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A Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c0233801.png" /> with identity over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c0233802.png" /> in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c0233803.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c0233804.png" />.
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Each maximal ideal of a commutative Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c0233805.png" /> is the kernel of some continuous multiplicative linear functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c0233806.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c0233807.png" />, that is, a homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c0233808.png" /> into the field of complex numbers. Conversely, every multiplicative linear functional on a commutative Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c0233809.png" /> is continuous, has norm 1 and its kernel is a maximal ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338010.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338011.png" /> be the set of all multiplicative linear functionals on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338012.png" />. An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338013.png" /> is invertible if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338014.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338015.png" />. Furthermore, the spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338016.png" /> consists precisely of the numbers of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338017.png" />. If a continuous linear functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338018.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338019.png" /> has the property that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338020.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338021.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338022.png" /> is multiplicative; this is not true, in general, for an algebra over the field of real numbers.
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A Banach algebra  $  A $
 +
with identity over the field  $  \mathbf C $
 +
in which  $  x y = y x $
 +
for all  $  x , y \in A $.
 +
 
 +
Each maximal ideal of a commutative Banach algebra $  A $
 +
is the kernel of some continuous multiplicative linear functional $  \phi $
 +
on $  A $,  
 +
that is, a homomorphism of $  A $
 +
into the field of complex numbers. Conversely, every multiplicative linear functional on a commutative Banach algebra $  A $
 +
is continuous, has norm 1 and its kernel is a maximal ideal in $  A $.  
 +
Let $  \Phi $
 +
be the set of all multiplicative linear functionals on $  A $.  
 +
An element $  a \in A $
 +
is invertible if and only if $  \phi ( a) \neq 0 $
 +
for all $  \phi \in \Phi $.  
 +
Furthermore, the spectrum $  \sigma ( a) $
 +
consists precisely of the numbers of the form $  \phi ( a) $.  
 +
If a continuous linear functional $  \psi $
 +
on $  A $
 +
has the property that $  \psi ( a) \in \sigma ( a) $
 +
for all $  a \in A $,  
 +
then $  \psi $
 +
is multiplicative; this is not true, in general, for an algebra over the field of real numbers.
  
 
==Examples of maximal ideals in commutative Banach algebras.==
 
==Examples of maximal ideals in commutative Banach algebras.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338023.png" /> be the algebra of all continuous functions on a compactum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338024.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338025.png" /> is a fixed point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338026.png" />, then the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338027.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338028.png" /> is a maximal ideal, and all maximal ideals in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338029.png" /> have this form. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338030.png" /> is a compact set in the complex plane and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338031.png" /> is the closed subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338032.png" /> consisting of all functions that can be approximated uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338033.png" /> by rational functions with poles outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338034.png" />, then the maximal ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338035.png" /> are obtained in the same way as in the case of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338036.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338037.png" /> be the group algebra of a discrete Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338038.png" />, and suppose that to every element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338039.png" /> corresponds its Fourier transform <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338040.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338041.png" /> is a multiplicative linear functional on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338042.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338043.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338044.png" /> in the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338045.png" /> of characters of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338046.png" />; therefore the maximal ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338047.png" /> are in one-to-one correspondence with the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338048.png" />. As applied to the group of integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338049.png" />, this last example leads to a proof of the well-known Wiener theorem: If the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338050.png" /> has an absolutely convergent Fourier series and does not vanish on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338051.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338052.png" /> also has an absolutely convergent Fourier series.
+
Let $  A = C ( X) $
 +
be the algebra of all continuous functions on a compactum $  X $.  
 +
If $  x _ {0} $
 +
is a fixed point of $  X $,  
 +
then the set of all $  f \in A $
 +
for which $  f ( x _ {0} ) = 0 $
 +
is a maximal ideal, and all maximal ideals in $  C ( X) $
 +
have this form. If $  X $
 +
is a compact set in the complex plane and $  A = R ( X) $
 +
is the closed subalgebra of $  C ( X) $
 +
consisting of all functions that can be approximated uniformly on $  X $
 +
by rational functions with poles outside $  X $,  
 +
then the maximal ideals of $  R ( X) $
 +
are obtained in the same way as in the case of $  C ( X) $.  
 +
Let $  L _ {1} ( G) $
 +
be the group algebra of a discrete Abelian group $  G $,  
 +
and suppose that to every element $  f \in L _ {1} ( G) $
 +
corresponds its Fourier transform $  \widehat{f}  $.  
 +
If $  \phi $
 +
is a multiplicative linear functional on $  L _ {1} ( G) $,  
 +
then $  \phi ( f) = \widehat{f}  ( \chi _ {0} ) $
 +
for some $  \chi _ {0} $
 +
in the group $  \widehat{G}  $
 +
of characters of $  G $;  
 +
therefore the maximal ideals of $  L _ {1} ( G) $
 +
are in one-to-one correspondence with the elements of $  \widehat{G}  $.  
 +
As applied to the group of integers $  \mathbf Z $,  
 +
this last example leads to a proof of the well-known Wiener theorem: If the function $  \widehat{f}  ( t) $
 +
has an absolutely convergent Fourier series and does not vanish on $  [ 0 , 2 \pi ] $,
 +
then $  1 / \widehat{f}  ( t) $
 +
also has an absolutely convergent Fourier series.
 +
 
 +
Since a multiplicative linear functional has norm 1, each such a functional belongs to the unit sphere of the dual of  $  A $.
 +
The set  $  \Phi $
 +
of all multiplicative linear functionals on  $  A $
 +
is closed in the weak topology on the dual space. Since the unit ball is compact in the weak topology on the dual space,  $  \Phi $
 +
is also compact in this topology; it is called the maximal ideal space of the algebra  $  A $
 +
and it is denoted by  $  \mathfrak M $.
  
Since a multiplicative linear functional has norm 1, each such a functional belongs to the unit sphere of the dual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338053.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338054.png" /> of all multiplicative linear functionals on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338055.png" /> is closed in the weak topology on the dual space. Since the unit ball is compact in the weak topology on the dual space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338056.png" /> is also compact in this topology; it is called the maximal ideal space of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338057.png" /> and it is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338058.png" />.
+
If a commutative Banach algebra  $  A $
 +
contains a non-trivial idempotent, that is, an element  $  f \in A $
 +
such that  $  f \neq 0 $,
 +
$  f \neq e $
 +
and  $  f ^ { 2 } = f $,
 +
then the maximal ideal space of $  A $
 +
is disconnected. Conversely, if the maximal ideal space $  X $
 +
of the algebra  $  A $
 +
is the union of two disjoint closed sets  $  X _ {0} $
 +
and  $  X _ {1} $,  
 +
then there is an element  $  f \in A $
 +
such that  $  \widehat{f}  \mid  _ {X _ {0}  } = 0 $
 +
and  $  \widehat{f}  \mid  _ {X _ {1}  } = 1 $ (Shilov's theorem). In particular, the maximal ideal space of a commutative Banach algebra is connected if and only if this algebra cannot be represented as a direct sum of two non-trivial ideals.
  
