User:Ulf Rehmann/PreTeX:Banach algebra

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$ \newcommand{\C}{\mathbf{C}} \newcommand{\Z}{\mathbf{Z}} \newcommand{\norm}[1]{\left\|#1\right\|} \newcommand{\abs}[1]{\left|#1\right|} $ 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 $ $ all elements have an inverse (or even a left inverse), then $ $ 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 $ $ consists of invertible elements, the closure of any non-trivial ideal is again an ideal which does not coincide with $ $. 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 $ $ (cf. Example 1) each closed ideal has the form $ $, where $ $ is a closed set in $ $. If $ $ 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 $ $.

An element $ $ 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 $ $ coincides with the intersection of all maximal right ideals; this intersection is called the radical of the algebra $ $ and is denoted by $ $. An element $ $ belongs to $ $ if and only if $ $ for any $ $. Algebras for which $ $ are said to be semi-simple. The algebras $ $ and the group algebras $ $ 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 $ $ is the function

$$ $$ defined on the set of all $ $ for which a (two-sided) inverse to $ $ exists. The domain of existence of the resolvent contains all points $ $ with $ $. The maximal domain of existence of the resolvent is an open set; the resolvent is continuous on this set and is even analytic, moreover $ $. In addition, Hilbert's identity

$$ $$ is valid. The complement of the domain of existence of the resolvent is called the spectrum of the element $ $ and is denoted by $ $. For each $ $ the set $ $ is non-empty, closed and bounded.

If $ $, then the sets $ $ and $ $ need not coincide, but

$$ $$ The number

$$ $$ is called the spectral radius of the element $ $; Gel'fand's formula

$$ $$ where the limit on the right-hand side always exists, is valid. If $ $, then $ $; the converse is true, generally speaking, only in commutative Banach algebras whose radical coincides with the set of generalized nilpotents, i.e. elements $ $ for which $ $. In any Banach algebra the relationships $ $, $ $ and $ $ are true. If $ $ is commutative, then $ $ and $ $ are valid.

Examples of non-commutative algebras in which generalized non-zero nilpotents are absent are known. However, if $ $ for any $ $, then the Banach algebra $ $ is commutative. The condition $ $ for all $ $ is also sufficient for an algebra $ $ with a unit to be commutative.

An algebra $ $ is said to be an algebra with involution if an operation $ $ is defined on $ $ that satisfies the conditions

$$ $$ for all $ $. The mapping $ $ is said to be an involution in $ $. A linear functional $ $ on an algebra $ $ with an involution is said to be positive if $ $ for any $ $. If the linear functional $ $ is positive, then

$$ $$ for all $ $. If the involution in $ $ is an isometry, i.e. if $ $ for all $ $, then

$$ $$ A Banach algebra $ $ with involution is said to be completely symmetric if $ $ for any $ $; $ $ is said to be a $ $-algebra (a completely-regular algebra) if $ $ for any $ $. Any $ $-algebra is completely symmetric. Examples of completely-symmetric algebras include the group algebras $ $ of commutative or compact groups. Examples of $ $-algebras include the algebras $ $ (the involution in $ $ 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 $ $-algebra is isometrically isomorphic (involution being preserved) with one of these algebras (the Gel'fand–Naimark theorem). In particular, any commutative $ $-algebra $ $ is isometrically isomorphic (involution being preserved) with one of the algebras $ $ (this theorem includes the Stone–Weierstrass theorem).

An element $ $ of a Banach algebra with involution is said to be Hermitian if $ $. For a Banach algebra with an involution to be a $ $-algebra it is necessary and sufficient that the condition $ $ be fulfilled for all Hermitian elements $ $. If, in a Banach algebra with an involution, $ $ (upper bound over all Hermitian elements), then this algebra is topologically $ $-isomorphic with a $ $-algebra. If, in an arbitrary Banach algebra, $ $ for all real $ $ for a certain fixed element $ $, then $ $ coincides with the spectral radius, i.e. $ $.

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

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