space is a -closed sub- algebra that contains the constants and separates the points from . ${\displaystyle X}$ ${\displaystyle \|\cdot \|}$ ${\displaystyle A\subset C(X)}$ ${\displaystyle X}$

Examples

• The algebras of the form , compact Hausdorff space, are themselves uniform algebras. ${\displaystyle C(X)}$ ${\displaystyle X}$
• For a compact subset be the sub-algebra of all functions that can be uniformly approximated by polynomials . If the unit circle is the discalgebra . ${\displaystyle X\subset \mathbb {C} ^{n}}$ ${\displaystyle P(X)\subset C(X)}$ ${\displaystyle X}$ ${\displaystyle S^{1}}$ ${\displaystyle P(S^{1})}$
• For a compact subset be the sub-algebra of all functions that can be uniformly approximated by rational and in a neighborhood of holomorphic functions. ${\displaystyle X\subset \mathbb {C} ^{n}}$ ${\displaystyle R(X)\subset C(X)}$ ${\displaystyle X}$ ${\displaystyle X}$
• For a compact subset be the subalgebra of all functions that are holomorphic , where the interior of denotes. Is the unit circle , then the disk algebra. So it is, but note that we are dealing with uniform algebras over different sets. ${\displaystyle X\subset \mathbb {C} ^{n}}$ ${\displaystyle A(X)\subset C(X)}$ ${\displaystyle \mathrm {int} (X)}$ ${\displaystyle \mathrm {int} (X)}$ ${\displaystyle X}$ ${\displaystyle D\subset \mathbb {C} }$ ${\displaystyle A(D)}$ ${\displaystyle P(S^{1})\cong A(D)}$

Remarks

The concept of uniform algebra depends very much on. The algebra is not a uniform algebra , because the points 0 and 1 are not separated by. But is isometrically isomorphic to the uniform algebra of the continuous functions on the unit circle , because ${\displaystyle X}$ ${\displaystyle A:=\{f\in C([0,1])\mid f(0)=f(1)\}}$ ${\displaystyle X=[0,1]}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle C(S^{1})}$ ${\displaystyle S^{1}}$

${\displaystyle \Phi :C(S^{1})\rightarrow A,\quad (\Phi (f))(t):=f(e^{2\pi \mathrm {i} t}),\quad t\in [0,1]}$

is obviously such an isomorphism.

If there is a uniform algebra , each defines a homomorphism ${\displaystyle A\subset C(X)}$ ${\displaystyle X}$ ${\displaystyle x\in X}$

${\displaystyle \delta _{x}:A\rightarrow \mathbb {C} ,\quad \delta _{x}(f):=f(x)}$.

Since the constant function contains 1 and , is , that is, is an element of Gelfand space . Since separating the points from is for two different points . thats why ${\displaystyle A}$ ${\displaystyle \delta _{x}(1)=1}$ ${\displaystyle \delta _{x}\not =0}$ ${\displaystyle \delta _{x}}$ ${\displaystyle X_{A}}$ ${\displaystyle A}$ ${\displaystyle X}$ ${\displaystyle \delta _{x_{1}}\not =\delta _{x_{2}}}$ ${\displaystyle x_{1},x_{2}\in X}$

${\displaystyle \delta :X\rightarrow X_{A},\quad x\mapsto \delta _{x}}$

a homeomorphic embedding that is generally not surjective.

By definition, this is a boundary of the Banach algebra and can therefore be regarded as a closed set of the Shilov boundary of . In analogy to the above examples, or in the theory of uniform algebras, one tries to generalize, among other things, the formation of terms from the theory of analytic functions for the remainder. ${\displaystyle X\cong \delta (X)\subset X_{A}}$ ${\displaystyle \partial _{A}}$ ${\displaystyle A}$ ${\displaystyle P(X)}$ ${\displaystyle A(X)}$ ${\displaystyle X_{A}\setminus X}$

Special cases

A uniform algebra is called antisymmetric if all real-valued functions from are constant. The disk algebra mentioned above is an example of an antisymmetric uniform algebra. ${\displaystyle A}$ ${\displaystyle A}$