If a commutative Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338059.png" /> contains a non-trivial idempotent, that is, an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338060.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338063.png" />, then the maximal ideal space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338064.png" /> is disconnected. Conversely, if the maximal ideal space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338065.png" /> of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338066.png" /> is the union of two disjoint closed sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338067.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338068.png" />, then there is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338069.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338070.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338071.png" /> (Shilov's theorem). In particular, the maximal ideal space of a commutative Banach algebra is connected if and only if this algebra cannot be represented as a direct sum of two non-trivial ideals.
+
Let  $  \epsilon _ {1} ( A) $
 +
be the subgroup of the group  $  \epsilon ( A) $
 +
of invertible elements of the algebra $  A $
 +
consisting of the exponentials, that is, of the elements of the form  $  \mathop{\rm exp}  a = \sum _ {0}  ^  \infty  a  ^ {n} / n ! $.  
 +
Then  $  \epsilon _ {1} ( A) $
 +
is the connected component of the identity in  $  \epsilon ( A) $.  
 +
For any compactum  $  X $
 +
there is a canonical isomorphism between the groups  $  H  ^ {1} ( X , \mathbf Z ) $
 +
and  $  \epsilon ( C) / \epsilon _ {1} ( C) $,
 +
where  $  C = C ( X) $
 +
is the algebra of all continuous functions on  $  X $ (the Brushlinskii–Eilenberg theorem). It turns out that this isomorphism naturally induces an isomorphism between  $  H  ^ {1} ( X , \mathbf Z ) $
 +
and $  \epsilon ( A) / \epsilon _ {1} ( A) $,  
 +
where  $  A $
 +
is any commutative Banach algebra whose maximal ideal space is  $  X $ (the Arens–Royden theorem). In some cases the groups  $  H  ^ {q} ( X , \mathbf Z ) $
 +
with  $  q $
 +
odd have a similar interpretation. The algebra  $  A $
 +
has the following canonical representation in the algebra  $  C ( \mathfrak M ) $.  
 +
The Gel'fand transform of an element  $  a \in A $
 +
is the function  $  \widehat{a}  $
 +
on  $  \mathfrak M $
 +
defined by the formula  $  \widehat{a}  ( x) = \phi _ {x} ( a) $,  
 +
where  $  \phi _ {x} $
 +
is the multiplicative linear functional corresponding to the point  $  x \in \mathfrak M $.  
 +
The kernel of the homomorphism  $  a \mapsto \widehat{a}  $
 +
is the set of all elements  $  a \in A $
 +
belonging to all maximal ideals, i.e. belonging to the radical of  $  A $.  
 +
If  $  A $
 +
is a semi-simple algebra, that is, if  $  \mathop{\rm Rad}  A = \{ 0 \} $,  
 +
then the homomorphism  $  a \mapsto \widehat{a}  $
 +
is an (algebraic) isomorphism of  $  A $
 +
to  $  C ( \mathfrak M ) $.  
 +
Semi-simple commutative Banach algebras are often called function algebras.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338072.png" /> be the subgroup of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338073.png" /> of invertible elements of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338074.png" /> consisting of the exponentials, that is, of the elements of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338075.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338076.png" /> is the connected component of the identity in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338077.png" />. For any compactum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338078.png" /> there is a canonical isomorphism between the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338079.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338080.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338081.png" /> is the algebra of all continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338082.png" /> (the Brushlinskii–Eilenberg theorem). It turns out that this isomorphism naturally induces an isomorphism between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338083.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338084.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338085.png" /> is any commutative Banach algebra whose maximal ideal space is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338086.png" /> (the Arens–Royden theorem). In some cases the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338087.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338088.png" /> odd have a similar interpretation. The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338089.png" /> has the following canonical representation in the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338090.png" />. The Gel'fand transform of an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338091.png" /> is the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338092.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338093.png" /> defined by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338094.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338095.png" /> is the multiplicative linear functional corresponding to the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338096.png" />. The kernel of the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338097.png" /> is the set of all elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338098.png" /> belonging to all maximal ideals, i.e. belonging to the radical of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c02338099.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380100.png" /> is a semi-simple algebra, that is, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380101.png" />, then the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380102.png" /> is an (algebraic) isomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380103.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380104.png" />. Semi-simple commutative Banach algebras are often called function algebras.
+
The Gel'fand transform is well suited to the study of semi-simple algebras: One of the fundamental results in the theory of commutative Banach algebras is the theorem that a semi-simple algebra can be represented as an algebra of continuous functions on the maximal ideal space. Far less is known about general algebras with a radical in comparison to semi-simple algebras. All ideals of the algebra of complex polynomials of degree  $  \leq  m $
 +
are known. This algebra consists of formal polynomials  $  \xi = a _ {0} + a _ {1} \lambda + \dots + a _ {m} \lambda  ^ {m} $,
 +
with the usual multiplication rule, subject to the relation  $  \lambda  ^ {m+1} = 0 $.  
 +
This algebra is finite-dimensional, all norms on it are equivalent and every ideal of it is closed. The set  $  I _ {k} $
 +
of those  $  \xi $
 +
for which  $  a _ {j} = 0 $
 +
for  $  j \leq  k $
 +
is a closed ideal; there are no other ideals in this algebra. Every algebra with a unique non-trivial ideal is isomorphic to the algebra of polynomials of the first degree. Until now (1987) it is not known whether the same is true for algebras with a unique non-trivial closed ideal.
  
