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Difference between revisions of "Banach algebra"

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{{MSC|46HXX|46JXX}}
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$
 
$
\newcommand{\C}{\mathbf{C}}
 
 
\newcommand{\norm}[1]{\left\|#1\right\|}
 
\newcommand{\norm}[1]{\left\|#1\right\|}
 
\newcommand{\abs}[1]{\left|#1\right|}
 
\newcommand{\abs}[1]{\left|#1\right|}
 +
\newcommand{\rad}{\mathrm{Rad}}
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\newcommand{\conj}[1]{\bar{#1}}
 
$
 
$
 
A [[Topological algebra|topological algebra]] $A$ over the field of complex numbers whose topology is  
 
A [[Topological algebra|topological algebra]] $A$ over the field of complex numbers whose topology is  
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algebra is said to be an algebra with a unit if $A$ contains an element $e$ such that $ex=xe=x$ for any  
 
algebra is said to be an algebra with a unit if $A$ contains an element $e$ such that $ex=xe=x$ for any  
 
$x\in A$. If a Banach algebra has no unit, a unit may be adjoined, i.e. it is possible to construct a  
 
$x\in A$. If a Banach algebra has no unit, a unit may be adjoined, i.e. it is possible to construct a  
Banach algebra $\tilde{A}$ with a unit element such that $\tilde{A}$ contains the initial algebra $A$ as  
+
Banach algebra $\tilde{A}$ with a unit element such that $\tilde{A}$ contains the initial algebra $A$ as a closed subalgebra of codimension one. In any Banach algebra $A$ with a unit element $e$ it is possible to change the norm for an equivalent one so that in the new norm the relationships  
a closed subalgebra of codimension one. In any Banach algebra $A$ with a unit element $e$ it is possible  
 
to change the norm for an equivalent one so that in the new norm the relationships  
 
 
$\norm{ab} \leq \norm{a}\norm{b}$, $\norm{e} = 1$ are valid. In what follows it is, as a rule, assumed  
 
$\norm{ab} \leq \norm{a}\norm{b}$, $\norm{e} = 1$ are valid. In what follows it is, as a rule, assumed  
 
that the algebra does contain a unit and that it satisfies the norm conditions given above.
 
that the algebra does contain a unit and that it satisfies the norm conditions given above.
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(f_1 * f_2)(h) = \Gint{f_1(g)f_2(g^{-1}h)}
 
(f_1 * f_2)(h) = \Gint{f_1(g)f_2(g^{-1}h)}
 
$$
 
$$
is considered as the multiplication in $L_1(G)$, then $L_1(G)$ becomes a Banach algebra; if $G$ is an Abelian locally compact group, then the Banach algebra $L_1(G)$ is commutative. The Banach algebra $L_1(G)$ is said to be the [[Group algebra|group algebra]] of $G$. The group algebra $L_1(G)$ has a unit (with respect to the convolution) if and only if $G$ is discrete.
+
is considered as the multiplication in $L_1(G)$, then $L_1(G)$ becomes a Banach algebra; if $G$ is an Abelian locally compact group, then the Banach algebra $L_1(G)$ is commutative. The Banach algebra $L_1(G)$ is said to be the
 +
[[Group algebra|group algebra]] of $G$. The group algebra $L_1(G)$ has a unit (with respect to the convolution) if and only if $G$ is discrete.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513043.png" /> is commutative it is possible to construct a faithful representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513044.png" />, given by the Fourier transform of each function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513045.png" />, i.e. by the function
+
If $G$ is commutative it is possible to construct a faithful representation of $L_1(G)$, given by the Fourier transform of each function $f \in L_1(G)$, i.e. by the function
 +
$$
 +
\hat{f}(\chi) = \Gint{\chi(g) f(g)}
 +
$$
 +
on the character group $\hat{G}$ of $G$. The set of functions $\hat{f}(\xi)$ forms a certain algebra $A(\hat{G})$ of continuous functions on $\hat{G}$ (with respect to the ordinary [[pointwise operation]]s), called the Fourier algebra of the locally compact Abelian group $\hat{G}$. In particular, if $G$ is the group of integers $\Z$, then $A(\hat{\Z})$ is the algebra of continuous functions on the circle which are expandable into an absolutely convergent trigonometric series.
  
<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/b/b015/b015130/b01513046.png" /></td> </tr></table>
+
5) Let $G$ be a topological group. A continuous complex-valued function $f(g)$ on $G$ is said to be almost periodic if the set of its shifts $f(g_0 g)$, $g_0 \in G$, forms a compact family with respect to uniform convergence on $G$. The set of almost-periodic functions forms a commutative Banach algebra with respect to the pointwise operations, with norm
 +
$$
 +
\norm{f} = \sup_{g \in G}\abs{f(g)}
 +
$$
  
on the character group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513047.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513048.png" />. The set of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513049.png" /> forms a certain algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513050.png" /> of continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513051.png" /> (with respect to the ordinary pointwise operations), called the Fourier algebra of the locally compact Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513052.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513053.png" /> is the group of integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513054.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513055.png" /> is the algebra of continuous functions on the circle which are expandable into an absolutely convergent trigonometric series.
+
6) The skew-field of quaternions does not form a Banach algebra over the field of complex numbers, since the product of elements of a Banach algebra $A$ should be compatible with multiplication by numbers: For all $\lambda \in \C$ and $x$, $y \in A$ the equation
 +
$$
 +
\lambda(xy) = (\lambda x)y = x(\lambda y),
 +
$$
 +
must be valid; it is not valid in the field of quaternions if $\lambda=i$, $x=j$, $y=k$.
  
5) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513056.png" /> be a topological group. A continuous complex-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513057.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513058.png" /> is said to be almost periodic if the set of its shifts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513060.png" />, forms a compact family with respect to uniform convergence on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513061.png" />. The set of almost-periodic functions forms a commutative Banach algebra with respect to the pointwise operations, with norm
+
Any Banach algebra with a unit is a topological algebra with continuous inverses. Moreover, if $\epsilon(A)$ is the set of elements of a Banach algebra $A$ which have a (two-sided) inverse with respect to multiplication, then $\epsilon(A)$ is a topological group in the topology induced by the imbedding $\epsilon(A)\subset A$. If $\norm{e-a} < 1$, then $a \in \epsilon(A)$, and
 +
$$
 +
a^{-1} = \sum_{n=0}^\infty\; b^n,
 +
$$
 +
where $b=e-a$, and the series is absolutely convergent. The set of elements invertible from the right (from the left) in $A$ also forms an open set in $A$.
  
<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/b/b015/b015130/b01513062.png" /></td> </tr></table>
+
If in a Banach algebra $A$ all elements have an inverse (or even a left inverse), then $A$ is isometrically isomorphic to the field of complex numbers (the Gel'fand–Mazur theorem).
  
6) The skew-field of quaternions does not form a Banach algebra over the field of complex numbers, since the product of elements of a Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513063.png" /> should be compatible with multiplication by numbers: For all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513064.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513065.png" /> the equation
+
Since a certain neighbourhood of the unit in a Banach algebra $A$ consists of invertible elements, the closure of any non-trivial ideal is again an ideal which does not coincide with $A$. In particular, a maximal (left, right, two-sided) ideal is closed.
  
<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/b/b015/b015130/b01513066.png" /></td> </tr></table>
+
An important task in the theory of Banach algebras is the description of closed ideals in Banach algebras. The problem can be simply solved in a number of cases. In the algebra $C(X)$ (cf. Example 1) each closed ideal has the form $\left\{ f \in C(X) : f |_Y = 0 \right\}$, where $Y$ is a closed set in $X$. If $A$ is the algebra of all bounded linear operators on a separable infinite-dimensional Hilbert space, then the ideal of completely-continuous operators is the only closed two-sided ideal in $A$.
  
must be valid; it is not valid in the field of quaternions if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513067.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513069.png" />.
+
An element $a \in A$ has a left (right) inverse if and only if it is not contained in any maximal left (right) ideal. The intersection of all maximal left ideals in $A$ coincides with the intersection of all maximal right ideals; this intersection is called the radical of the algebra $A$ and is denoted by $\rad A$. An element $a_0 \in A$ belongs to $\rad A $ if and only if $e + a a_0 \in \epsilon(A)$ for any $a \in A$. Algebras for which $\rad A = 0$ are said to be semi-simple. The algebras $C(X)$ and the group algebras $L_1(G)$ are semi-simple. All irreducible (i.e. not having a non-trivial invariant subspace) closed subalgebras of the algebra of all bounded linear operators on a Banach space are semi-simple.
  
Any Banach algebra with a unit is a topological algebra with continuous inverses. Moreover, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513070.png" /> is the set of elements of a Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513071.png" /> which have a (two-sided) inverse with respect to multiplication, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513072.png" /> is a topological group in the topology induced by the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513073.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513074.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513075.png" />, and
+
The resolvent of an element $a \in A$ is the function
 
+
$$
<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/b/b015/b015130/b01513076.png" /></td> </tr></table>
+
\lambda \rightarrow a_\lambda = (a - \lambda e)^{-1}
 
+
$$
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513077.png" />, and the series is absolutely convergent. The set of elements invertible from the right (from the left) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513078.png" /> also forms an open set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513079.png" />.
+
defined on the set of all $\lambda \in \C$ for which a (two-sided) inverse to $a - \lambda e$ exists. The domain of existence of the resolvent contains all points $\lambda$ with $\abs{\lambda} \geq \norm{a} $. The maximal domain of existence of the resolvent is an open set; the resolvent is continuous on this set and is even analytic, moreover $da_\lambda/d\lambda = a_\lambda^2$. In addition, Hilbert's identity
 
+
$$
If in a Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513080.png" /> all elements have an inverse (or even a left inverse), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513081.png" /> is isometrically isomorphic to the field of complex numbers (the Gel'fand–Mazur theorem).
+
a_{\lambda_2} - a_{\lambda_1} = (\lambda_2 - \lambda_1)a_{\lambda_1}a_{\lambda_2}
 
+
$$
Since a certain neighbourhood of the unit in a Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513082.png" /> consists of invertible elements, the closure of any non-trivial ideal is again an ideal which does not coincide with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513083.png" />. In particular, a maximal (left, right, two-sided) ideal is closed.
+
is valid. The complement of the domain of existence of the resolvent is called the spectrum of the element $a$ and is denoted by $\sigma(a)$. For each $a \in A$ the set $\sigma(a)$ is non-empty, closed and bounded.
 
