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{{Short description|Particular kind of algebraic structure}}
{{Short description|Particular kind of algebraic structure}}
In [[mathematics]], especially [[functional analysis]], a '''Banach algebra''', named after [[Stefan Banach]], is an [[associative algebra]] ''A'' over the [[real number|real]] or [[complex number|complex]] numbers (or over a [[nonarchimedean field|non-Archimedean]] complete [[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
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 [[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>

:<math> \forall x, y \in A, \|x \, y\| \ \leq \|x \| \, \| y\|. </math>


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


A Banach algebra is called ''unital'' if it has an [[identity element]] for the multiplication whose norm is 1, and ''commutative'' if its multiplication is [[commutative]].
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 has an [[identity element]] 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: for 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.
Any Banach algebra <math>A</math> (whether it has an [[identity element]] 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: for 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.
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|''p''-adic number]]s. This is part of [[p-adic analysis|''p''-adic analysis]].
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 ==
==Examples==

The prototypical example of a Banach algebra is <math>C_0(X)</math>, the space of (complex-valued) continuous functions on a [[locally compact]] ([[Hausdorff space|Hausdorff]]) space that [[vanish at infinity]]. <math>C_0(X)</math> is unital if and only if ''X'' 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 prototypical example of a Banach algebra is <math>C_0(X)</math>, the space of (complex-valued) continuous functions on a [[locally compact]] ([[Hausdorff space|Hausdorff]]) space 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 real (or complex) numbers is a Banach algebra with norm given by the [[absolute value]].
* The set of all real or complex ''n''-by-''n'' [[matrix (mathematics)|matrices]] becomes a [[unital algebra|unital]] Banach algebra if we equip it with a sub-multiplicative [[matrix norm]].
* 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 '''R'''<sup>''n''</sup> (or '''C'''<sup>''n''</sup>) with norm ||''x''|| = max |''x''<sub>''i''</sub>| and define multiplication componentwise: (''x''<sub>1</sub>,...,''x''<sub>''n''</sub>)(''y''<sub>1</sub>,...,''y''<sub>''n''</sub>) = (''x''<sub>1</sub>''y''<sub>1</sub>,...,''x''<sub>''n''</sub>''y''<sub>''n''</sub>).
* 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 [[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 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 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 ''E'' (with functional composition as multiplication and the [[operator norm]] as norm) is a unital Banach algebra. The set of all [[compact operator]]s on ''E'' is a Banach algebra and closed ideal. It is without identity if {{math|dim ''E'' {{=}} ∞}}.<ref>{{harvnb|Conway|1990|loc=Example VII.1.8.}}</ref>
* 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 ''G'' is a [[locally compact]] [[Hausdorff space|Hausdorff]] [[topological group]] and ''μ'' is its [[Haar measure]], then the Banach space L<sup>1</sup>(''G'') of all ''μ''-integrable functions on ''G'' becomes a Banach algebra under the [[convolution]] ''xy''(''g'') = ''x''(''h'') ''y''(''h''<sup>−1</sup>''g'') d''μ''(''h'') for ''x'', ''y'' in L<sup>1</sup>(''G'').<ref name="harvnb conway 1990 example VII.1.9.">{{harvnb|Conway|1990|loc=Example VII.1.9.}}</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 ''C''(''X'') with the supremum norm and that contains the constants and separates the points of ''X'' (which must be a compact Hausdorff space).
* [[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 ''X''.
* [[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]].
* [[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." />
* [[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." />
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* 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]].
* 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 ==
==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.
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>
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 '''1''', then '''1''' cannot be a [[commutator (ring theory)|commutator]]; i.e., <math>xy - yx \ne \mathbf{1}</math>&thinsp; for any ''x'', ''y''&nbsp;∈&nbsp;''A''. This is because ''xy'' and ''yx'' have the same [[spectrum (functional analysis)|spectrum]] except possibly 0.
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:
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:
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* 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 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.
* 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]], ''i.e.'', considering extensions ''B'' of Banach algebras ''A'' some elements that are singular in the given algebra ''A'' have a multiplicative inverse element in a Banach algebra extension ''B''. Topological divisors of zero in ''A'' are permanently singular in any Banach extension ''B'' of&nbsp;''A''.
* 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 ==
==Spectral theory==
{{Main|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 ''x''&nbsp;∈&nbsp;''A'', denoted by <math>\sigma(x)</math>, consists of all those complex [[scalar (mathematics)|scalar]]s ''λ'' such that ''x''&nbsp;&minus;&nbsp;''λ'''''1''' is not invertible in ''A''. The spectrum of any element ''x'' is a closed subset of the closed disc in '''C''' with radius ||''x''|| and center 0, and thus is [[Compact space|compact]]. Moreover, the spectrum <math>\sigma(x)</math> of an element ''x'' is [[non-empty]] and satisfies the [[spectral radius]] formula:
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>\sup \{ |\lambda| : \lambda \in \sigma(x) \} = \lim_{n \to \infty} \|x^n\|^{1/n}.</math>
<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:
Given ''x''&nbsp;&isin;&nbsp;''A'', the [[holomorphic functional calculus]] allows to define ''ƒ''(''x'')&nbsp;∈ ''A'' for any function ''ƒ'' [[holomorphic function|holomorphic]] in a neighborhood of <math>\sigma(x).</math> Furthermore, the spectral mapping theorem holds:
<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.
:<math>\sigma(f(x)) = f(\sigma(x)).</math><ref>{{harvnb|Takesaki|1979|loc=Proposition 2.8.}}</ref>