The Gel'fand transform is well suited to the study of semi-simple algebras: One of the fundamental results in the theory of commutative Banach algebras is the theorem that a semi-simple algebra can be represented as an algebra of continuous functions on the maximal ideal space. Far less is known about general algebras with a radical in comparison to semi-simple algebras. All ideals of the algebra of complex polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380105.png" /> are known. This algebra consists of formal polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380106.png" />, with the usual multiplication rule, subject to the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380107.png" />. This algebra is finite-dimensional, all norms on it are equivalent and every ideal of it is closed. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380108.png" /> of those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380109.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380110.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380111.png" /> is a closed ideal; there are no other ideals in this algebra. Every algebra with a unique non-trivial ideal is isomorphic to the algebra of polynomials of the first degree. Until now (1987) it is not known whether the same is true for algebras with a unique non-trivial closed ideal.
+
The natural infinite-dimensional analogues of algebras of polynomials are algebras of power series  $  \xi = a _ {0} + a _ {1} \lambda + a _ {2} \lambda  ^ {2} + \dots $,
 +
with the usual operations and norm  $  \| \xi \| = \sum _ {k=0}  ^  \infty  | a _ {k} | \alpha _ {k} $,
 +
where  $  \alpha _ {k} $
 +
is a sequence of positive numbers satisfying  $  \alpha _ {k+l} \leq  \alpha _ {k} \alpha _ {l} $.  
 +
If  $  \alpha _ {k}  ^ {1/k} \rightarrow 0 $
 +
as  $  k \rightarrow \infty $,
 +
then the unique non-trivial homomorphism into the field of complex numbers is given by  $  \xi \rightarrow a _ {0} $.  
 +
Thus, $  I _ {1} $
 +
is the unique maximal ideal and this ideal coincides with the radical. The ideals  $  I _ {k} $,  
 +
defined in the same way as in the finite-dimensional case, constitute a countable set of closed ideals. If the sequence  $  \{ \alpha _ {k+} 1 / \alpha _ {k} \} $
 +
is monotone, then this set of ideals contains all closed ideals. In general, an algebra may contain uncountably many distinct closed ideals.
  
The natural infinite-dimensional analogues of algebras of polynomials are algebras of power series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380112.png" />, with the usual operations and norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380113.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380114.png" /> is a sequence of positive numbers satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380115.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380116.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380117.png" />, then the unique non-trivial homomorphism into the field of complex numbers is given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380118.png" />. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380119.png" /> is the unique maximal ideal and this ideal coincides with the radical. The ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380120.png" />, defined in the same way as in the finite-dimensional case, constitute a countable set of closed ideals. If the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380121.png" /> is monotone, then this set of ideals contains all closed ideals. In general, an algebra may contain uncountably many distinct closed ideals.
+
By suitably choosing the sequence $  \{ \alpha _ {k} \} $
 +
in the algebra under consideration (without non-trivial nilpotents), it is possible to define a non-zero derivation, that is, a bounded linear operator  $  D $
 +
such that  $  D ( \xi \eta ) = ( D \xi ) \eta + \xi ( D \eta ) $.  
 +
There are no non-trivial continuous derivations on a semi-simple algebra, since in any (not necessarily commutative) algebra the identity
  
By suitably choosing the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380122.png" /> in the algebra under consideration (without non-trivial nilpotents), it is possible to define a non-zero derivation, that is, a bounded linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380123.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380124.png" />. There are no non-trivial continuous derivations on a semi-simple algebra, since in any (not necessarily commutative) algebra the identity
+
$$
 +
( D \xi ) ^ {n}  =
 +
\frac{1}{n!}
  
<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/c/c023/c023380/c023380125.png" /></td> </tr></table>
+
\sum _ { k= 1} ^ { n }
 +
( - 1 )  ^ {k+n}
 +
\left ( \begin{array}{c}
 +
n \\
 +
k
 +
\end{array}
 +
\right )
 +
\xi  ^ {n-k} D  ^ {n} \xi  ^ {k}
 +
$$
  
holds if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380126.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380127.png" /> commute. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380128.png" /> is continuous, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380129.png" /> is a generalized nilpotent.
+
holds if $  \xi $
 +
and $  D \xi $
 +
commute. In particular, if $  D $
 +
is continuous, then $  D \xi $
 +
is a generalized nilpotent.
  
 
Any finite-dimensional algebra decomposes into the direct sum of the radical and a semi-simple algebra. In the infinite-dimensional case this assertion ceases to be true in general, even for commutative Banach algebras. In addition, it is necessary to distinguish between the cases of algebraic and strong (topological) decomposability.
 
Any finite-dimensional algebra decomposes into the direct sum of the radical and a semi-simple algebra. In the infinite-dimensional case this assertion ceases to be true in general, even for commutative Banach algebras. In addition, it is necessary to distinguish between the cases of algebraic and strong (topological) decomposability.
Line 26: Line 173:
 
It turns out that there are no conditions that can be imposed merely on the radical that will ensure even algebraic decomposability: the radical may be one-dimensional and may annihilate some maximal ideal but it need not be a direct summand, even in the algebraic sense.
 
It turns out that there are no conditions that can be imposed merely on the radical that will ensure even algebraic decomposability: the radical may be one-dimensional and may annihilate some maximal ideal but it need not be a direct summand, even in the algebraic sense.
  
On the other hand, if the radical is finite-dimensional and the quotient algebra is an algebra of continuous functions (or an algebra of operators on a Hilbert space), then it is strongly decomposable. If the quotient algebra is an algebra of continuous functions and its annihilator radical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380130.png" /> (i.e. the square of every element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380131.png" /> is zero) has a Banach complement, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380132.png" /> is strongly decomposable. Instead of the condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380133.png" /> has a complement one can require that the space of maximal ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380134.png" /> satisfy the first axiom of countability at every point.
+
On the other hand, if the radical is finite-dimensional and the quotient algebra is an algebra of continuous functions (or an algebra of operators on a Hilbert space), then it is strongly decomposable. If the quotient algebra is an algebra of continuous functions and its annihilator radical $  R $ (i.e. the square of every element of $  R $
 +
is zero) has a Banach complement, then $  A $
 +
is strongly decomposable. Instead of the condition that $  R $
 +
has a complement one can require that the space of maximal ideals of $  A $
 +
satisfy the first axiom of countability at every point.
  
 
Completely investigated is also the case when the quotient algebra by the radical is the algebra of continuous functions on a totally-disconnected compactum: A necessary and sufficient condition for decomposability is that the idempotents of the original algebra be uniformly bounded.
 