 
An important task in the theory of Banach algebras is the description of closed ideals in Banach algebras. The problem can be simply solved in a number of cases. In the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513084.png" /> (cf. Example 1) each closed ideal has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513085.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513086.png" /> is a closed set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513087.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513088.png" /> is the algebra of all bounded linear operators on a separable infinite-dimensional Hilbert space, then the ideal of completely-continuous operators is the only closed two-sided ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513089.png" />.
 
 
 
An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513090.png" /> has a left (right) inverse if and only if it is not contained in any maximal left (right) ideal. The intersection of all maximal left ideals in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513091.png" /> coincides with the intersection of all maximal right ideals; this intersection is called the radical of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513092.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513093.png" />. An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513094.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513095.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513096.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513097.png" />. Algebras for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513098.png" /> are said to be semi-simple. The algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b01513099.png" /> and the group algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130100.png" /> are semi-simple. All irreducible (i.e. not having a non-trivial invariant subspace) closed subalgebras of the algebra of all bounded linear operators on a Banach space are semi-simple.
 
 
 
The resolvent of an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130101.png" /> is the function
 
 
 
<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/b/b015/b015130/b015130102.png" /></td> </tr></table>
 
 
 
defined on the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130103.png" /> for which a (two-sided) inverse to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130104.png" /> exists. The domain of existence of the resolvent contains all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130105.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130106.png" />. The maximal domain of existence of the resolvent is an open set; the resolvent is continuous on this set and is even analytic, moreover <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130107.png" />. In addition, Hilbert's identity
 
 
 
<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/b/b015/b015130/b015130108.png" /></td> </tr></table>
 
 
 
is valid. The complement of the domain of existence of the resolvent is called the spectrum of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130109.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130110.png" />. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130111.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130112.png" /> is non-empty, closed and bounded.
 
 
 
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130113.png" />, then the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130114.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130115.png" /> need not coincide, but
 
 
 
<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/b/b015/b015130/b015130116.png" /></td> </tr></table>
 
  
 +
If $a$, $b\in A$ then the sets $\sigma(ab)$ and $\sigma(ba)$ need not coincide, but
 +
$$
 +
\sigma(ab) \cup \left\{0\right\} = \sigma(ba) \cup \left\{0\right\}
 +
$$
 
The number
 
The number
 +
$$
 +
\abs{a} = \max_{\lambda \in \sigma(a)}\abs{\lambda}
 +
$$
 +
is called the spectral radius of the element $a$; Gel'fand's formula
 +
$$
 +
\abs{a} = \lim \norm{a^n}^{1/n},
 +
$$
 +
where the limit on the right-hand side always exists, is valid. If $a \in \rad A$, then $\abs{a}=0$; the converse is true, generally speaking, only in commutative Banach algebras whose radical coincides with the set of generalized nilpotents, i.e. elements $a$ for which $\abs{a}=0$. In any Banach algebra the relationships $\abs{a^k}=\abs{a}^k$, $\abs{\lambda a}=\abs{\lambda}\abs{a}$ and $\abs{a} \leq \norm{a}$ are true. If $A$ is commutative, then $\abs{ab} \leq \abs{a}\abs{b}$ and $\abs{a+b} \leq \abs{a} + \abs{b}$ are valid.
  
<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/b/b015/b015130/b015130117.png" /></td> </tr></table>
+
Examples of non-commutative algebras in which generalized non-zero nilpotents are absent are known. However, if $\norm{a^2} = \norm{a}^2$ for any $a \in A$, then the Banach algebra $A$ is commutative. The condition $\norm{ab}=\norm{ba}$ for all $a$, $b \in A$ is also sufficient for an algebra $A$ with a unit to be commutative.
  
is called the spectral radius of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130118.png" />; Gel'fand's formula
+
An algebra $A$ is said to be an algebra with involution if an operation $a \rightarrow a^*$ is defined on $A$ that satisfies the conditions
 
+
$$
<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/b/b015/b015130/b015130119.png" /></td> </tr></table>
+
(\lambda a + \mu b)^* = \conj{\lambda}a^* + \conj{\mu}b^*, \quad
 
+
(a^*)^* = a, \quad
where the limit on the right-hand side always exists, is valid. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130120.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130121.png" />; the converse is true, generally speaking, only in commutative Banach algebras whose radical coincides with the set of generalized nilpotents, i.e. elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130122.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130123.png" />. In any Banach algebra the relationships <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130124.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130125.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130126.png" /> are true. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130127.png" /> is commutative, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130128.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130129.png" /> are valid.
+
(ab)^* = b^* a^*,
 
