View source for User:Ulf Rehmann/PreTeX:Banach algebra

{{MSC|46HXX|46JXX}}
{{TEX|done}}

$
\newcommand{\C}{\mathbf{C}}
\newcommand{\Z}{\mathbf{Z}}
\newcommand{\norm}[1]{\left\|#1\right\|}
\newcommand{\abs}[1]{\left|#1\right|}
\newcommand{\rad}{\mathrm{Rad}}
\newcommand{\conj}[1]{\bar{#1}}
$
A [[Topological algebra|topological algebra]] $A$ over the field of complex numbers whose topology is 
defined by a norm which converts $A$ into a [[Banach space|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|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|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|$C^*$-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.

====References====
{|
|-
|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}} 
|-
|}