A uniform algebra is called maximal if there is no real uniform algebra between and . According to Wermer's maximality theorem , disk algebra is maximal. Disk algebra also occurs as a uniform algebra and is obviously not maximal in . The concept of maximality therefore depends on . ${\displaystyle A\subset C(X)}$ ${\displaystyle A}$ ${\displaystyle C(X)}$ ${\displaystyle X}$ ${\displaystyle P(S^{1})\subset C(S^{1})}$ ${\displaystyle A(D)\subset C(D)}$ ${\displaystyle C(D)}$ ${\displaystyle X}$

A uniform algebra is called Dirichlet if the vector space of the real parts of the functions from is a dense subset in . Is additional , it is called a Dirichlet algebra . ${\displaystyle A\subset C(X)}$ ${\displaystyle X}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathrm {Re} (A):=\{\mathrm {Re} (f)\mid f\in A\}}$ ${\displaystyle A}$ ${\displaystyle \mathrm {Re} (C(X))}$ ${\displaystyle X=\partial _{A}}$ ${\displaystyle A}$

A uniform algebra is called logmodular if the set of logarithms of the values ​​of functions that can be inverted in is dense in . Dirichlet algebras are logmodular. ${\displaystyle A\subset C(X)}$ ${\displaystyle \mathrm {log} |A^{-1}|:=\{\mathrm {log} \circ |f|\mid f\in A^{-1}\}}$ ${\displaystyle A}$ ${\displaystyle \mathrm {Re} (C(X))}$

generalization

The above example begs the question, when is a commutative Banach algebra with a unit element isometrically isomorphic to a uniform algebra. Using the Gelfand transformation and the spectral radius formula , it is easy to see that a commutative Banach algebra with one element is isometrically isomorphic to a uniform algebra if and only if for all . This is even used as a definition in H. Goldmann's textbook given below. If the Banach algebra is finitely generated, its Gelfand space can be identified with the common spectrum of a generating system and therefore with a compact subset of . ${\displaystyle A:=\{f\in C([0,1])\mid f(0)=f(1)\}}$ ${\displaystyle A}$ ${\displaystyle \|a^{2}\|=\|a\|^{2}}$ ${\displaystyle a\in A}$ ${\displaystyle \mathbb {C} ^{n}}$

This characterization can be used for a generalization to Fréchet algebras . A Fréchet algebra is called a uniform Fréchet algebra if the Fréchet space topology is given by a sequence of submultiplicative semi - norms for which holds for all and . ${\displaystyle A}$ ${\displaystyle (p_{n})_{n\in \mathbb {N} }}$ ${\displaystyle p_{n}(a^{2})=p_{n}(a)^{2}}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle a\in A}$

Individual evidence

1. ↑ TW Gamelin: Uniform Algebras , Chelsea Publishing Company 1969, Chapter II
2. ↑ SA Gregoryan, TV Tonev: Shift-invariant Uniform Algebras on Groups , Birkhauser-Verlag 2006, ISBN 3-7643-7606-6 , Chapter 1.2: Uniform Algebras
3. ^ J. Agler, JE McCarthy: Pick Interpolation and Hilbert Function Spaces , American Mathematical Society 2002, ISBN 0-8218-2898-3 , definition 13.13
4. ↑ TW Gamelin: Uniform Algebras , Chelsea Publishing Company 1969, Chapter II.5: Maximal Subalgebras
5. ↑ TW Gamelin: Uniform Algebras , Chelsea Publishing Company 1969, Chapter II.4: Logmodular Algebras
6. ^ H. Goldmann: Uniform Fréchet Algebras , Elsevier Science Publishing Company, ISBN 0-444-88488-2 , definition 1.1.2
7. ↑ Winfried Kaballo: Advanced Course Functional Analysis and Operator Theory , Springer-Verlag 2014, ISBN 978-3-642-37794-5 , chap. 13.4: Uniform algebras and common spectra .
8. ^ H. Goldmann: Uniform Fréchet Algebras , Elsevier Science Publishing Company, ISBN 0-444-88488-2 , definition 4.1.2