+
$$
Examples of non-commutative algebras in which generalized non-zero nilpotents are absent are known. However, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130130.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130131.png" />, then the Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130132.png" /> is commutative. The condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130133.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130134.png" /> is also sufficient for an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130135.png" /> with a unit to be commutative.
+
for all $a$, $b \in A$, $\lambda$, $\mu \in \C$. The mapping $a \rightarrow a^*$ is said to be an involution in $A$. A linear functional $\psi$ on an algebra $A$ with an involution is said to be positive if $\psi(aa^*) \geq 0$ for any $a\in A$. If the linear functional $\psi$ is positive, then
 
+
$$
An algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130136.png" /> is said to be an algebra with involution if an operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130137.png" /> is defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130138.png" /> that satisfies the conditions
+
\abs{\psi(a)}^2 \leq \psi(e)\psi(aa^*)
 
+
$$
<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/b/b015/b015130/b015130139.png" /></td> </tr></table>
+
for all $a \in A$. If the involution in $A$ is an isometry, i.e. if $\norm{a^*}=\norm{a}$ for all $a \in A$, then
 
+
$$
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130140.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130141.png" /> is said to be an involution in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130142.png" />. A linear functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130143.png" /> on an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130144.png" /> with an involution is said to be positive if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130145.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130146.png" />. If the linear functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130147.png" /> is positive, then
+
\psi(a^*a) \leq \psi(e) \abs{a^*a}.
 
+
$$
<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/b/b015/b015130/b015130148.png" /></td> </tr></table>
+
A Banach algebra $A$ with involution is said to be completely symmetric if $e + a^*a \in \epsilon(A)$ for any $a \in A$; $A$ is said to be a [[C*-algebra|$C^*$-algebra]] (a completely-regular algebra) if $\norm{a^*a} = \norm{a}^2$ for any $a \in A$. Any $C^*$-algebra is completely symmetric. Examples of completely-symmetric algebras include the group algebras $L_1(G)$ of commutative or compact groups. Examples of $C^*$-algebras include the algebras $C(X)$ (the involution in $C(X)$ is defined as transition to the complex conjugate function) and closed subalgebras of the algebra of bounded linear operators in a Hilbert space containing both the operator and the adjoint operator (involution is defined as transition to the adjoint operator). Any $C^*$-algebra is isometrically isomorphic (involution being preserved) with one of these algebras (the Gel'fand–Naimark theorem). In particular, any commutative $C^*$-algebra $A$ is isometrically isomorphic (involution being preserved) with one of the algebras $C(X)$ (this theorem includes the Stone–Weierstrass theorem).
 
 
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130149.png" />. If the involution in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130150.png" /> is an isometry, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130151.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130152.png" />, then
 
 
 
<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/b/b015/b015130/b015130153.png" /></td> </tr></table>
 
 
 
A Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130154.png" /> with involution is said to be completely symmetric if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130155.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130156.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130157.png" /> is said to be a [[C*-algebra|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130158.png" />-algebra]] (a completely-regular algebra) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130159.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130160.png" />. Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130161.png" />-algebra is completely symmetric. Examples of completely-symmetric algebras include the group algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130162.png" /> of commutative or compact groups. Examples of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130163.png" />-algebras include the algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130164.png" /> (the involution in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130165.png" /> is defined as transition to the complex conjugate function) and closed subalgebras of the algebra of bounded linear operators in a Hilbert space containing both the operator and the adjoint operator (involution is defined as transition to the adjoint operator). Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130166.png" />-algebra is isometrically isomorphic (involution being preserved) with one of these algebras (the Gel'fand–Naimark theorem). In particular, any commutative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130167.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130168.png" /> is isometrically isomorphic (involution being preserved) with one of the algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130169.png" /> (this theorem includes the Stone–Weierstrass theorem).
 
  
An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130170.png" /> of a Banach algebra with involution is said to be Hermitian if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130171.png" />. For a Banach algebra with an involution to be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130172.png" />-algebra it is necessary and sufficient that the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130173.png" /> be fulfilled for all Hermitian elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130174.png" />. If, in a Banach algebra with an involution, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130175.png" /> (upper bound over all Hermitian elements), then this algebra is topologically <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130176.png" />-isomorphic with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130177.png" />-algebra. If, in an arbitrary Banach algebra, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130178.png" /> for all real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130179.png" /> for a certain fixed element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130180.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130181.png" /> coincides with the spectral radius, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015130/b015130182.png" />.
+
An element $a$ of a Banach algebra with involution is said to be Hermitian if $a^* = a$. For a Banach algebra with an involution to be a $C^*$-algebra it is necessary and sufficient that the condition $\norm{e^{ia}} = 1$ be fulfilled for all Hermitian elements $a$. If, in a Banach algebra with an involution, $\sup\norm{e^{ia}} < \infty$ (upper bound over all Hermitian elements), then this algebra is topologically $*$-isomorphic with a $C^*$-algebra. If, in an arbitrary Banach algebra, $\norm{e^{ita}} = 1$ for all real $t$ for a certain fixed element $a$, then $\norm{a}$ coincides with the spectral radius, i.e. $\norm{a} = \abs{a}$.
  
