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{{Short description|Particular kind of algebraic structure}}
In [[mathematics]], especially [[functional analysis]], a '''Banach algebra''', named after [[Stefan Banach]], is an [[associative algebra]] <math>A</math> over the [[real number|real]] or [[complex number|complex]] numbers (or over a [[nonarchimedean field|non-Archimedean]] complete [[Norm (mathematics)|normed field]]) that at the same time is also a [[Banach space]], that is, a [[normed space]] that is [[complete metric space|complete]] in the [[metric (mathematics)|metric]] induced by the norm. The norm is required to satisfy
<math display=block>\|x \, y\| \ \leq  \|x\| \, \|y\| \quad \text{ for all } x, y \in A.</math>

This ensures that the multiplication operation is [[continuous function (topology)|continuous]] with respect to the metric topology.

A Banach algebra is called ''unital'' if it has an [[identity element]] for the multiplication whose norm is <math>1,</math> and ''commutative'' if its multiplication is [[commutative]].
Any Banach algebra <math>A</math> (whether it is unital or not) can be embedded [[isometry|isometrically]] into a unital Banach algebra <math>A_e</math> so as to form a [[closed set|closed]] [[ideal (algebra)|ideal]] of <math>A_e</math>. Often one assumes ''a priori'' that the algebra under consideration is unital because one can develop much of the theory by considering <math>A_e</math> and then applying the outcome in the original algebra. However, this is not the case all the time. For example, one cannot define all the [[trigonometric function]]s 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, the [[Spectrum (functional analysis)|spectrum]] 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 [[p-adic number|<math>p</math>-adic number]]s. This is part of [[p-adic analysis|<math>p</math>-adic analysis]].

==Examples==

The prototypical example of a Banach algebra is <math>C_0(X)</math>, the space of (complex-valued) continuous functions, defined on a [[locally compact Hausdorff space]] <math>X</math>, that [[vanish at infinity]]. <math>C_0(X)</math> is unital if and only if <math>X</math> is [[compactness|compact]]. The [[complex conjugation]] being an [[involution (mathematics)|involution]], <math>C_0(X)</math> is in fact a [[C*-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 the [[absolute value]].
* The set of all real or complex <math>n</math>-by-<math>n</math> [[matrix (mathematics)|matrices]] becomes a [[unital algebra|unital]] Banach algebra if we equip it with a sub-multiplicative [[matrix norm]].
* Take the Banach space <math>\R^n</math> (or <math>\Complex^n</math>) with norm <math>\|x\| = \max_{} |x_i|</math> and define multiplication componentwise: <math>\left(x_1, \ldots, x_n\right) \left(y_1, \ldots, y_n\right) = \left(x_1 y_1, \ldots, x_n y_n\right).</math>
* The [[quaternion]]s 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 the [[supremum]] norm) is a unital Banach algebra.
* The algebra of all bounded [[continuous function (topology)|continuous]] real- or complex-valued functions on some [[locally compact space]] (again with pointwise operations and supremum norm) is a Banach algebra.
* The algebra of all [[continuous function (topology)|continuous]] [[linear transformation|linear]] operators on a Banach space <math>E</math> (with functional composition as multiplication and the [[operator norm]] as norm) is a unital Banach algebra. The set of all [[compact operator]]s on <math>E</math> is a Banach algebra and closed ideal. It is without identity if <math>\dim E = \infty.</math><ref>{{harvnb|Conway|1990|loc=Example VII.1.8.}}</ref>
* If <math>G</math> is a [[locally compact]] [[Hausdorff space|Hausdorff]] [[topological group]] and <math>\mu</math> is its [[Haar measure]], then the Banach space <math>L^1(G)</math> of all <math>\mu</math>-integrable functions on <math>G</math> becomes a Banach algebra under the [[convolution]] <math>x y(g) = \int x(h) y\left(h^{-1} g\right) d \mu(h)</math> for <math>x, y \in L^1(G).</math><ref name="harvnb conway 1990 example VII.1.9.">{{harvnb|Conway|1990|loc=Example VII.1.9.}}</ref>
* [[Uniform algebra]]: A Banach algebra that is a subalgebra of the complex algebra <math>C(X)</math> with the supremum norm and that contains the constants and separates the points of <math>X</math> (which must be a compact Hausdorff space).
* [[Uniform algebra|Natural Banach function algebra]]: A uniform algebra all of whose characters are evaluations at points of <math>X.</math>
* [[C*-algebra]]: A Banach algebra that is a closed *-subalgebra of the algebra of bounded operators on some [[Hilbert space]].
* [[Measure algebra]]: A Banach algebra consisting of all [[Radon measure]]s on some [[locally compact group]], where the product of two measures is given by [[Convolution#Measures|convolution of measures]].<ref name="harvnb conway 1990 example VII.1.9." />
* The algebra of the [[quaternion]]s <math>\H</math> 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.
* An [[affinoid algebra]] is a certain kind of Banach algebra over a nonarchimedean field. Affinoid algebras are the basic building blocks in [[Rigid analytic space|rigid analytic geometry]].