Completely investigated is also the case when the quotient algebra by the radical is the algebra of continuous functions on a totally-disconnected compactum: A necessary and sufficient condition for decomposability is that the idempotents of the original algebra be uniformly bounded.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380135.png" /> be a bounded domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380136.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380137.png" /> be the closed subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380138.png" /> consisting of the functions holomorphic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380139.png" />. It is known that under fairly general hypotheses concerning <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380140.png" />, any maximal ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380141.png" />, corresponding to a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380142.png" />, is finitely generated; namely, it is generated by the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380143.png" />. This statement has the following local converse. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380144.png" /> be a semi-simple commutative Banach algebra with maximal ideal space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380145.png" />. If the maximal ideal corresponding to a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380146.png" /> is generated by a finite set of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380147.png" />, then the maximal ideals corresponding to the points in some neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380148.png" /> are generated by elements of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380149.png" />; the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380150.png" /> is one-to-one in some neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380151.png" /> and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380152.png" /> is, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380153.png" />, holomorphic in some fixed neighbourhood of the origin in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380154.png" />. Furthermore, in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380155.png" /> a certain natural analytic structure can be introduced.
+
Let $  V $
 +
be a bounded domain in $  \mathbf C  ^ {n} $
 +
and let $  A $
 +
be the closed subalgebra of $  C ( \overline{V} ) $
 +
consisting of the functions holomorphic on $  V $.  
 +
It is known that under fairly general hypotheses concerning $  V $,  
 +
any maximal ideal of $  A $,  
 +
corresponding to a point $  z  ^ {0} = ( z _ {1}  ^ {0}, \dots, z _ {n}  ^ {0} ) \in V $,  
 +
is finitely generated; namely, it is generated by the functions $  f _ {i} = z _ {i} - z _ {i}  ^ {0} $.  
 +
This statement has the following local converse. Let $  A $
 +
be a semi-simple commutative Banach algebra with maximal ideal space $  X $.  
 +
If the maximal ideal corresponding to a point $  x _ {0} \in X $
 +
is generated by a finite set of elements $  f _ {1}, \dots, f _ {n} \in A $,  
 +
then the maximal ideals corresponding to the points in some neighbourhood of $  x _ {0} $
 +
are generated by elements of the form $  f _ {i} - \lambda _ {i} e $;  
 +
the mapping $  \psi : x \mapsto ( f _ {1} ( x), \dots, f _ {n} ( x) ) $
 +
is one-to-one in some neighbourhood of $  x _ {0} $
 +
and the function $  g \circ \psi  ^ {-1} $
 +
is, for any $  g \in A $,  
 +
holomorphic in some fixed neighbourhood of the origin in $  \mathbf C  ^ {n} $.  
 +
Furthermore, in a neighbourhood of $  x _ {0} $
 +
a certain natural analytic structure can be introduced.
  
A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380156.png" /> of elements of an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380157.png" /> is called a system of generators if the smallest closed algebra with identity in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380158.png" /> that contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380159.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380160.png" /> itself. The identity is usually not included in the set of generators. If there is a finite system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380161.png" /> with the above properties, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380162.png" /> is called a finitely-generated algebra. The smallest numbers of elements in a system of generators is called the number of generators of the algebra.
+
A set $  S $
 +
of elements of an algebra $  A $
 +
is called a system of generators if the smallest closed algebra with identity in $  A $
 +
that contains $  S $
 +
is $  A $
 +
itself. The identity is usually not included in the set of generators. If there is a finite system $  S $
 +
with the above properties, then $  A $
 +
is called a finitely-generated algebra. The smallest numbers of elements in a system of generators is called the number of generators of the algebra.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380163.png" /> is a system of generators of an algebra, then the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380164.png" /> induces a homomorphism of the maximal ideal space of this algebra onto some polynomially-convex compact set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380165.png" />. Each polynomially-convex compact set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380166.png" /> is the maximal ideal space of some Banach algebra (for example, the algebra of uniform limits of polynomials on this set).
+
If $  f _ {1}, \dots, f _ {n} $
 +
is a system of generators of an algebra, then the mapping $  x \mapsto ( \widehat{f}  _ {1} ( x), \dots, \widehat{f}  _ {n} ( x) ) $
 +
induces a homomorphism of the maximal ideal space of this algebra onto some polynomially-convex compact set in $  \mathbf C  ^ {n} $.  
 +
Each polynomially-convex compact set in $  \mathbf C  ^ {n} $
 +
is the maximal ideal space of some Banach algebra (for example, the algebra of uniform limits of polynomials on this set).
  
The maximal ideal space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380167.png" /> of an algebra with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380168.png" /> generators satisfies the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380169.png" /> and possesses a number of other properties; for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380170.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380171.png" />. Hence it follows, in particular, that the number of generators in the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380172.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380173.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380174.png" />-dimensional unit sphere, is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380175.png" />; a similar result holds for an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380176.png" />-dimensional compact manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380177.png" />. For any finite cellular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380178.png" />-dimensional polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380179.png" />, the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380180.png" /> has a system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380181.png" /> generators.
+
The maximal ideal space $  X $
 +
of an algebra with $  n $
 +
generators satisfies the condition $  \mathop{\rm dim}  X \leq  2 n $
 +
and possesses a number of other properties; for example, $  H  ^ {i} ( X , \mathbf C ) = 0 $
 +
for $  i \geq  n $.  
 +
Hence it follows, in particular, that the number of generators in the algebra $  C ( S  ^ {n} ) $,  
 +
where $  S  ^ {n} $
 +
is the $  n $-dimensional unit sphere, is equal to $  n + 1 $;  
 +
a similar result holds for an arbitrary $  n $-dimensional compact manifold $  X $.  
 +
For any finite cellular $  n $-dimensional polyhedron $  X $,  
 +
the algebra $  C ( X) $
 +
has a system of $  n + 1 $
 +
generators.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380182.png" /> be an algebra with maximal ideal space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380183.png" />. The smallest closet set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380184.png" /> on which all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380185.png" /> attain their maximum is called the Shilov boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380186.png" />. For any commutative Banach algebra with identity this set exists and is unique.
+
Let $  A $
 +
be an algebra with maximal ideal space $  X $.  
 +
The smallest closet set $  \Gamma \subset  X $
 +
on which all functions $  | \widehat{f}  | $
 +
attain their maximum is called the ''[[Shilov boundary]]'' of $  A $.  
 +
For any commutative Banach algebra with identity this set exists and is unique.
  
A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380187.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380188.png" /> if and only if for any neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380189.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380190.png" /> there is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380191.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380192.png" />, but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380193.png" /> outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380194.png" />. Furthermore, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380195.png" /> is an open subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380196.png" /> and if there exist a closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380197.png" /> and an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380198.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380199.png" /> for points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380200.png" />, then the intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380201.png" /> is non-empty.
+
A point $  x _ {0} \in X $
 +
belongs to $  \Gamma $
 +
if and only if for any neighbourhood $  U $
 +
of $  x _ {0} $
 +
there is an element $  f \in A $
 +
for which $  \max _ {X}  | \widehat{f}  | = 1 $,  
 +
but $  | \widehat{f}  ( x) | < 1 $
 +
outside $  U $.  
 +
Furthermore, if $  U $
 +
is an open subset of $  X $
 +
and if there exist a closed set $  F \subset  U $
 +
and an element $  g \in A $
 +
such that $  | \widehat{g}  ( x) | < \max _ {F}  | \widehat{g}  | $
 +
for points $  x \in U \setminus  F $,  
 +
then the intersection $  \Gamma \cap U $
 +
is non-empty.
  