 
The theory of Banach algebras, and of commutative Banach algebras in particular, has numerous applications in various branches of functional analysis and in a number of other mathematical disciplines.
 
The theory of Banach algebras, and of commutative Banach algebras in particular, has numerous applications in various branches of functional analysis and in a number of other mathematical disciplines.
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Spectral theories" , Addison-Wesley  (1977)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  T.W. Gamelin,  "Uniform algebras" , Prentice-Hall  (1969)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R.C. Gunning,  H. Rossi,  "Analytic functions of several complex variables" , Prentice-Hall  (1965)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.M. Gel'fand,  "Normierte Ringe"  ''Mat. Sb.'' , '''9 (51)''' :  1  (1941)  pp. 3–24</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.M. Gleason,  "Function algebras" , ''Proc. Sem. on analytic functions'' , '''2'''  (1958)  pp. 213–226</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  K. Hoffman,  "Banach spaces of analytic functions" , Prentice-Hall  (1962)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  E.A. Gorin,  "Maximal subalgebras of commutative Banach algebras with involution"  ''Math. Notes'' , '''1''' :  2  (1967)  pp. 173–178  ''Mat. Zametki'' , '''1''' :  2  (1967)  pp. 173–178</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators" , '''1–3''' , Interscience  (1958–1971)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  W. Zelazko,  "Banach algebras" , Elsevier  (1973)  (Translated from Polish)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  I. Kaplansky,  "Functional analysis" , ''Surveys in applied mathematics'' , '''4. Some aspects of analysis and probability''' , Wiley  (1958)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  L.H. Loomis,  "An introduction to abstract harmonic analysis" , v. Nostrand  (1953)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  M.A. Naimark,  "Normed rings" , Reidel  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> , ''Some questions in approximation theory'' , Moscow  (1963)  (In Russian; translated from English)</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top">  C.E. Rickart,  "General theory of Banach algebras" , v. Nostrand  (1960)</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top">  H.L. Royden,  "Function algebras"  ''Bull. Amer. Math. Soc.'' , '''69''' :  3  (1963)  pp. 281–298</TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top">  R.R. Phelps,  "Lectures on Choquet's theorem" , v. Nostrand  (1966)</TD></TR><TR><TD valign="top">[17]</TD> <TD valign="top">  E. Hille,  R.S. Phillips,  "Functional analysis and semi-groups" , Amer. Math. Soc.  (1957)</TD></TR></table>
 
 
 
  
 