==Properties==

Several [[list of functions|elementary functions]] that are defined via [[power series]] may be defined in any unital Banach algebra; examples include the [[exponential function]] and the [[trigonometric functions]], and more generally any [[entire function]]. (In particular, the exponential map can be used to define [[abstract index group]]s.) The formula for the [[geometric series]] remains valid in general unital Banach algebras. The [[binomial theorem]] also holds for two commuting elements of a Banach algebra.

The set of [[invertible element]]s in any unital Banach algebra is an [[open set]], and the inversion operation on this set is continuous (and hence is a homeomorphism), so that it forms a [[topological group]] under multiplication.<ref>{{harvnb|Conway|1990|loc=Theorem VII.2.2.}}</ref>

If a Banach algebra has unit <math>\mathbf{1},</math> then <math>\mathbf{1}</math> cannot be a [[commutator (ring theory)|commutator]]; that is, <math>xy - yx \neq \mathbf{1}</math>&thinsp; for any <math>x, y \in A.</math> This is because <math>x y</math> and <math>y x</math> have the same [[spectrum (functional analysis)|spectrum]] except possibly <math>0.</math>

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 a [[division 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 the [[Gelfand–Mazur theorem]].)
* Every unital real Banach algebra with no [[zero divisor]]s, and in which every [[principal ideal]] is [[closed set|closed]], is isomorphic to the reals, the complexes, or the quaternions.<ref>{{Cite journal|last1=García|first1=Miguel Cabrera|last2=Palacios|first2=Angel Rodríguez|date=1995|title=A New Simple Proof of the Gelfand-Mazur-Kaplansky Theorem|url=https://www.jstor.org/stable/2160559|journal=Proceedings of the American Mathematical Society|volume=123|issue=9|pages=2663–2666|doi=10.2307/2160559|jstor=2160559|issn=0002-9939}}</ref>
* Every commutative real unital [[Noetherian ring|Noetherian]] 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 are [[topological divisor of zero|topological divisors of zero]], that is, considering extensions <math>B</math> of Banach algebras <math>A</math> some elements that are singular in the given algebra <math>A</math> have a multiplicative inverse element in a Banach algebra extension <math>B.</math>  Topological divisors of zero in <math>A</math> are permanently singular in any Banach extension <math>B</math> of <math>A.</math>

==Spectral theory==
{{Main|Spectral theory}}

Unital Banach algebras over the complex field provide a general setting to develop spectral theory. The ''spectrum'' of an element <math>x \in A,</math> denoted by <math>\sigma(x)</math>, consists of all those complex [[scalar (mathematics)|scalar]]s <math>\lambda</math> such that <math>x - \lambda \mathbf{1}</math> is not invertible in <math>A.</math>  The spectrum of any element <math>x</math> is a closed subset of the closed disc in <math>\Complex</math> with radius <math>\|x\|</math> and center <math>0,</math> and thus is  [[Compact space|compact]]. Moreover, the spectrum <math>\sigma(x)</math> of an element <math>x</math> is [[non-empty]] and satisfies the [[spectral radius]] formula:
<math display=block>\sup \{|\lambda| : \lambda \in \sigma(x)\} = \lim_{n \to \infty} \|x^n\|^{1/n}.</math>

Given <math>x \in A,</math> the [[holomorphic functional calculus]] allows to define <math>f(x) \in A</math> for any function <math>f</math> [[holomorphic function|holomorphic]] in a neighborhood of <math>\sigma(x).</math>  Furthermore, the spectral mapping theorem holds:<ref>{{harvnb|Takesaki|1979|loc=Proposition 2.8.}}</ref>
<math display=block>\sigma(f(x)) = f(\sigma(x)).</math>

When the Banach algebra <math>A</math> is the algebra <math>L(X)</math> of bounded linear operators on a complex Banach space <math>X</math> (for example, the algebra of square matrices), the notion of the spectrum in <math>A</math> coincides with the usual one in [[operator theory]]. For <math>f \in C(X)</math> (with a compact Hausdorff space <math>X</math>), one sees that:
<math display=block>\sigma(f) = \{f(t) : t \in X\}.</math>

The norm of a normal element <math>x</math> of a C*-algebra coincides with its spectral radius.  This generalizes an analogous fact for normal operators.

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

==Ideals and characters==

Let <math>A</math> be a unital ''commutative'' Banach algebra over <math>\Complex.</math> Since <math>A</math> is then a commutative ring with unit, every non-invertible element of <math>A</math> belongs to some [[maximal ideal]] of <math>A.</math> Since a maximal ideal <math>\mathfrak m</math> in <math>A</math> is closed, <math>A / \mathfrak m</math> is a Banach algebra that is a field, and it follows from the Gelfand–Mazur theorem that there is a bijection between the set of all maximal ideals of <math>A</math> and the set <math>\Delta(A)</math> of all nonzero homomorphisms from <math>A</math> to <math>\Complex.</math> The set <math>\Delta(A)</math> is called the "[[structure space]]" or "character space" of <math>A,</math> and its members "characters".