Any multiplicative linear functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380202.png" /> is continuous with respect to the norm defined by the spectral radius; moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380203.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380204.png" /> is the maximal ideal space. In this inequality, according to the definition of the Shilov boundary, one can replace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380205.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380206.png" />; therefore there exists a positive regular measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380207.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380208.png" /> "representing"  the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380209.png" />, that is, such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380210.png" /> the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380211.png" /> holds. In the case of the algebra of functions analytic on the disc, this formula reduces to the classical Poisson formula. Among the representing measures there exists a measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380212.png" /> satisfying the Jensen inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380213.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380214.png" />.
+
Any multiplicative linear functional $  \phi $
 +
is continuous with respect to the norm defined by the spectral radius; moreover, $  | \phi ( f) | \leq  \max _ {X}  | \widehat{f}  | $,  
 +
where $  X $
 +
is the maximal ideal space. In this inequality, according to the definition of the Shilov boundary, one can replace $  X $
 +
by $  \Gamma $;  
 +
therefore there exists a positive regular measure $  \mu $
 +
on  $  \Gamma $"
 +
representing"  the functional $  \phi $,  
 +
that is, such that for all $  f \in A $
 +
the equation $  \phi ( f) = \int _  \Gamma  \widehat{f}  d \mu $
 +
holds. In the case of the algebra of functions analytic on the disc, this formula reduces to the classical Poisson formula. Among the representing measures there exists a measure $  \mu $
 +
satisfying the Jensen inequality $  \mathop{\rm ln}  | \phi ( f) | \leq  \int _  \Gamma  \mathop{\rm ln}  \widehat{f}  d \mu $
 +
for all $  f \in A $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380215.png" /> be a commutative Banach algebra with identity and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380216.png" /> be a closed subalgebra. The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380217.png" /> is called a maximal subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380218.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380219.png" /> contains no closed proper subalgebra properly containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380220.png" />. In each sufficiently large algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380221.png" /> there are maximal subalgebras with identity, and even closed subalgebras of codimension 1. In fact, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380222.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380223.png" /> are two distinct homomorphisms of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380224.png" /> into the field of complex numbers and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380225.png" />, then the kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380226.png" /> is a closed subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380227.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380228.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380229.png" />. Similarly, the kernel of a  "point derivation13B10point derivation" , that is, a functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380230.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380231.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380232.png" /> is a multiplicative functional, is a subalgebra of codimension 1. In the complex case these examples exhaust all subalgebras of codimension 1. In particular, every such subalgebra of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380233.png" /> does not separate points of the compactum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380234.png" />, since on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380235.png" /> there are no derivations (not even discontinuous ones). All subalgebras of finite codimension have a similar description.
+
Let $  B $
 +
be a commutative Banach algebra with identity and let $  A $
 +
be a closed subalgebra. The algebra $  A $
 +
is called a maximal subalgebra of $  B $
 +
if $  B $
 +
contains no closed proper subalgebra properly containing $  A $.  
 +
In each sufficiently large algebra $  B $
 +
there are maximal subalgebras with identity, and even closed subalgebras of codimension 1. In fact, if $  \phi _ {1} $
 +
and $  \phi _ {2} $
 +
are two distinct homomorphisms of the algebra $  B $
 +
into the field of complex numbers and if $  \psi = \phi _ {1} - \phi _ {2} $,  
 +
then the kernel of $  \psi $
 +
is a closed subalgebra $  A $
 +
of $  B $
 +
for which $  \mathop{\rm dim}  B / A = 1 $.  
 +
Similarly, the kernel of a  "point derivation13B10point derivation" , that is, a functional $  \psi $
 +
such that $  \psi ( f g ) = \psi ( f  ) \phi ( g) + \psi ( g) \phi ( f  ) $,  
 +
where $  \phi $
 +
is a multiplicative functional, is a subalgebra of codimension 1. In the complex case these examples exhaust all subalgebras of codimension 1. In particular, every such subalgebra of the algebra $  C ( X) $
 +
does not separate points of the compactum $  X $,  
 +
since on $  C ( X) $
 +
there are no derivations (not even discontinuous ones). All subalgebras of finite codimension have a similar description.
  
The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380236.png" /> of continuous functions on the unit circle that have an analytic continuation inside the unit disc is a maximal subalgebra of the algebra of continuous functions on the unit circle. This statement can be regarded as a generalization of the Stone–Weierstrass approximation theorem, which asserts that a closed subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380237.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380238.png" />, containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380239.png" /> and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380240.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380241.png" />. The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380242.png" /> is a closed subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380243.png" />; this subalgebra is maximal.
+
The algebra $  A $
 +
of continuous functions on the unit circle that have an analytic continuation inside the unit disc is a maximal subalgebra of the algebra of continuous functions on the unit circle. This statement can be regarded as a generalization of the Stone–Weierstrass approximation theorem, which asserts that a closed subalgebra of $  C ( \Gamma ) $,  
 +
where $  \Gamma = \{ {z } : {| z | = 1 } \} $,  
 +
containing $  A $
 +
and the function $  \overline{z} $
 +
coincides with $  C ( \Gamma ) $.  
 +
The algebra $  L _ {1}  ^ {+} ( \mathbf R ) = \{ {f \in L _ {1} ( \mathbf R ) } : {f ( t) = 0 \textrm{ when }  t < 0 } \} $
 +
is a closed subalgebra of $  L _ {1} ( \mathbf R ) $;  
 +
this subalgebra is maximal.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380244.png" /> be a irrational number and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380245.png" /> be the algebra of all continuous functions on the two-dimensional torus with Fourier coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380246.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380247.png" />. This algebra is a maximal subalgebra of the algebra of all continuous functions on the torus. The torus is the Shilov boundary of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380248.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380249.png" /> is a Dirichlet algebra. If the torus is realized as the skeleton of the unit bidisc in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380250.png" />, then the maximal ideal space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380251.png" /> is identified with the subset of the bidisc described by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380252.png" />. The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380253.png" /> does not belong to the Shilov boundary, but is a one-point Gleason part. (Two multiplicative functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380254.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380255.png" /> on a uniform algebra belong to the same Gleason part, by definition, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380256.png" />.) The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380257.png" /> is analytic on the maximal ideal space (in the sense of uniqueness: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380258.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380259.png" /> for the points of a non-empty open set), even though the real dimension of the maximal ideal space is equal to 3. The algebra of continuous functions on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380260.png" />-dimensional torus having an extension inside the corresponding polydisc is not maximal when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023380/c023380261.png" />, but it is maximal in the class of subalgebras that are invariant with respect to holomorphic automorphisms of the torus.
+
Let $  \alpha $
 +
be a irrational number and let $  A _  \alpha  $
 +
be the algebra of all continuous functions on the two-dimensional torus with Fourier coefficients $  c _ {mn} = 0 $
 +
for $  m + n \alpha < 0 $.  
 +
This algebra is a maximal subalgebra of the algebra of all continuous functions on the torus. The torus is the Shilov boundary of the algebra $  A _  \alpha  $,  
 +
and $  A _  \alpha  $
 +
is a Dirichlet algebra. If the torus is realized as the skeleton of the unit bidisc in $  \mathbf C  ^ {2} $,  
 +
then the maximal ideal space of $  A _  \alpha  $
 +
is identified with the subset of the bidisc described by the equation $  | z _ {1} | = | z _ {2} |  ^  \alpha  $.  
 +
The point $  ( 0 , 0 ) $
 +
does not belong to the Shilov boundary, but is a one-point Gleason part. (Two multiplicative functionals $  \phi _ {1} $
 +
and $  \phi _ {2} $
 +
on a uniform algebra belong to the same Gleason part, by definition, if $  \| \phi _ {1} - \phi _ {2} \| < 2 $.)  
 +
The algebra $  A _  \alpha  $
 +
is analytic on the maximal ideal space (in the sense of uniqueness: $  f = 0 $
 +
if $  \widehat{f}  ( \xi ) = 0 $
 +
for the points of a non-empty open set), even though the real dimension of the maximal ideal space is equal to 3. The algebra of continuous functions on the $  n $-dimensional torus having an extension inside the corresponding polydisc is not maximal when $  n > 1 $,  
 +
but it is maximal in the class of subalgebras that are invariant with respect to holomorphic automorphisms of the torus.
  