====Comments====
 
====Comments====
Line 129: Line 130:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R.V. Kadison,  J.R. Ringrose,  "Fundamentals of the theory of operator algebras" , '''1''' , Acad. Press  (1983)</TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|Bo}}||valign="top"|  N. Bourbaki,  "Elements of mathematics. Spectral theories", Addison-Wesley  (1977)  (Translated from French)  {{MR|0583191}}  {{ZBL|1106.46004}}
 +
|-
 +
|valign="top"|{{Ref|DuSc}}||valign="top"|  N. Dunford,  J.T. Schwartz,  "Linear operators", '''1–3''', Interscience  (1958–1971)  {{MR|0117523}}  {{ZBL|0084.10402}}
 +
|-
 +
|valign="top"|{{Ref|Ga}}||valign="top"|  T.W. Gamelin,  "Uniform algebras", Prentice-Hall  (1969)  {{MR|0410387}}  {{ZBL|0213.40401}}
 +
|-
 +
|valign="top"|{{Ref|Ge}}||valign="top"|  I.M. Gel'fand,  "Normierte Ringe"  ''Mat. Sb.'', '''9 (51)''' :  1  (1941)  pp. 3–24 
 +
|-
 +
|valign="top"|{{Ref|Gl}}||valign="top"|  A.M. Gleason,  "Function algebras", ''Proc. Sem. on analytic functions'', '''2'''  (1958)  pp. 213–226  {{ZBL|0095.10103}}
 +
|-
 +
|valign="top"|{{Ref|Go}}||valign="top"|  E.A. Gorin,  "Maximal subalgebras of commutative Banach algebras with involution"  ''Math. Notes'', '''1''' :  2  (1967)  pp. 173–178  ''Mat. Zametki'', '''1''' :  2  (1967)  pp. 173–178  {{MR|0208412}}  {{ZBL|0172.17901}}
 +
|-
 +
|valign="top"|{{Ref|GuRo}}||valign="top"|  R.C. Gunning,  H. Rossi,  "Analytic functions of several complex variables", Prentice-Hall  (1965)  {{MR|0180696}}  {{ZBL|0141.08601}}
 +
|-
 +
|valign="top"|{{Ref|HiPh}}||valign="top"|  E. Hille,  R.S. Phillips,  "Functional analysis and semi-groups", Amer. Math. Soc.  (1957)  {{MR|0089373}}  {{ZBL|0392.46001}} {{ZBL|0033.06501}}
 +
|-
 +
|valign="top"|{{Ref|Ho}}||valign="top"|  K. Hoffman,  "Banach spaces of analytic functions", Prentice-Hall  (1962)  {{MR|0133008}}  {{ZBL|0117.34001}}
 +
|-
 +
|valign="top"|{{Ref|Ka}}||valign="top"|  I. Kaplansky,  "Functional analysis", ''Surveys in applied mathematics'', '''4. Some aspects of analysis and probability''', Wiley  (1958)  {{MR|0101475}}  {{ZBL|0087.31102}}
 +
|-
 +
|valign="top"|{{Ref|KaRi}}||valign="top"| R.V. Kadison,  J.R. Ringrose,  "Fundamentals of the theory of operator algebras", '''1''', Acad. Press  (1983) {{MR|0719020}}  {{ZBL|0518.46046}}
 +
|-
 +
|valign="top"|{{Ref|Lo}}||valign="top"|  L.H. Loomis,  "An introduction to abstract harmonic analysis", v. Nostrand  (1953)  {{MR|0054173}}  {{ZBL|0052.11701}}
 +
|-
 +
|valign="top"|{{Ref|Na}}||valign="top"|  M.A. Naimark,  "Normed rings", Reidel  (1984)  (Translated from Russian)  {{MR|1292007}} {{MR|0355601}} {{MR|0355602}} {{MR|0205093}} {{MR|0110956}} {{MR|0090786}} {{MR|0026763}}  {{ZBL|0218.46042}} {{ZBL|0137.31703}} {{ZBL|0089.10102}} {{ZBL|0073.08902}}
 +
|-
 +
|valign="top"|{{Ref|Ph}}||valign="top"|  R.R. Phelps,  "Lectures on Choquet's theorem", v. Nostrand  (1966)  {{MR|0193470}}  {{ZBL|0135.36203}}
 +
|-
 +
|valign="top"|{{Ref|Ri}}||valign="top"|  C.E. Rickart,  "General theory of Banach algebras", v. Nostrand  (1960)  {{MR|0115101}}  {{ZBL|0095.09702}}
 +
|-
 +
|valign="top"|{{Ref|Ro}}||valign="top"|  H.L. Royden,  "Function algebras"  ''Bull. Amer. Math. Soc.'', '''69''' :  3  (1963)  pp. 281–298  {{MR|0149327}}  {{ZBL|0111.11802}}
 +
|-
 +
|valign="top"|{{Ref|Ze}}||valign="top"|  W. Zelazko,  "Banach algebras", Elsevier  (1973)  (Translated from Polish)  {{MR|0448079}}  {{ZBL|0248.46037}}
 +
|-
 +
|}

Latest revision as of 22:48, 29 November 2014

2020 Mathematics Subject Classification: Primary: 46HXX Secondary: 46JXX [MSN][ZBL]

$ \newcommand{\norm}[1]{\left\|#1\right\|} \newcommand{\abs}[1]{\left|#1\right|} \newcommand{\rad}{\mathrm{Rad}} \newcommand{\conj}[1]{\bar{#1}} $ A topological algebra $A$ over the field of complex numbers whose topology is defined by a norm which converts $A$ into a Banach space, the multiplication of the elements being separately continuous for both factors. A Banach algebra is said to be commutative if $xy=yx$ for all $x$, $y\in A$ (cf. Commutative Banach algebra). A Banach algebra is said to be an algebra with a unit if $A$ contains an element $e$ such that $ex=xe=x$ for any $x\in A$. If a Banach algebra has no unit, a unit may be adjoined, i.e. it is possible to construct a Banach algebra $\tilde{A}$ with a unit element such that $\tilde{A}$ contains the initial algebra $A$ as a closed subalgebra of codimension one. In any Banach algebra $A$ with a unit element $e$ it is possible to change the norm for an equivalent one so that in the new norm the relationships $\norm{ab} \leq \norm{a}\norm{b}$, $\norm{e} = 1$ are valid. In what follows it is, as a rule, assumed that the algebra does contain a unit and that it satisfies the norm conditions given above.

Examples.

1) Let $X$ be a compact topological space and let $C(X)$ be the set of all continuous complex-valued functions on $X$. $C(X)$ will then be a Banach algebra with respect to the usual operations, with norm $$ \norm{f} = \max_{X}\abs{f}. $$

2) The set of all bounded linear operators on a Banach space forms a Banach algebra with respect to the usual operations of addition and multiplication of linear operators with the operator norm.

3) Let $V$ be a bounded domain in $n$-dimensional complex space $\C^n$. The set of bounded holomorphic functions on $V$ is a Banach algebra with respect to the usual operations, with the natural sup-norm: $$ \norm{f} = \sup_V\abs{f}. $$ This Banach algebra contains the closed subalgebra of bounded holomorphic functions on $V$ that have a continuous extension to the closure of $V$. The simplest example is the algebra of functions that are continuous in the disc $\abs{z} \leq 1$ and analytic in the disc $\abs{z} < 1$.