A character <math>\chi</math> is a linear functional on <math>A</math> that is at the same time multiplicative, <math>\chi(a b) = \chi(a) \chi(b),</math> and satisfies <math>\chi(\mathbf{1}) = 1.</math>  Every character is automatically continuous from <math>A</math> to <math>\Complex,</math> 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 <math>A</math> (that is, the topology induced by the weak-* topology of <math>A^*</math>), the character space, <math>\Delta(A),</math> is a Hausdorff compact space.

For any <math>x \in A,</math>
<math display=block>\sigma(x) = \sigma(\hat x)</math>
where <math>\hat x</math> is the [[Gelfand representation]] of <math>x</math> defined as follows: <math>\hat x</math> is the continuous function from <math>\Delta(A)</math> to <math>\Complex</math> given by <math>\hat x(\chi) = \chi(x).</math>  The spectrum of <math>\hat x,</math> in the formula above, is the spectrum as element of the algebra <math>C(\Delta(A))</math> of complex continuous functions on the compact space <math>\Delta(A).</math>  Explicitly,
<math display=block>\sigma(\hat x) = \{\chi(x) : \chi \in \Delta(A)\}.</math>

As an algebra, a unital commutative Banach algebra is [[semisimple algebra|semisimple]] (that is, its [[Jacobson 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 <math>A</math> is a commutative unital C*-algebra, the Gelfand representation is then an isometric *-isomorphism between <math>A</math> and <math>C(\Delta(A)).</math>{{efn-la|Proof: Since every element of a commutative C*-algebra is normal, the Gelfand representation is isometric; in particular, it is injective and its image is closed. But the image of the Gelfand representation is dense by the [[Stone–Weierstrass theorem]].}}

==Banach *-algebras==

A Banach *-algebra <math>A</math> is a Banach algebra over the field of [[complex number]]s, together with a map <math>{}^* : A \to A</math> that has the following properties:
# <math>\left(x^*\right)^* = x</math> for all <math>x \in A</math> (so the map is an [[Involution (mathematics)|involution]]).
# <math>(x + y)^* = x^* + y^*</math> for all <math>x, y \in A.</math>
# <math>(\lambda x)^* = \bar{\lambda}x^*</math> for every <math>\lambda \in \Complex</math> and every <math>x \in A;</math> here, <math>\bar{\lambda}</math> denotes the [[complex conjugate]] of <math>\lambda.</math>
# <math>(x y)^* = y^* x^*</math> for all <math>x, y \in A.</math>

In other words, a Banach *-algebra is a Banach algebra over <math>\Complex</math> that is also a [[*-algebra]].

In most natural examples, one also has that the involution is [[isometry|isometric]], that is, 
<math display=block>\|x^*\| = \|x\| \quad \text{ for all } x \in A.</math>
Some authors include this isometric property in the definition of a Banach *-algebra.

A Banach *-algebra satisfying <math>\|x^* x\| = \|x^*\| \|x\|</math> is a [[C*-algebra]].

==See also==

* {{annotated link|Approximate identity}}
* {{annotated link|Kaplansky's conjecture}}
* {{annotated link|Operator algebra}}
* {{annotated link|Shilov boundary}}

==Notes==
{{notelist-la}}

==References==

{{reflist}}
{{Refbegin}}
* {{cite book|first=B|last=Bollobás|author-link=Béla Bollobás|title=Linear Analysis|publisher=Cambridge University Press|year=1990|isbn=0-521-38729-9|url-access=registration|url=https://archive.org/details/linearanalysisin0000boll}}
* {{cite book|first1=F. F.|last1=Bonsall|author-link1=Frank Bonsall|first2=J.|last2=Duncan|title=Complete Normed Algebras|publisher=Springer-Verlag|location=New York|year=1973|isbn=0-387-06386-2}}
* {{cite book|last=Conway|first=J. B.|author-link=John B. Conway|title=A Course in Functional Analysis|year=1990|series=Graduate Texts in Mathematics|volume=96|publisher=[[Springer Verlag]]|isbn=0-387-97245-5}}
* {{cite book|first1=H. G.|last1=Dales|first2=P.|last2=Aeina|first3=J|last3=Eschmeier|first4=K.|last4=Laursen|first5=G. A.|last5=Willis|authorlink5=George A. Willis|title=Introduction to Banach Algebras, Operators and Harmonic Analysis|publisher=Cambridge University Press|year=2003|isbn=0-521-53584-0|doi=10.1017/CBO9780511615429}}
* {{cite book|first=R. D.|last=Mosak|title=Banach algebras|series=Chicago Lectures in Mathematics|year=1975|publisher=University of Chicago Press)|isbn=0-226-54203-3}}
* {{cite book|last=Takesaki|first=M.|author-link=Masamichi Takesaki|title=Theory of Operator Algebras I|year=1979|edition=1st|series=Encyclopaedia of Mathematical Sciences|volume=124|publisher=Springer-Verlag|location=Berlin Heidelberg|issn=0938-0396|isbn=978-3-540-42248-8}}
{{Refend}}

{{Spectral theory}}
{{Functional analysis}}
{{Banach spaces}}
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