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A subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u0951901.png" />, closed with respect to the [[Topology of uniform convergence|topology of uniform convergence]], of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u0951902.png" /> of continuous functions on a compactum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u0951903.png" /> that contains all constant functions and separates the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u0951904.png" />. The last condition means that for each pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u0951905.png" /> of distinct points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u0951906.png" /> there is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u0951907.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u0951908.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u0951909.png" />. Uniform algebras are usually provided with the sup norm:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519010.png" /></td> </tr></table>
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Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519011.png" />. Every Banach algebra with an identity (even without assuming commutativity) and with a norm satisfying the latter condition is isomorphic to a uniform algebra.
+
A subalgebra  $  A $,
 +
closed with respect to the [[Topology of uniform convergence|topology of uniform convergence]], of the algebra $  C ( X) $
 +
of continuous functions on a compactum  $  X $
 +
that contains all constant functions and separates the points of  $  X $.
 +
The last condition means that for each pair  $  x, y $
 +
of distinct points in  $  X $
 +
there is a function  $  f $
 +
in  $  A $
 +
for which  $  f ( x) \neq f ( y) $.  
 +
Uniform algebras are usually provided with the sup norm:
  
The uniform algebras form an important subclass of the class of commutative Banach algebras (cf. [[Commutative Banach algebra|Commutative Banach algebra]]) over the field of complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519012.png" />.
+
$$
 +
\| f \|  =  \sup _ { X }  | f ( x) |.
 +
$$
  
To each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519013.png" /> corresponds a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519014.png" />, defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519015.png" />. Therefore <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519016.png" /> is in a natural way topologically imbedded in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519017.png" /> of maximal ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519018.png" />, and under the corresponding identification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519019.png" /> contains the Shilov boundary (cf. [[Boundary (in the theory of uniform algebras)|Boundary (in the theory of uniform algebras)]]). In the study of uniform algebras a major role is played by peak points (that is, points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519020.png" /> at which the strict maximum modulus of at least one element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519021.png" /> is attained), by multiplicative probability measures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519022.png" /> (that is, representing measures of homomorphisms from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519023.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519024.png" />) and by measures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519025.png" /> that are orthogonal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519026.png" />. Many concrete results relating to uniform algebras touch on the relations between these notions.
+
Here  $  \| f ^ { 2 } \| = \| f \|  ^ {2} $.  
 +
Every Banach algebra with an identity (even without assuming commutativity) and with a norm satisfying the latter condition is isomorphic to a uniform algebra.
  
A uniform algebra is called symmetric if with each function its complex conjugate belongs to the algebra. According to the [[Stone–Weierstrass theorem|Stone–Weierstrass theorem]], each symmetric uniform algebra on a compactum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519027.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519028.png" />. The so-called anti-symmetric uniform algebras, containing no real-valued functions apart from the constants, form a kind of opposite class. A typical example is the algebra of all functions that are analytic in the open unit disc of the complex plane and continuous on its closure (the disc algebra). The Shilov–Bishop theorem: Each uniform algebra can be obtained from anti-symmetric uniform algebras by  "glueing"  in a certain way. Even more refined classification theorems are known. At the same time arbitrary uniform algebras do not reduce to algebras of analytic functions of the type of the disc algebra. For example, it is possible to construct a uniform algebra on a one-dimensional compactum, which coincides with its space of maximal ideals, such that all points of the compactum are peak points and at the same time only the zero element of the algebra can be zero on a non-empty open set.
+
The uniform algebras form an important subclass of the class of commutative Banach algebras (cf. [[Commutative Banach algebra|Commutative Banach algebra]]) over the field of complex numbers  $  \mathbf C $.
 +
 
 +
To each point  $  x \in X $
 +
corresponds a homomorphism  $  \phi _ {x} :  A \rightarrow \mathbf C $,
 +
defined by  $  \phi _ {x} ( f  ) = f ( x) $.
 +
Therefore  $  X $
 +
is in a natural way topologically imbedded in the space  $  \mathop{\rm MSpec} ( A) $
 +
of maximal ideals of  $  A $,
 +
and under the corresponding identification  $  X $
 +
contains the Shilov boundary (cf. [[Boundary (in the theory of uniform algebras)|Boundary (in the theory of uniform algebras)]]). In the study of uniform algebras a major role is played by peak points (that is, points of  $  X $
 +
at which the strict maximum modulus of at least one element of  $  A $
 +
is attained), by multiplicative probability measures on  $  X $(
 +
that is, representing measures of homomorphisms from  $  A $
 +
to  $  \mathbf C $)
 +
and by measures on  $  X $
 +
that are orthogonal to  $  A $.
 +
Many concrete results relating to uniform algebras touch on the relations between these notions.
 +
 