 
For references see [[Banach algebra|Banach algebra]].
 
For references see [[Banach algebra|Banach algebra]].

Latest revision as of 03:47, 25 February 2022


A Banach algebra $ A $ with identity over the field $ \mathbf C $ in which $ x y = y x $ for all $ x , y \in A $.

Each maximal ideal of a commutative Banach algebra $ A $ is the kernel of some continuous multiplicative linear functional $ \phi $ on $ A $, that is, a homomorphism of $ A $ into the field of complex numbers. Conversely, every multiplicative linear functional on a commutative Banach algebra $ A $ is continuous, has norm 1 and its kernel is a maximal ideal in $ A $. Let $ \Phi $ be the set of all multiplicative linear functionals on $ A $. An element $ a \in A $ is invertible if and only if $ \phi ( a) \neq 0 $ for all $ \phi \in \Phi $. Furthermore, the spectrum $ \sigma ( a) $ consists precisely of the numbers of the form $ \phi ( a) $. If a continuous linear functional $ \psi $ on $ A $ has the property that $ \psi ( a) \in \sigma ( a) $ for all $ a \in A $, then $ \psi $ is multiplicative; this is not true, in general, for an algebra over the field of real numbers.

Examples of maximal ideals in commutative Banach algebras.

Let $ A = C ( X) $ be the algebra of all continuous functions on a compactum $ X $. If $ x _ {0} $ is a fixed point of $ X $, then the set of all $ f \in A $ for which $ f ( x _ {0} ) = 0 $ is a maximal ideal, and all maximal ideals in $ C ( X) $ have this form. If $ X $ is a compact set in the complex plane and $ A = R ( X) $ is the closed subalgebra of $ C ( X) $ consisting of all functions that can be approximated uniformly on $ X $ by rational functions with poles outside $ X $, then the maximal ideals of $ R ( X) $ are obtained in the same way as in the case of $ C ( X) $. Let $ L _ {1} ( G) $ be the group algebra of a discrete Abelian group $ G $, and suppose that to every element $ f \in L _ {1} ( G) $ corresponds its Fourier transform $ \widehat{f} $. If $ \phi $ is a multiplicative linear functional on $ L _ {1} ( G) $, then $ \phi ( f) = \widehat{f} ( \chi _ {0} ) $ for some $ \chi _ {0} $ in the group $ \widehat{G} $ of characters of $ G $; therefore the maximal ideals of $ L _ {1} ( G) $ are in one-to-one correspondence with the elements of $ \widehat{G} $. As applied to the group of integers $ \mathbf Z $, this last example leads to a proof of the well-known Wiener theorem: If the function $ \widehat{f} ( t) $ has an absolutely convergent Fourier series and does not vanish on $ [ 0 , 2 \pi ] $, then $ 1 / \widehat{f} ( t) $ also has an absolutely convergent Fourier series.

Since a multiplicative linear functional has norm 1, each such a functional belongs to the unit sphere of the dual of $ A $. The set $ \Phi $ of all multiplicative linear functionals on $ A $ is closed in the weak topology on the dual space. Since the unit ball is compact in the weak topology on the dual space, $ \Phi $ is also compact in this topology; it is called the maximal ideal space of the algebra $ A $ and it is denoted by $ \mathfrak M $.

If a commutative Banach algebra $ A $ contains a non-trivial idempotent, that is, an element $ f \in A $ such that $ f \neq 0 $, $ f \neq e $ and $ f ^ { 2 } = f $, then the maximal ideal space of $ A $ is disconnected. Conversely, if the maximal ideal space $ X $ of the algebra $ A $ is the union of two disjoint closed sets $ X _ {0} $ and $ X _ {1} $, then there is an element $ f \in A $ such that $ \widehat{f} \mid _ {X _ {0} } = 0 $ and $ \widehat{f} \mid _ {X _ {1} } = 1 $ (Shilov's theorem). In particular, the maximal ideal space of a commutative Banach algebra is connected if and only if this algebra cannot be represented as a direct sum of two non-trivial ideals.

Let $ \epsilon _ {1} ( A) $ be the subgroup of the group $ \epsilon ( A) $ of invertible elements of the algebra $ A $ consisting of the exponentials, that is, of the elements of the form $ \mathop{\rm exp} a = \sum _ {0} ^ \infty a ^ {n} / n ! $. Then $ \epsilon _ {1} ( A) $ is the connected component of the identity in $ \epsilon ( A) $. For any compactum $ X $ there is a canonical isomorphism between the groups $ H ^ {1} ( X , \mathbf Z ) $ and $ \epsilon ( C) / \epsilon _ {1} ( C) $, where $ C = C ( X) $ is the algebra of all continuous functions on $ X $ (the Brushlinskii–Eilenberg theorem). It turns out that this isomorphism naturally induces an isomorphism between $ H ^ {1} ( X , \mathbf Z ) $ and $ \epsilon ( A) / \epsilon _ {1} ( A) $, where $ A $ is any commutative Banach algebra whose maximal ideal space is $ X $ (the Arens–Royden theorem). In some cases the groups $ H ^ {q} ( X , \mathbf Z ) $ with $ q $ odd have a similar interpretation. The algebra $ A $ has the following canonical representation in the algebra $ C ( \mathfrak M ) $. The Gel'fand transform of an element $ a \in A $ is the function $ \widehat{a} $ on $ \mathfrak M $ defined by the formula $ \widehat{a} ( x) = \phi _ {x} ( a) $, where $ \phi _ {x} $ is the multiplicative linear functional corresponding to the point $ x \in \mathfrak M $. The kernel of the homomorphism $ a \mapsto \widehat{a} $ is the set of all elements $ a \in A $ belonging to all maximal ideals, i.e. belonging to the radical of $ A $. If $ A $ is a semi-simple algebra, that is, if $ \mathop{\rm Rad} A = \{ 0 \} $, then the homomorphism $ a \mapsto \widehat{a} $ is an (algebraic) isomorphism of $ A $ to $ C ( \mathfrak M ) $. Semi-simple commutative Banach algebras are often called function algebras.