4) Let $G$ be a locally compact group and let $L_1(G)$ be the space (of equivalence classes) of all functions that are measurable with respect to the Haar measure on $G$ and that are absolutely integrable with respect is this measure, with norm $$ \newcommand{\groupint}[3]{\int_#1 #2\,d#3} \newcommand{\Gint}[1]{\groupint{G}{#1}{g}} \norm{f} = \Gint{\abs{f(g)}} $$ (left Haar integral).

If the convolution operation $$ (f_1 * f_2)(h) = \Gint{f_1(g)f_2(g^{-1}h)} $$ is considered as the multiplication in $L_1(G)$, then $L_1(G)$ becomes a Banach algebra; if $G$ is an Abelian locally compact group, then the Banach algebra $L_1(G)$ is commutative. The Banach algebra $L_1(G)$ is said to be the group algebra of $G$. The group algebra $L_1(G)$ has a unit (with respect to the convolution) if and only if $G$ is discrete.

If $G$ is commutative it is possible to construct a faithful representation of $L_1(G)$, given by the Fourier transform of each function $f \in L_1(G)$, i.e. by the function $$ \hat{f}(\chi) = \Gint{\chi(g) f(g)} $$ on the character group $\hat{G}$ of $G$. The set of functions $\hat{f}(\xi)$ forms a certain algebra $A(\hat{G})$ of continuous functions on $\hat{G}$ (with respect to the ordinary pointwise operations), called the Fourier algebra of the locally compact Abelian group $\hat{G}$. In particular, if $G$ is the group of integers $\Z$, then $A(\hat{\Z})$ is the algebra of continuous functions on the circle which are expandable into an absolutely convergent trigonometric series.

5) Let $G$ be a topological group. A continuous complex-valued function $f(g)$ on $G$ is said to be almost periodic if the set of its shifts $f(g_0 g)$, $g_0 \in G$, forms a compact family with respect to uniform convergence on $G$. The set of almost-periodic functions forms a commutative Banach algebra with respect to the pointwise operations, with norm $$ \norm{f} = \sup_{g \in G}\abs{f(g)} $$

6) The skew-field of quaternions does not form a Banach algebra over the field of complex numbers, since the product of elements of a Banach algebra $A$ should be compatible with multiplication by numbers: For all $\lambda \in \C$ and $x$, $y \in A$ the equation $$ \lambda(xy) = (\lambda x)y = x(\lambda y), $$ must be valid; it is not valid in the field of quaternions if $\lambda=i$, $x=j$, $y=k$.

Any Banach algebra with a unit is a topological algebra with continuous inverses. Moreover, if $\epsilon(A)$ is the set of elements of a Banach algebra $A$ which have a (two-sided) inverse with respect to multiplication, then $\epsilon(A)$ is a topological group in the topology induced by the imbedding $\epsilon(A)\subset A$. If $\norm{e-a} < 1$, then $a \in \epsilon(A)$, and $$ a^{-1} = \sum_{n=0}^\infty\; b^n, $$ where $b=e-a$, and the series is absolutely convergent. The set of elements invertible from the right (from the left) in $A$ also forms an open set in $A$.

If in a Banach algebra $A$ all elements have an inverse (or even a left inverse), then $A$ is isometrically isomorphic to the field of complex numbers (the Gel'fand–Mazur theorem).

Since a certain neighbourhood of the unit in a Banach algebra $A$ consists of invertible elements, the closure of any non-trivial ideal is again an ideal which does not coincide with $A$. In particular, a maximal (left, right, two-sided) ideal is closed.

An important task in the theory of Banach algebras is the description of closed ideals in Banach algebras. The problem can be simply solved in a number of cases. In the algebra $C(X)$ (cf. Example 1) each closed ideal has the form $\left\{ f \in C(X) : f |_Y = 0 \right\}$, where $Y$ is a closed set in $X$. If $A$ is the algebra of all bounded linear operators on a separable infinite-dimensional Hilbert space, then the ideal of completely-continuous operators is the only closed two-sided ideal in $A$.

An element $a \in A$ has a left (right) inverse if and only if it is not contained in any maximal left (right) ideal. The intersection of all maximal left ideals in $A$ coincides with the intersection of all maximal right ideals; this intersection is called the radical of the algebra $A$ and is denoted by $\rad A$. An element $a_0 \in A$ belongs to $\rad A $ if and only if $e + a a_0 \in \epsilon(A)$ for any $a \in A$. Algebras for which $\rad A = 0$ are said to be semi-simple. The algebras $C(X)$ and the group algebras $L_1(G)$ are semi-simple. All irreducible (i.e. not having a non-trivial invariant subspace) closed subalgebras of the algebra of all bounded linear operators on a Banach space are semi-simple.

