Inmathematics, especiallyfunctional analysis, aBanach algebra, named afterStefan Banach, is anassociative algebra${\displaystyle A}$ over thereal orcomplex numbers (or over anon-Archimedean completenormed field) that at the same time is also aBanach space, that is, anormed space that iscomplete in themetric induced by the norm. The norm is required to satisfy$${\displaystyle \Vert xy\Vert \le \Vert x\Vert \Vert y\Vert \text{ for all }x,y\in A.}$$

This ensures that the multiplication operation iscontinuous with respect to themetric topology.

A Banach algebra is calledunital if it has anidentity element for the multiplication whose norm is${\displaystyle 1,}$ andcommutative if its multiplication iscommutative.Any Banach algebra${\displaystyle A}$ (whether it is unital or not) can be embeddedisometrically into a unital Banach algebra${\displaystyle A_e}$ so as to form aclosedideal of${\displaystyle A_e}$. Often one assumesa priori that the algebra under consideration is unital because one can develop much of the theory by considering${\displaystyle A_e}$ and then applying the outcome in the original algebra. However, this is not the case all the time. For example, one cannot define all thetrigonometric functions in a Banach algebra without identity.

The theory of real Banach algebras can be very different from the theory of complex Banach algebras. For example, thespectrum of an element of a nontrivial complex Banach algebra can never be empty, whereas in a real Banach algebra it could be empty for some elements.

Banach algebras can also be defined over fields of${\displaystyle p}$-adic numbers. This is part of${\displaystyle p}$-adic analysis.

The prototypical example of a Banach algebra is${\displaystyle C_0(X)}$, the space of (complex-valued) continuous functions, defined on alocally compact Hausdorff space${\displaystyle X}$, thatvanish at infinity.${\displaystyle C_0(X)}$ is unital if and only if${\displaystyle X}$ iscompact. Thecomplex conjugation being aninvolution,${\displaystyle C_0(X)}$ is in fact aC*-algebra. More generally, every C*-algebra is a Banach algebra by definition.

• The set of real (or complex) numbers is a Banach algebra with norm given by theabsolute value.
• The set of all real or complex${\displaystyle n}$-by-${\displaystyle n}$matrices becomes aunital Banach algebra if we equip it with a sub-multiplicativematrix norm.
• Take the Banach space${\displaystyle {\mathbb{R}}^n}$ (or${\displaystyle {\mathbb{C}}^n}$) with norm${\displaystyle \Vert x\Vert =\underset{}{max}|x_i|}$ and define multiplication componentwise:${\displaystyle \left(x_1,\dots ,x_n\right)\left(y_1,\dots ,y_n\right)=\left(x_1{y}_1,\dots ,x_n{y}_n\right).}$
• Thequaternions form a 4-dimensional real Banach algebra, with the norm being given by the absolute value of quaternions.
• The algebra of all bounded real- or complex-valued functions defined on some set (with pointwise multiplication and thesupremum norm) is a unital Banach algebra.
• The algebra of all boundedcontinuous real- or complex-valued functions on somelocally compact space (again with pointwise operations and supremum norm) is a Banach algebra.
• The algebra of allcontinuouslinear operators on a Banach space${\displaystyle E}$ (with functional composition as multiplication and theoperator norm as norm) is a unital Banach algebra. The set of allcompact operators on${\displaystyle E}$ is a Banach algebra and closed ideal. It is without identity if${\displaystyle \dim E=\mathrm{\infty }.}$^[1]
• If${\displaystyle G}$ is alocally compactHausdorfftopological group and${\displaystyle \mu }$ is itsHaar measure, then the Banach space${\displaystyle L^1(G)}$ of all${\displaystyle \mu }$-integrable functions on${\displaystyle G}$ becomes a Banach algebra under theconvolution${\displaystyle xy(g)=\int x(h)y\left(h^{-1}g\right)d\mu (h)}$ for${\displaystyle x,y\in L^1(G).}$^[2]
• Uniform algebra: A Banach algebra that is a subalgebra of the complex algebra${\displaystyle C(X)}$ with the supremum norm and that contains the constants and separates the points of${\displaystyle X}$ (which must be a compact Hausdorff space).
• Natural Banach function algebra: A uniform algebra all of whose characters are evaluations at points of${\displaystyle X.}$
• C*-algebra: A Banach algebra that is a closed *-subalgebra of the algebra of bounded operators on someHilbert space.
• Measure algebra: A Banach algebra consisting of allRadon measures on somelocally compact group, where the product of two measures is given byconvolution of measures.^[2]
• The algebra of thequaternions${\displaystyle \mathbb{H}}$ is a real Banach algebra, but it is not a complex algebra (and hence not a complex Banach algebra) for the simple reason that the center of the quaternions is the real numbers, which cannot contain a copy of the complex numbers.
• Anaffinoid algebra is a certain kind of Banach algebra over a nonarchimedean field. Affinoid algebras are the basic building blocks inrigid analytic geometry.