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
When the Banach algebra ''A'' is the algebra L(''X'') of bounded linear operators on a complex Banach space ''X''&thinsp; (e.g., the algebra of square matrices), the notion of the spectrum in ''A'' coincides with the usual one in [[operator theory]]. For ''ƒ''&nbsp;&isin; ''C''(''X'') (with a compact Hausdorff space&nbsp;''X''), one sees 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==
:<math>\sigma(f) = \{ f(t) : t \in X \}.</math>


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


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.
Let ''A''&thinsp; be a complex unital Banach algebra in which every non-zero element ''x'' is invertible (a division algebra). For every ''a''&nbsp;&isin; ''A'', there is ''λ''&nbsp;&isin; '''C''' such that
''a''&nbsp;&minus;&nbsp;''λ'''''1''' is not invertible (because the spectrum of ''a'' is not empty) hence ''a''&nbsp;=&nbsp;''λ'''''1'''&nbsp;: this algebra ''A'' is naturally isomorphic to '''C''' (the complex case of the Gelfand–Mazur theorem).


For any <math>x \in A,</math>
== Ideals and characters ==
<math display=block>\sigma(x) = \sigma(\hat x)</math>
Let ''A''&thinsp; be a unital ''commutative'' Banach algebra over '''C'''. Since ''A'' is then a commutative ring with unit, every non-invertible element of ''A'' belongs to some [[maximal ideal]] of ''A''. Since a maximal ideal <math>\mathfrak m</math> in ''A'' 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 ''A'' and the set Δ(''A'') of all nonzero homomorphisms from ''A''&thinsp; to '''C'''. The set Δ(''A'') is called the "[[structure space]]" or "character space" of ''A'', and its members "characters".
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]].}}
A character ''χ'' is a linear functional on ''A'' that is at the same time multiplicative, ''χ''(''ab'') = ''χ''(''a'') ''χ''(''b''), and satisfies ''χ''('''1''') = 1. Every character is automatically continuous from ''A''&thinsp; to '''C''', since the kernel of a character is a maximal ideal, which is closed. Moreover, the norm (''i.e.'', operator norm) of a character is one. Equipped with the topology of pointwise convergence on ''A'' (''i.e.'', the topology induced by the weak-* topology of&nbsp;''A''<sup></sup>), the character space, Δ(''A''), is a Hausdorff compact space.


==Banach *-algebras==
For any ''x'' ∈ ''A'',


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>\sigma(x) = \sigma(\hat x)</math>
# <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]].
where <math>\hat x</math> is the [[Gelfand representation]] of ''x'' defined as follows: <math>\hat x</math> is the continuous function from Δ(''A'') to '''C''' given by <math>\hat x(\chi) = \chi(x).</math>&thinsp; The spectrum of <math>\hat x,</math> in the formula above, is the spectrum as element of the algebra ''C''(Δ(''A'')) of complex continuous functions on the compact space Δ(''A''). Explicitly,

:<math>\sigma(\hat x) = \{ \chi(x) : \chi \in \Delta(A) \}</math>.

As an algebra, a unital commutative Banach algebra is [[semisimple algebra|semisimple]] (i.e., 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 ''A'' is a commutative unital C*-algebra, the Gelfand representation is then an isometric *-isomorphism between ''A'' and ''C''(Δ(''A'')) .{{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 ''A'' is a Banach algebra over the field of [[complex number]]s, together with a map * : ''A'' ''A'' that has the following properties:
# (''x''*)* = ''x'' for all ''x'' in ''A'' (so the map is an [[Involution (mathematics)|involution]]).
# (''x'' + ''y'')* = ''x''* + ''y''* for all ''x'', ''y'' in ''A''.
# <math>(\lambda x)^* = \bar{\lambda}x^*</math> for every ''λ'' in '''C''' and every ''x'' in ''A''; here, <math>\bar{\lambda}</math> denotes the complex conjugate of ''λ''.
# (''xy'')* = ''y''* ''x''* for all ''x'', ''y'' in ''A''.

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


In most natural examples, one also has that the involution is [[isometry|isometric]], that is,
In most natural examples, one also has that the involution is [[isometry|isometric]], that is,
: ||''x''*|| = ||''x''|| for all ''x'' in ''A''.
<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.
Some authors include this isometric property in the definition of a Banach *-algebra.


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


== See also ==
==See also==

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


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


== References ==
==References==

{{Reflist}}
{{reflist}}
{{Refbegin}}
{{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|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}}
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{{Refend}}
{{Refend}}


{{Functional Analysis}}
{{Spectral theory}}
{{Functional analysis}}
{{SpectralTheory}}
{{Banach spaces}}
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Revision as of 01:26, 19 August 2022

In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach space, that is, a normed space that is complete in the metric induced by the norm. The norm is required to satisfy

This ensures that the multiplication operation is continuous.

