# Banach algebra

2010 Mathematics Subject Classification: Primary: 46HXX Secondary: 46JXX [MSN][ZBL]


### 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 $\abs{z} < 1$.


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.

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

#### References

 [Bo] N. Bourbaki, "Elements of mathematics. Spectral theories", Addison-Wesley (1977) (Translated from French) MR0583191 Zbl 1106.46004 [DuSc] N. Dunford, J.T. Schwartz, "Linear operators", 1–3, Interscience (1958–1971) MR0117523 Zbl 0084.10402 [Ga] T.W. Gamelin, "Uniform algebras", Prentice-Hall (1969) MR0410387 Zbl 0213.40401 [Ge] I.M. Gel'fand, "Normierte Ringe" Mat. Sb., 9 (51) : 1 (1941) pp. 3–24 [Gl] A.M. Gleason, "Function algebras", Proc. Sem. on analytic functions, 2 (1958) pp. 213–226 Zbl 0095.10103 [Go] 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 MR0208412 Zbl 0172.17901 [GuRo] R.C. Gunning, H. Rossi, "Analytic functions of several complex variables", Prentice-Hall (1965) MR0180696 Zbl 0141.08601 [HiPh] E. Hille, R.S. Phillips, "Functional analysis and semi-groups", Amer. Math. Soc. (1957) MR0089373 Zbl 0392.46001 Zbl 0033.06501 [Ho] K. Hoffman, "Banach spaces of analytic functions", Prentice-Hall (1962) MR0133008 Zbl 0117.34001 [Ka] I. Kaplansky, "Functional analysis", Surveys in applied mathematics, 4. Some aspects of analysis and probability, Wiley (1958) MR0101475 Zbl 0087.31102 [KaRi] R.V. Kadison, J.R. Ringrose, "Fundamentals of the theory of operator algebras", 1, Acad. Press (1983) MR0719020 Zbl 0518.46046 [Lo] L.H. Loomis, "An introduction to abstract harmonic analysis", v. Nostrand (1953) MR0054173 Zbl 0052.11701 [Na] M.A. Naimark, "Normed rings", Reidel (1984) (Translated from Russian) MR1292007 MR0355601 MR0355602 MR0205093 MR0110956 MR0090786 MR0026763 Zbl 0218.46042 Zbl 0137.31703 Zbl 0089.10102 Zbl 0073.08902 [Ph] R.R. Phelps, "Lectures on Choquet's theorem", v. Nostrand (1966) MR0193470 Zbl 0135.36203 [Ri] C.E. Rickart, "General theory of Banach algebras", v. Nostrand (1960) MR0115101 Zbl 0095.09702 [Ro] H.L. Royden, "Function algebras" Bull. Amer. Math. Soc., 69 : 3 (1963) pp. 281–298 MR0149327 Zbl 0111.11802 [Ze] W. Zelazko, "Banach algebras", Elsevier (1973) (Translated from Polish) MR0448079 Zbl 0248.46037
How to Cite This Entry:
Banach algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Banach_algebra&oldid=35168
This article was adapted from an original article by E.A. Gorin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article