The Gel'fand transform is well suited to the study of semi-simple algebras: One of the fundamental results in the theory of commutative Banach algebras is the theorem that a semi-simple algebra can be represented as an algebra of continuous functions on the maximal ideal space. Far less is known about general algebras with a radical in comparison to semi-simple algebras. All ideals of the algebra of complex polynomials of degree $ \leq m $ are known. This algebra consists of formal polynomials $ \xi = a _ {0} + a _ {1} \lambda + \dots + a _ {m} \lambda ^ {m} $, with the usual multiplication rule, subject to the relation $ \lambda ^ {m+1} = 0 $. This algebra is finite-dimensional, all norms on it are equivalent and every ideal of it is closed. The set $ I _ {k} $ of those $ \xi $ for which $ a _ {j} = 0 $ for $ j \leq k $ is a closed ideal; there are no other ideals in this algebra. Every algebra with a unique non-trivial ideal is isomorphic to the algebra of polynomials of the first degree. Until now (1987) it is not known whether the same is true for algebras with a unique non-trivial closed ideal.

The natural infinite-dimensional analogues of algebras of polynomials are algebras of power series $ \xi = a _ {0} + a _ {1} \lambda + a _ {2} \lambda ^ {2} + \dots $, with the usual operations and norm $ \| \xi \| = \sum _ {k=0} ^ \infty | a _ {k} | \alpha _ {k} $, where $ \alpha _ {k} $ is a sequence of positive numbers satisfying $ \alpha _ {k+l} \leq \alpha _ {k} \alpha _ {l} $. If $ \alpha _ {k} ^ {1/k} \rightarrow 0 $ as $ k \rightarrow \infty $, then the unique non-trivial homomorphism into the field of complex numbers is given by $ \xi \rightarrow a _ {0} $. Thus, $ I _ {1} $ is the unique maximal ideal and this ideal coincides with the radical. The ideals $ I _ {k} $, defined in the same way as in the finite-dimensional case, constitute a countable set of closed ideals. If the sequence $ \{ \alpha _ {k+} 1 / \alpha _ {k} \} $ is monotone, then this set of ideals contains all closed ideals. In general, an algebra may contain uncountably many distinct closed ideals.

By suitably choosing the sequence $ \{ \alpha _ {k} \} $ in the algebra under consideration (without non-trivial nilpotents), it is possible to define a non-zero derivation, that is, a bounded linear operator $ D $ such that $ D ( \xi \eta ) = ( D \xi ) \eta + \xi ( D \eta ) $. There are no non-trivial continuous derivations on a semi-simple algebra, since in any (not necessarily commutative) algebra the identity

$$ ( D \xi ) ^ {n} = \frac{1}{n!} \sum _ { k= 1} ^ { n } ( - 1 ) ^ {k+n} \left ( \begin{array}{c} n \\ k \end{array} \right ) \xi ^ {n-k} D ^ {n} \xi ^ {k} $$

holds if $ \xi $ and $ D \xi $ commute. In particular, if $ D $ is continuous, then $ D \xi $ is a generalized nilpotent.

Any finite-dimensional algebra decomposes into the direct sum of the radical and a semi-simple algebra. In the infinite-dimensional case this assertion ceases to be true in general, even for commutative Banach algebras. In addition, it is necessary to distinguish between the cases of algebraic and strong (topological) decomposability.

It turns out that there are no conditions that can be imposed merely on the radical that will ensure even algebraic decomposability: the radical may be one-dimensional and may annihilate some maximal ideal but it need not be a direct summand, even in the algebraic sense.

On the other hand, if the radical is finite-dimensional and the quotient algebra is an algebra of continuous functions (or an algebra of operators on a Hilbert space), then it is strongly decomposable. If the quotient algebra is an algebra of continuous functions and its annihilator radical $ R $ (i.e. the square of every element of $ R $ is zero) has a Banach complement, then $ A $ is strongly decomposable. Instead of the condition that $ R $ has a complement one can require that the space of maximal ideals of $ A $ satisfy the first axiom of countability at every point.

Completely investigated is also the case when the quotient algebra by the radical is the algebra of continuous functions on a totally-disconnected compactum: A necessary and sufficient condition for decomposability is that the idempotents of the original algebra be uniformly bounded.

Let $ V $ be a bounded domain in $ \mathbf C ^ {n} $ and let $ A $ be the closed subalgebra of $ C ( \overline{V} ) $ consisting of the functions holomorphic on $ V $. It is known that under fairly general hypotheses concerning $ V $, any maximal ideal of $ A $, corresponding to a point $ z ^ {0} = ( z _ {1} ^ {0}, \dots, z _ {n} ^ {0} ) \in V $, is finitely generated; namely, it is generated by the functions $ f _ {i} = z _ {i} - z _ {i} ^ {0} $. This statement has the following local converse. Let $ A $ be a semi-simple commutative Banach algebra with maximal ideal space $ X $. If the maximal ideal corresponding to a point $ x _ {0} \in X $ is generated by a finite set of elements $ f _ {1}, \dots, f _ {n} \in A $, then the maximal ideals corresponding to the points in some neighbourhood of $ x _ {0} $ are generated by elements of the form $ f _ {i} - \lambda _ {i} e $; the mapping $ \psi : x \mapsto ( f _ {1} ( x), \dots, f _ {n} ( x) ) $ is one-to-one in some neighbourhood of $ x _ {0} $ and the function $ g \circ \psi ^ {-1} $ is, for any $ g \in A $, holomorphic in some fixed neighbourhood of the origin in $ \mathbf C ^ {n} $. Furthermore, in a neighbourhood of $ x _ {0} $ a certain natural analytic structure can be introduced.

A set $ S $ of elements of an algebra $ A $ is called a system of generators if the smallest closed algebra with identity in $ A $ that contains $ S $ is $ A $ itself. The identity is usually not included in the set of generators. If there is a finite system $ S $ with the above properties, then $ A $ is called a finitely-generated algebra. The smallest numbers of elements in a system of generators is called the number of generators of the algebra.

