Banach algebra

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.

References

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