 +
A uniform algebra is called symmetric if with each function its complex conjugate belongs to the algebra. According to the [[Stone–Weierstrass theorem|Stone–Weierstrass theorem]], each symmetric uniform algebra on a compactum $  X $
 +
coincides with $  C ( X) $.  
 +
The so-called anti-symmetric uniform algebras, containing no real-valued functions apart from the constants, form a kind of opposite class. A typical example is the algebra of all functions that are analytic in the open unit disc of the complex plane and continuous on its closure (the disc algebra). The Shilov–Bishop theorem: Each uniform algebra can be obtained from anti-symmetric uniform algebras by  "glueing"  in a certain way. Even more refined classification theorems are known. At the same time arbitrary uniform algebras do not reduce to algebras of analytic functions of the type of the disc algebra. For example, it is possible to construct a uniform algebra on a one-dimensional compactum, which coincides with its space of maximal ideals, such that all points of the compactum are peak points and at the same time only the zero element of the algebra can be zero on a non-empty open set.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  T.W. Gamelin,  "Uniform algebras" , Prentice-Hall  (1969)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  T.W. Gamelin,  "Uniform algebras" , Prentice-Hall  (1969)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
Instead of  "uniform algebra"  the terminology  "function algebra"  is also used.
 
Instead of  "uniform algebra"  the terminology  "function algebra"  is also used.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519029.png" />, the maximal ideal space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519030.png" />. A representing measure for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519031.png" /> is a positive measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519032.png" /> such that
+
Let $  \xi \in  \mathop{\rm MSpec} ( A) $,  
 +
the maximal ideal space of $  A $.  
 +
A representing measure for $  \xi $
 +
is a positive measure on $  X $
 +
such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519033.png" /></td> </tr></table>
+
$$
 +
\xi ( f  )  = \int\limits f  d \mu ,\  f \in A .
 +
$$
  
They exist by the Riesz representation theorem (cf. (the editorial comments to the second article) [[Riesz theorem(2)|Riesz theorem]]). Of course, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519034.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519035.png" /> is a probability measure. A Jensen measure for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519036.png" /> is a positive measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519037.png" /> such that the Jensen inequality
+
They exist by the Riesz representation theorem (cf. (the editorial comments to the second article) [[Riesz theorem(2)|Riesz theorem]]). Of course, then $  \int d \mu = 1( f  ) = 1 $,  
 +
so that $  \mu $
 +
is a probability measure. A Jensen measure for $  \xi $
 +
is a positive measure on $  X $
 +
such that the Jensen inequality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519038.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm log}  | \xi ( f  ) |  \leq  \int\limits  \mathop{\rm log}  | f |  d \mu ,\ \
 +
f \in A ,
 +
$$
  
holds. A Jensen measure is a representing measure, and a Jensen measure for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519039.png" /> always exists.
+
holds. A Jensen measure is a representing measure, and a Jensen measure for $  \xi $
 +
always exists.
  
A measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519040.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519041.png" /> is orthogonal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519042.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519043.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095190/u09519044.png" />.
+
A measure $  \mu $
 +
on $  X $
 +
is orthogonal to $  A $
 +
if $  \int f  d \mu = 0 $
 +
for all $  f \in A $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.W. Gamelin,  "Uniform algebras and Jensen measures" , Cambridge Univ. Press  (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.M. Leibowitz,  "Lectures on complex functions algebras" , Foresman  (1970)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E.L. Stout,  "The theory of uniform algebras" , Bogden &amp; Quigley  (1971)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Wermer,  "Banach algebras and several complex variables" , Springer  (1975)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  I. Suciu,  "Function algebras" , Ed. Acad. Romania  (1973)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A. Browder,  "Introduction to function algebras" , Benjamin  (1969)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.W. Gamelin,  "Uniform algebras and Jensen measures" , Cambridge Univ. Press  (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.M. Leibowitz,  "Lectures on complex functions algebras" , Foresman  (1970)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E.L. Stout,  "The theory of uniform algebras" , Bogden &amp; Quigley  (1971)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Wermer,  "Banach algebras and several complex variables" , Springer  (1975)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  I. Suciu,  "Function algebras" , Ed. Acad. Romania  (1973)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A. Browder,  "Introduction to function algebras" , Benjamin  (1969)</TD></TR></table>