Severalelementary functions that are defined viapower series may be defined in any unital Banach algebra; examples include theexponential function and thetrigonometric functions, and more generally anyentire function. (In particular, the exponential map can be used to defineabstract index groups.) The formula for thegeometric series remains valid in general unital Banach algebras. Thebinomial theorem also holds for two commuting elements of a Banach algebra.

The set ofinvertible elements in any unital Banach algebra is anopen set, and the inversion operation on this set is continuous (and hence is a homeomorphism), so that it forms atopological group under multiplication.^[3]

If a Banach algebra has unit${\displaystyle \mathbf{1},}$ then${\displaystyle \mathbf{1}}$ cannot be acommutator; that is,${\displaystyle xy-yx\ne \mathbf{1}}$ for any${\displaystyle x,y\in A.}$ This is because${\displaystyle xy}$ and${\displaystyle yx}$ have the samespectrum except possibly${\displaystyle 0.}$

The various algebras of functions given in the examples above have very different properties from standard examples of algebras such as the reals. For example:

• Every real Banach algebra that is adivision algebra is isomorphic to the reals, the complexes, or the quaternions. Hence, the only complex Banach algebra that is a division algebra is the complexes. (This is known as theGelfand–Mazur theorem.)
• Every unital real Banach algebra with nozero divisors, and in which everyprincipal ideal isclosed, is isomorphic to the reals, the complexes, or the quaternions.^[4]
• Every commutative real unitalNoetherian Banach algebra with no zero divisors is isomorphic to the real or complex numbers.
• Every commutative real unital Noetherian Banach algebra (possibly having zero divisors) is finite-dimensional.
• Permanently singular elements in Banach algebras aretopological divisors of zero, that is, considering extensions${\displaystyle B}$ of Banach algebras${\displaystyle A}$ some elements that are singular in the given algebra${\displaystyle A}$ have a multiplicative inverse element in a Banach algebra extension${\displaystyle B.}$ Topological divisors of zero in${\displaystyle A}$ are permanently singular in any Banach extension${\displaystyle B}$ of${\displaystyle A.}$

Unital Banach algebras over the complex field provide a general setting to develop spectral theory. Thespectrum of an element${\displaystyle x\in A,}$ denoted by${\displaystyle \sigma (x)}$, consists of all those complexscalars${\displaystyle \lambda }$ such that${\displaystyle x-\lambda \mathbf{1}}$ is not invertible in${\displaystyle A.}$ The spectrum of any element${\displaystyle x}$ is a closed subset of the closed disc in${\displaystyle \mathbb{C}}$ with radius${\displaystyle \Vert x\Vert }$ and center${\displaystyle 0,}$ and thus iscompact. Moreover, the spectrum${\displaystyle \sigma (x)}$ of an element${\displaystyle x}$ isnon-empty and satisfies thespectral radius formula:$${\displaystyle sup\{|\lambda |:\lambda \in \sigma (x)\}=\underset{n\to \mathrm{\infty }}{lim}\Vert x^n{\Vert }^{1/n}.}$$

Given${\displaystyle x\in A,}$ theholomorphic functional calculus allows to define${\displaystyle f(x)\in A}$ for any function${\displaystyle f}$holomorphic in a neighborhood of${\displaystyle \sigma (x).}$ Furthermore, the spectral mapping theorem holds:^[5]$${\displaystyle \sigma (f(x))=f(\sigma (x)).}$$

When the Banach algebra${\displaystyle A}$ is the algebra${\displaystyle L(X)}$ of bounded linear operators on a complex Banach space${\displaystyle X}$ (for example, the algebra of square matrices), the notion of the spectrum in${\displaystyle A}$ coincides with the usual one inoperator theory. For${\displaystyle f\in C(X)}$ (with a compact Hausdorff space${\displaystyle X}$), one sees that:$${\displaystyle \sigma (f)=\{f(t):t\in X\}.}$$

The norm of a normal element${\displaystyle x}$ of a C*-algebra coincides with its spectral radius. This generalizes an analogous fact for normal operators.

Let${\displaystyle A}$ be a complex unital Banach algebra in which every non-zero element${\displaystyle x}$ is invertible (a division algebra). For every${\displaystyle a\in A,}$ there is${\displaystyle \lambda \in \mathbb{C}}$ such that${\displaystyle a-\lambda \mathbf{1}}$ is not invertible (because the spectrum of${\displaystyle a}$ is not empty) hence${\displaystyle a=\lambda \mathbf{1}:}$ this algebra${\displaystyle A}$ is naturally isomorphic to${\displaystyle \mathbb{C}}$ (the complex case of the Gelfand–Mazur theorem).

