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| 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 which at the same time is also a [[Banach space]], i.e. normed and complete. The algebra multiplication and the Banach space norm are required to be related by the following inequality:
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| :<math> \forall x, y \in A : \|x \, y\| \ \leq \|x \| \, \| y\| </math>
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| (i.e., the norm of the product is less than or equal to the product of the norms). This ensures that the multiplication operation is [[continuous function (topology)|continuous]]. This property is found in the real and complex numbers; for instance, |-6×5| ≤ |-6|×|5|.
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| If in the above we relax [[Banach space]] to [[normed space]] the analogous structure is called a '''normed algebra'''.
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| 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]].
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| Any Banach algebra <math>A</math> (whether it has an [[identity element]] or not) can be embedded isometrically into a unital Banach algebra <math>A_e</math> so as to form a closed 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 functions in a Banach algebra without identity.
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| The theory of real Banach algebras can be very different from the theory of complex Banach algebras. For example, the [[Spectrum of an operator|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.
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| Banach algebras can also be defined over fields of [[p-adic number]]s. This is part of [[p-adic analysis]].
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| == Examples ==
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| 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 that vanish at infinity. <math>C_0(X)</math> is unital if and only if ''X'' is compact. The complex conjugation being an involution, <math>C_0(X)</math> is in fact a [[C*-algebra]]. More generally, every C*-algebra is a Banach algebra.
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| * The set of real (or complex) numbers is a Banach algebra with norm given by the [[absolute value]].
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| * 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]].
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| * 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>).
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| * The [[quaternion]]s form a 4-dimensional real Banach algebra, with the norm being given by the absolute value of quaternions.
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| * 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.
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| * 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.
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| * 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 closed ideal in this algebra.
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| * If ''G'' is a [[locally compact]] [[Hausdorff space|Hausdorff]] [[topological group]] and μ 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'') dμ(''h'') for ''x'', ''y'' in L<sup>1</sup>(''G'').
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| * [[Uniform algebra]]: A Banach algebra that is a subalgebra of C(X) with the supremum norm and that contains the constants and separates the points of X (which must be a compact Hausdorff space).
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| * [[Uniform algebra|Natural Banach function algebra]]: A uniform algebra whose all characters are evaluations at points of X.
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| * [[C*-algebra]]: A Banach algebra that is a closed *-subalgebra of the algebra of bounded operators on some [[Hilbert space]].
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| * [[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]].
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| == Properties ==
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| Several [[list of functions|elementary functions]] which are defined via [[power series]] may be defined in any unital Banach algebra; examples include the [[exponential function]] and the [[trigonometric function]]s, 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.
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| 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 homeomorphism) so that it forms a [[topological group]] under multiplication.
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| 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>  for any ''x'', ''y'' ∈ ''A''.
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| 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 real Banach algebra which is a [[division algebra]] is isomorphic to the reals, the complexes, or the quaternions. Hence, the only complex Banach algebra which is a division algebra is the complexes. (This is known as the [[Gelfand-Mazur theorem]].)
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| * 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.
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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.
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| * Every commutative real unital Noetherian Banach algebra (possibly having zero divisors) is finite-dimensional.
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| * Permanently singular elements in Banach algebras are [[topological divisior 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 all Banach extension ''B'' of ''A''.
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| == Spectral theory ==
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| {{main|Spectral theory}}
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| Unital Banach algebras over the complex field provide a general setting to develop spectral theory. The ''spectrum'' of an element ''x'' ∈ ''A'', denoted by <math>\sigma(x)</math>, consists of all those complex [[scalar (mathematics)|scalar]]s ''λ'' such that ''x'' − ''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:
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| :<math>\sup \{ |\lambda| : \lambda \in \sigma(x) \} = \lim_{n \to \infty} \|x^n\|^{1/n}.</math>
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| Given ''x'' ∈ ''A'', the [[holomorphic functional calculus]] allows to define ''ƒ''(''x'') ∈ ''A'' for any function ''ƒ'' [[holomorphic function|holomorphic]] in a neighborhood of <math>\sigma(x).</math> Furthermore, the spectral mapping theorem holds:
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| :<math>\sigma(f(x)) = f(\sigma(x)).</math><ref>Takesaki, Theory of Operator Algebras I. Proposition 2.8.</ref>
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| When the Banach algebra ''A'' is the algebra L(''X'') of bounded linear operators on a complex Banach space ''X''  (e.g., the algebra of square matrices), the notion of the spectrum in ''A'' coincides with the usual one in the operator theory. For ''ƒ'' ∈ ''C''(''X'') (with a compact Hausdorff space ''X''), one sees that:
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| :<math>\sigma(f) = \{ f(t) : t \in X \}.</math>
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| The norm of a normal element ''x'' of a C*-algebra coincides with its spectral radius. This generalizes an analogous fact for normal operators.
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| Let ''A''  be a complex unital Banach algebra in which every non-zero element ''x'' is invertible (a division algebra). For every ''a'' ∈ ''A'', there is ''λ'' ∈ '''C''' such that
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| ''a'' − ''λ'''''1''' is not invertible (because the spectrum of ''a'' is not empty) hence ''a'' = ''λ'''''1''' : this algebra ''A'' is naturally isomorphic to '''C''' (the complex case of the Gelfand-Mazur theorem).
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| == Ideals and characters ==
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| Let ''A''  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''  to '''C'''. The set Δ(''A'') is called the "[[structure space]]" or "character space" of ''A'', and its members "characters."
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| A character χ is a linear functional on ''A'' which is at the same time multiplicative, χ(''ab'') = χ(''a'') χ(''b''), and satisfies ''χ''('''1''') = 1. Every character is automatically continuous from ''A''  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 ''A''<sup>∗</sup>), the character space, Δ(''A''), is a Hausdorff compact space.
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| For any ''x'' ∈ ''A'',
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| :<math>\sigma(x) = \sigma(\hat x)</math> | |
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| 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>  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,
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| :<math>\sigma(\hat x) = \{ \chi(x) : \chi \in \Delta(A) \}</math>.
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| 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'')) .<ref>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]].</ref>
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| == See also ==
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| * [[Operator algebras]]
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| * [[Shilov boundary]]
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| * [[Automatic continuity]]
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| * [[Kaplansky's conjecture]]
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| * [[Approximate identity]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * {{cite book | author=Béla Bollobás | authorlink=Béla Bollobás | title=Linear Analysis | publisher=Cambridge University Press | year=1990 | isbn=0-521-38729-9 }}
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| * {{cite book | author=Frank F. Bonsall, John Duncan | title=Complete Normed Algebras | publisher=Springer-Verlag, New York | year=1973 | isbn=0-387-06386-2}}
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| * {{cite book | author=H. Garth Dales, Pietro Aeina, Jörg Eschmeier, Kjeld Laursen, George A. Willis | title=Introduction to Banach Algebras, Operators and Harmonic Analysis | series=Cambridge University Press | year=2003 | isbn=0-521-53584-0 }}
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| * {{cite book | author=Richard D. Mosak | title=Banach algebras | series=Chicago Lectures in Mathematics | year=1975 | isbn=0-226-54203-3 }}
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| {{DEFAULTSORT:Banach Algebra}}
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| [[Category:Banach algebras| ]]
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| [[Category:Fourier analysis]]
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| [[Category:Science and technology in Poland]]
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