If $ f _ {1}, \dots, f _ {n} $ is a system of generators of an algebra, then the mapping $ x \mapsto ( \widehat{f} _ {1} ( x), \dots, \widehat{f} _ {n} ( x) ) $ induces a homomorphism of the maximal ideal space of this algebra onto some polynomially-convex compact set in $ \mathbf C ^ {n} $. Each polynomially-convex compact set in $ \mathbf C ^ {n} $ is the maximal ideal space of some Banach algebra (for example, the algebra of uniform limits of polynomials on this set).

The maximal ideal space $ X $ of an algebra with $ n $ generators satisfies the condition $ \mathop{\rm dim} X \leq 2 n $ and possesses a number of other properties; for example, $ H ^ {i} ( X , \mathbf C ) = 0 $ for $ i \geq n $. Hence it follows, in particular, that the number of generators in the algebra $ C ( S ^ {n} ) $, where $ S ^ {n} $ is the $ n $-dimensional unit sphere, is equal to $ n + 1 $; a similar result holds for an arbitrary $ n $-dimensional compact manifold $ X $. For any finite cellular $ n $-dimensional polyhedron $ X $, the algebra $ C ( X) $ has a system of $ n + 1 $ generators.

Let $ A $ be an algebra with maximal ideal space $ X $. The smallest closet set $ \Gamma \subset X $ on which all functions $ | \widehat{f} | $ attain their maximum is called the Shilov boundary of $ A $. For any commutative Banach algebra with identity this set exists and is unique.

A point $ x _ {0} \in X $ belongs to $ \Gamma $ if and only if for any neighbourhood $ U $ of $ x _ {0} $ there is an element $ f \in A $ for which $ \max _ {X} | \widehat{f} | = 1 $, but $ | \widehat{f} ( x) | < 1 $ outside $ U $. Furthermore, if $ U $ is an open subset of $ X $ and if there exist a closed set $ F \subset U $ and an element $ g \in A $ such that $ | \widehat{g} ( x) | < \max _ {F} | \widehat{g} | $ for points $ x \in U \setminus F $, then the intersection $ \Gamma \cap U $ is non-empty.

Any multiplicative linear functional $ \phi $ is continuous with respect to the norm defined by the spectral radius; moreover, $ | \phi ( f) | \leq \max _ {X} | \widehat{f} | $, where $ X $ is the maximal ideal space. In this inequality, according to the definition of the Shilov boundary, one can replace $ X $ by $ \Gamma $; therefore there exists a positive regular measure $ \mu $ on $ \Gamma $" representing" the functional $ \phi $, that is, such that for all $ f \in A $ the equation $ \phi ( f) = \int _ \Gamma \widehat{f} d \mu $ holds. In the case of the algebra of functions analytic on the disc, this formula reduces to the classical Poisson formula. Among the representing measures there exists a measure $ \mu $ satisfying the Jensen inequality $ \mathop{\rm ln} | \phi ( f) | \leq \int _ \Gamma \mathop{\rm ln} \widehat{f} d \mu $ for all $ f \in A $.

Let $ B $ be a commutative Banach algebra with identity and let $ A $ be a closed subalgebra. The algebra $ A $ is called a maximal subalgebra of $ B $ if $ B $ contains no closed proper subalgebra properly containing $ A $. In each sufficiently large algebra $ B $ there are maximal subalgebras with identity, and even closed subalgebras of codimension 1. In fact, if $ \phi _ {1} $ and $ \phi _ {2} $ are two distinct homomorphisms of the algebra $ B $ into the field of complex numbers and if $ \psi = \phi _ {1} - \phi _ {2} $, then the kernel of $ \psi $ is a closed subalgebra $ A $ of $ B $ for which $ \mathop{\rm dim} B / A = 1 $. Similarly, the kernel of a "point derivation13B10point derivation" , that is, a functional $ \psi $ such that $ \psi ( f g ) = \psi ( f ) \phi ( g) + \psi ( g) \phi ( f ) $, where $ \phi $ is a multiplicative functional, is a subalgebra of codimension 1. In the complex case these examples exhaust all subalgebras of codimension 1. In particular, every such subalgebra of the algebra $ C ( X) $ does not separate points of the compactum $ X $, since on $ C ( X) $ there are no derivations (not even discontinuous ones). All subalgebras of finite codimension have a similar description.

The algebra $ A $ of continuous functions on the unit circle that have an analytic continuation inside the unit disc is a maximal subalgebra of the algebra of continuous functions on the unit circle. This statement can be regarded as a generalization of the Stone–Weierstrass approximation theorem, which asserts that a closed subalgebra of $ C ( \Gamma ) $, where $ \Gamma = \{ {z } : {| z | = 1 } \} $, containing $ A $ and the function $ \overline{z} $ coincides with $ C ( \Gamma ) $. The algebra $ L _ {1} ^ {+} ( \mathbf R ) = \{ {f \in L _ {1} ( \mathbf R ) } : {f ( t) = 0 \textrm{ when } t < 0 } \} $ is a closed subalgebra of $ L _ {1} ( \mathbf R ) $; this subalgebra is maximal.

Let $ \alpha $ be a irrational number and let $ A _ \alpha $ be the algebra of all continuous functions on the two-dimensional torus with Fourier coefficients $ c _ {mn} = 0 $ for $ m + n \alpha < 0 $. This algebra is a maximal subalgebra of the algebra of all continuous functions on the torus. The torus is the Shilov boundary of the algebra $ A _ \alpha $, and $ A _ \alpha $ is a Dirichlet algebra. If the torus is realized as the skeleton of the unit bidisc in $ \mathbf C ^ {2} $, then the maximal ideal space of $ A _ \alpha $ is identified with the subset of the bidisc described by the equation $ | z _ {1} | = | z _ {2} | ^ \alpha $. The point $ ( 0 , 0 ) $ does not belong to the Shilov boundary, but is a one-point Gleason part. (Two multiplicative functionals $ \phi _ {1} $ and $ \phi _ {2} $ on a uniform algebra belong to the same Gleason part, by definition, if $ \| \phi _ {1} - \phi _ {2} \| < 2 $.) The algebra $ A _ \alpha $ is analytic on the maximal ideal space (in the sense of uniqueness: $ f = 0 $ if $ \widehat{f} ( \xi ) = 0 $ for the points of a non-empty open set), even though the real dimension of the maximal ideal space is equal to 3. The algebra of continuous functions on the $ n $-dimensional torus having an extension inside the corresponding polydisc is not maximal when $ n > 1 $, but it is maximal in the class of subalgebras that are invariant with respect to holomorphic automorphisms of the torus.

For references see Banach algebra.

How to Cite This Entry:
Commutative Banach algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Commutative_Banach_algebra&oldid=14701
This article was adapted from an original article by E.A. Gorin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article