Latest revision as of 08:27, 6 June 2020


A subalgebra $ A $, closed with respect to the topology of uniform convergence, of the algebra $ C ( X) $ of continuous functions on a compactum $ X $ that contains all constant functions and separates the points of $ X $. The last condition means that for each pair $ x, y $ of distinct points in $ X $ there is a function $ f $ in $ A $ for which $ f ( x) \neq f ( y) $. Uniform algebras are usually provided with the sup norm:

$$ \| f \| = \sup _ { X } | f ( x) |. $$

Here $ \| f ^ { 2 } \| = \| f \| ^ {2} $. Every Banach algebra with an identity (even without assuming commutativity) and with a norm satisfying the latter condition is isomorphic to a uniform algebra.

The uniform algebras form an important subclass of the class of commutative Banach algebras (cf. Commutative Banach algebra) over the field of complex numbers $ \mathbf C $.

To each point $ x \in X $ corresponds a homomorphism $ \phi _ {x} : A \rightarrow \mathbf C $, defined by $ \phi _ {x} ( f ) = f ( x) $. Therefore $ X $ is in a natural way topologically imbedded in the space $ \mathop{\rm MSpec} ( A) $ of maximal ideals of $ A $, and under the corresponding identification $ X $ contains the Shilov boundary (cf. Boundary (in the theory of uniform algebras)). In the study of uniform algebras a major role is played by peak points (that is, points of $ X $ at which the strict maximum modulus of at least one element of $ A $ is attained), by multiplicative probability measures on $ X $( that is, representing measures of homomorphisms from $ A $ to $ \mathbf C $) and by measures on $ X $ that are orthogonal to $ A $. Many concrete results relating to uniform algebras touch on the relations between these notions.

A uniform algebra is called symmetric if with each function its complex conjugate belongs to the algebra. According to the Stone–Weierstrass theorem, each symmetric uniform algebra on a compactum $ X $ coincides with $ C ( X) $. The so-called anti-symmetric uniform algebras, containing no real-valued functions apart from the constants, form a kind of opposite class. A typical example is the algebra of all functions that are analytic in the open unit disc of the complex plane and continuous on its closure (the disc algebra). The Shilov–Bishop theorem: Each uniform algebra can be obtained from anti-symmetric uniform algebras by "glueing" in a certain way. Even more refined classification theorems are known. At the same time arbitrary uniform algebras do not reduce to algebras of analytic functions of the type of the disc algebra. For example, it is possible to construct a uniform algebra on a one-dimensional compactum, which coincides with its space of maximal ideals, such that all points of the compactum are peak points and at the same time only the zero element of the algebra can be zero on a non-empty open set.

References

[1] T.W. Gamelin, "Uniform algebras" , Prentice-Hall (1969)

Comments

Instead of "uniform algebra" the terminology "function algebra" is also used.

Let $ \xi \in \mathop{\rm MSpec} ( A) $, the maximal ideal space of $ A $. A representing measure for $ \xi $ is a positive measure on $ X $ such that

$$ \xi ( f ) = \int\limits f d \mu ,\ f \in A . $$

They exist by the Riesz representation theorem (cf. (the editorial comments to the second article) Riesz theorem). Of course, then $ \int d \mu = 1( f ) = 1 $, so that $ \mu $ is a probability measure. A Jensen measure for $ \xi $ is a positive measure on $ X $ such that the Jensen inequality

$$ \mathop{\rm log} | \xi ( f ) | \leq \int\limits \mathop{\rm log} | f | d \mu ,\ \ f \in A , $$

holds. A Jensen measure is a representing measure, and a Jensen measure for $ \xi $ always exists.

A measure $ \mu $ on $ X $ is orthogonal to $ A $ if $ \int f d \mu = 0 $ for all $ f \in A $.

References

[a1] T.W. Gamelin, "Uniform algebras and Jensen measures" , Cambridge Univ. Press (1978)
[a2] G.M. Leibowitz, "Lectures on complex functions algebras" , Foresman (1970)
[a3] E.L. Stout, "The theory of uniform algebras" , Bogden & Quigley (1971)
[a4] J. Wermer, "Banach algebras and several complex variables" , Springer (1975)
[a5] I. Suciu, "Function algebras" , Ed. Acad. Romania (1973)
[a6] A. Browder, "Introduction to function algebras" , Benjamin (1969)
How to Cite This Entry:
Uniform algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Uniform_algebra&oldid=13486
This article was adapted from an original article by E.A. Gorin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article