A Banach algebra is called unital if it has an identity element for the multiplication whose norm is and commutative if its multiplication is commutative. Any Banach algebra (whether it has an identity element or not) can be embedded isometrically into a unital Banach algebra so as to form a closed ideal of . Often one assumes a priori that the algebra under consideration is unital: for one can develop much of the theory by considering 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 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, the 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 -adic numbers. This is part of -adic analysis.

Examples

The prototypical example of a Banach algebra is , the space of (complex-valued) continuous functions on a locally compact (Hausdorff) space that vanish at infinity. is unital if and only if is compact. The complex conjugation being an involution, 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 -by- matrices becomes a unital Banach algebra if we equip it with a sub-multiplicative matrix norm.
  • Take the Banach space (or ) with norm and define multiplication componentwise:
  • The quaternions 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 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 linear operators on a Banach space (with functional composition as multiplication and the operator norm as norm) is a unital Banach algebra. The set of all compact operators on is a Banach algebra and closed ideal. It is without identity if [1]
  • If is a locally compact Hausdorff topological group and is its Haar measure, then the Banach space of all -integrable functions on becomes a Banach algebra under the convolution for [2]
  • Uniform algebra: A Banach algebra that is a subalgebra of the complex algebra with the supremum norm and that contains the constants and separates the points of (which must be a compact Hausdorff space).
  • Natural Banach function algebra: A uniform algebra all of whose characters are evaluations at points of
  • 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 measures on some locally compact group, where the product of two measures is given by convolution of measures.[2]
  • The algebra of the quaternions 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 geometry.

Properties

Several 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 groups.) 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 elements 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.[3]

If a Banach algebra has unit then cannot be a commutator; that is,   for any This is because and have the same spectrum except possibly

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 divisors, and in which every principal ideal is closed, is isomorphic to the reals, the complexes, or the quaternions.[4]
  • Every commutative real unital 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 divisors of zero, that is, considering extensions of Banach algebras some elements that are singular in the given algebra have a multiplicative inverse element in a Banach algebra extension Topological divisors of zero in are permanently singular in any Banach extension of

Spectral theory

Unital Banach algebras over the complex field provide a general setting to develop spectral theory. The spectrum of an element denoted by , consists of all those complex scalars such that is not invertible in The spectrum of any element is a closed subset of the closed disc in with radius and center and thus is compact. Moreover, the spectrum of an element is non-empty and satisfies the spectral radius formula:

Given the holomorphic functional calculus allows to define for any function holomorphic in a neighborhood of Furthermore, the spectral mapping theorem holds:[5]

When the Banach algebra is the algebra of bounded linear operators on a complex Banach space (for example, the algebra of square matrices), the notion of the spectrum in coincides with the usual one in operator theory. For (with a compact Hausdorff space ), one sees that:

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

Let be a complex unital Banach algebra in which every non-zero element is invertible (a division algebra). For every there is such that is not invertible (because the spectrum of is not empty) hence this algebra is naturally isomorphic to (the complex case of the Gelfand–Mazur theorem).

Ideals and characters

Let be a unital commutative Banach algebra over Since is then a commutative ring with unit, every non-invertible element of belongs to some maximal ideal of Since a maximal ideal in is closed, 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 and the set of all nonzero homomorphisms from to The set is called the "structure space" or "character space" of and its members "characters".

A character is a linear functional on that is at the same time multiplicative, and satisfies Every character is automatically continuous from to 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 (that is, the topology induced by the weak-* topology of ), the character space, is a Hausdorff compact space.

For any where is the Gelfand representation of defined as follows: is the continuous function from to given by The spectrum of in the formula above, is the spectrum as element of the algebra of complex continuous functions on the compact space Explicitly,

As an algebra, a unital commutative Banach algebra is 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 is a commutative unital C*-algebra, the Gelfand representation is then an isometric *-isomorphism between and [a]

Banach *-algebras

A Banach *-algebra is a Banach algebra over the field of complex numbers, together with a map that has the following properties:

  1. for all (so the map is an involution).
  2. for all
  3. for every and every here, denotes the complex conjugate of
  4. for all

In other words, a Banach *-algebra is a Banach algebra over that is also a *-algebra.

In most natural examples, one also has that the involution is isometric, that is, Some authors include this isometric property in the definition of a Banach *-algebra.

A Banach *-algebra satisfying is a C*-algebra.

See also

Notes

  1. ^ 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.

References

  1. ^ Conway 1990, Example VII.1.8.
  2. ^ a b Conway 1990, Example VII.1.9.
  3. ^ Conway 1990, Theorem VII.2.2.
  4. ^ García, Miguel Cabrera; Palacios, Angel Rodríguez (1995). "A New Simple Proof of the Gelfand-Mazur-Kaplansky Theorem". Proceedings of the American Mathematical Society. 123 (9): 2663–2666. doi:10.2307/2160559. ISSN 0002-9939. JSTOR 2160559.
  5. ^ Takesaki 1979, Proposition 2.8.