The resolvent of an element $a \in A$ is the function $$ \lambda \rightarrow a_\lambda = (a - \lambda e)^{-1} $$ defined on the set of all $\lambda \in \C$ for which a (two-sided) inverse to $a - \lambda e$ exists. The domain of existence of the resolvent contains all points $\lambda$ with $\abs{\lambda} \geq \norm{a} $. The maximal domain of existence of the resolvent is an open set; the resolvent is continuous on this set and is even analytic, moreover $da_\lambda/d\lambda = a_\lambda^2$. In addition, Hilbert's identity $$ a_{\lambda_2} - a_{\lambda_1} = (\lambda_2 - \lambda_1)a_{\lambda_1}a_{\lambda_2} $$ is valid. The complement of the domain of existence of the resolvent is called the spectrum of the element $a$ and is denoted by $\sigma(a)$. For each $a \in A$ the set $\sigma(a)$ is non-empty, closed and bounded.

If $a$, $b\in A$ then the sets $\sigma(ab)$ and $\sigma(ba)$ need not coincide, but $$ \sigma(ab) \cup \left\{0\right\} = \sigma(ba) \cup \left\{0\right\} $$ The number $$ \abs{a} = \max_{\lambda \in \sigma(a)}\abs{\lambda} $$ is called the spectral radius of the element $a$; Gel'fand's formula $$ \abs{a} = \lim \norm{a^n}^{1/n}, $$ where the limit on the right-hand side always exists, is valid. If $a \in \rad A$, then $\abs{a}=0$; the converse is true, generally speaking, only in commutative Banach algebras whose radical coincides with the set of generalized nilpotents, i.e. elements $a$ for which $\abs{a}=0$. In any Banach algebra the relationships $\abs{a^k}=\abs{a}^k$, $\abs{\lambda a}=\abs{\lambda}\abs{a}$ and $\abs{a} \leq \norm{a}$ are true. If $A$ is commutative, then $\abs{ab} \leq \abs{a}\abs{b}$ and $\abs{a+b} \leq \abs{a} + \abs{b}$ are valid.

Examples of non-commutative algebras in which generalized non-zero nilpotents are absent are known. However, if $\norm{a^2} = \norm{a}^2$ for any $a \in A$, then the Banach algebra $A$ is commutative. The condition $\norm{ab}=\norm{ba}$ for all $a$, $b \in A$ is also sufficient for an algebra $A$ with a unit to be commutative.

An algebra $A$ is said to be an algebra with involution if an operation $a \rightarrow a^*$ is defined on $A$ that satisfies the conditions $$ (\lambda a + \mu b)^* = \conj{\lambda}a^* + \conj{\mu}b^*, \quad (a^*)^* = a, \quad (ab)^* = b^* a^*, $$ for all $a$, $b \in A$, $\lambda$, $\mu \in \C$. The mapping $a \rightarrow a^*$ is said to be an involution in $A$. A linear functional $\psi$ on an algebra $A$ with an involution is said to be positive if $\psi(aa^*) \geq 0$ for any $a\in A$. If the linear functional $\psi$ is positive, then $$ \abs{\psi(a)}^2 \leq \psi(e)\psi(aa^*) $$ for all $a \in A$. If the involution in $A$ is an isometry, i.e. if $\norm{a^*}=\norm{a}$ for all $a \in A$, then $$ \psi(a^*a) \leq \psi(e) \abs{a^*a}. $$ A Banach algebra $A$ with involution is said to be completely symmetric if $e + a^*a \in \epsilon(A)$ for any $a \in A$; $A$ is said to be a $C^*$-algebra (a completely-regular algebra) if $\norm{a^*a} = \norm{a}^2$ for any $a \in A$. Any $C^*$-algebra is completely symmetric. Examples of completely-symmetric algebras include the group algebras $L_1(G)$ of commutative or compact groups. Examples of $C^*$-algebras include the algebras $C(X)$ (the involution in $C(X)$ is defined as transition to the complex conjugate function) and closed subalgebras of the algebra of bounded linear operators in a Hilbert space containing both the operator and the adjoint operator (involution is defined as transition to the adjoint operator). Any $C^*$-algebra is isometrically isomorphic (involution being preserved) with one of these algebras (the Gel'fand–Naimark theorem). In particular, any commutative $C^*$-algebra $A$ is isometrically isomorphic (involution being preserved) with one of the algebras $C(X)$ (this theorem includes the Stone–Weierstrass theorem).

An element $a$ of a Banach algebra with involution is said to be Hermitian if $a^* = a$. For a Banach algebra with an involution to be a $C^*$-algebra it is necessary and sufficient that the condition $\norm{e^{ia}} = 1$ be fulfilled for all Hermitian elements $a$. If, in a Banach algebra with an involution, $\sup\norm{e^{ia}} < \infty$ (upper bound over all Hermitian elements), then this algebra is topologically $*$-isomorphic with a $C^*$-algebra. If, in an arbitrary Banach algebra, $\norm{e^{ita}} = 1$ for all real $t$ for a certain fixed element $a$, then $\norm{a}$ coincides with the spectral radius, i.e. $\norm{a} = \abs{a}$.

The theory of Banach algebras, and of commutative Banach algebras in particular, has numerous applications in various branches of functional analysis and in a number of other mathematical disciplines.

Comments

Gel'fand's formula is also called the spectral radius formula.

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How to Cite This Entry:
Banach algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Banach_algebra&oldid=24288
This article was adapted from an original article by E.A. Gorin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article