Let${\displaystyle A}$ be a unitalcommutative Banach algebra over${\displaystyle \mathbb{C}.}$ Since${\displaystyle A}$ is then a commutative ring with unit, every non-invertible element of${\displaystyle A}$ belongs to somemaximal ideal of${\displaystyle A.}$ Since a maximal ideal${\displaystyle \mathfrak{m}}$ in${\displaystyle A}$ is closed,${\displaystyle A/\mathfrak{m}}$ is a Banach algebra that is a field, and it follows from theGelfand–Mazur theorem that there is a bijection between the set of all maximal ideals of${\displaystyle A}$ and the set${\displaystyle \mathrm{\Delta }(A)}$ of all nonzero homomorphisms from${\displaystyle A}$ to${\displaystyle \mathbb{C}.}$ The set${\displaystyle \mathrm{\Delta }(A)}$ is called thestructure space orcharacter space of${\displaystyle A}$.

Acharacter${\displaystyle \chi \in \mathrm{\Delta }(A)}$ is a linear functional on${\displaystyle A}$ that is at the same time multiplicative,${\displaystyle \chi (ab)=\chi (a)\chi (b),}$ and satisfies${\displaystyle \chi (\mathbf{1})=1.}$ Every character is automatically continuous from${\displaystyle A}$ to${\displaystyle \mathbb{C},}$ since the kernel of a character is a maximal ideal, which is closed. Moreover, the norm (that is, operator norm) of a character is one. Equipped with the topology of pointwise convergence on${\displaystyle A}$ (that is, the topology induced by theweak-* topology of${\displaystyle A^{\ast }}$), the character space,${\displaystyle \mathrm{\Delta }(A),}$ is a compact Hausdorff space.

For any${\displaystyle x\in A,}$$${\displaystyle \sigma (x)=\sigma (\hat{x})}$$where${\displaystyle \hat{x}}$ is theGelfand representation of${\displaystyle x}$ defined as follows:${\displaystyle \hat{x}}$ is the continuous function from${\displaystyle \mathrm{\Delta }(A)}$ to${\displaystyle \mathbb{C}}$ given by${\displaystyle \hat{x}(\chi )=\chi (x).}$ The spectrum of${\displaystyle \hat{x},}$ in the formula above, is the spectrum as element of the algebra${\displaystyle C(\mathrm{\Delta }(A))}$ of complex continuous functions on the compact space${\displaystyle \mathrm{\Delta }(A).}$ Explicitly,$${\displaystyle \sigma (\hat{x})=\{\chi (x):\chi \in \mathrm{\Delta }(A)\}.}$$

As an algebra, a unital commutative Banach algebra issemisimple (that is, itsJacobson radical is zero) if and only if its Gelfand representation has trivial kernel. An important example of such an algebra is a commutative C*-algebra. In fact, when${\displaystyle A}$ is a commutative unital C*-algebra, the Gelfand representation is then an isometric *-isomorphism between${\displaystyle A}$ and${\displaystyle C(\mathrm{\Delta }(A)).}$^[a]

A Banach *-algebra${\displaystyle A}$ is a Banach algebra over the field ofcomplex numbers, together with a map${\displaystyle {}^{\ast }:A\to A}$ that has the following properties:

1. ${\displaystyle {\left(x^{\ast }\right)}^{\ast }=x}$ for all${\displaystyle x\in A}$ (so the map is aninvolution).
2. ${\displaystyle (x+y{)}^{\ast }=x^{\ast }+y^{\ast }}$ for all${\displaystyle x,y\in A.}$
3. ${\displaystyle (\lambda x{)}^{\ast }=\overline{\lambda }{x}^{\ast }}$ for every${\displaystyle \lambda \in \mathbb{C}}$ and every${\displaystyle x\in A;}$ here,${\displaystyle \overline{\lambda }}$ denotes thecomplex conjugate of${\displaystyle \lambda .}$
4. ${\displaystyle (xy{)}^{\ast }=y^{\ast }{x}^{\ast }}$ for all${\displaystyle x,y\in A.}$

In other words, a Banach *-algebra is a Banach algebra over${\displaystyle \mathbb{C}}$ that is also a*-algebra.

In most natural examples, one also has that the involution isisometric, that is,$${\displaystyle \Vert x^{\ast }\Vert =\Vert x\Vert \text{ for all }x\in A.}$$Some authors include this isometric property in the definition of a Banach *-algebra.

A Banach *-algebra satisfying${\displaystyle \Vert x^{\ast }x\Vert =\Vert x^{\ast }\Vert \Vert x\Vert }$ is aC*-algebra.

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