Banach algebra

In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebraover 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.This ensures that the multiplication operation is continuous with respect to the metric topology.A Banach algebra is called unital if it has an identity element for the multiplication whose norm isOften one assumes a priori that the algebra under consideration is unital because one can develop much of the theory by consideringand then applying the outcome in the original algebra.For example, one cannot define all the trigonometric functions in a Banach algebra without identity.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 ofThe prototypical example of a Banach algebra is(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.For example: Unital Banach algebras over the complex field provide a general setting to develop spectral theory.is non-empty and satisfies the spectral radius formula:Furthermore, the spectral mapping theorem holds:[5]of bounded linear operators on a complex Banach space(for example, the algebra of square matrices), the notion of the spectrum incoincides with the usual one in operator theory.This generalizes an analogous fact for normal operators.be a complex unital Banach algebra in which every non-zero elementis then a commutative ring with unit, every non-invertible element ofis 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 ofsince the kernel of a character is a maximal ideal, which is closed.of complex continuous functions on the compact spaceAs 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.is a commutative unital C*-algebra, the Gelfand representation is then an isometric *-isomorphism betweenis a Banach algebra over the field of complex numbers, together with a mapSome authors include this isometric property in the definition of a Banach *-algebra.
mathematicsfunctional analysisStefan Banachassociative algebracomplexnon-Archimedeannormed fieldBanach spacenormed spacecompletemetriccontinuousidentity elementcommutativeisometricallyclosedtrigonometric functionsspectrum p {\displaystyle p} -adic numbers p {\displaystyle p} -adic analysislocally compact Hausdorff spacevanish at infinitycompactcomplex conjugationinvolutionC*-algebraabsolute valuematricesunitalmatrix normquaternionssupremumlocally compact spacelinearoperator normcompact operatorslocally compactHausdorfftopological groupHaar measureconvolutionUniform algebraNatural Banach function algebraHilbert spaceMeasure algebraRadon measureslocally compact groupaffinoid algebrarigid analytic geometryelementary functionspower seriesexponential functionentire functionabstract index groupsgeometric seriesbinomial theoreminvertible elementsopen setcommutatordivision algebraGelfand–Mazur theoremzero divisorsprincipal idealNoetheriantopological divisors of zeroSpectral theoryscalarsnon-emptyspectral radiusholomorphic functional calculusholomorphicoperator theorymaximal idealstructure spaceGelfand representationsemisimpleJacobson radicalcomplex numberscomplex conjugate*-algebraisometricApproximate identityKaplansky's conjectureOperator algebraShilov boundaryStone–Weierstrass theoremBollobás, BBonsall, F. F.Conway, J. B.Springer VerlagWillis, G. 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Riesz extensionParseval's identityRiesz's lemmaSchauder fixed-pointAbstract Wiener spaceBanach manifoldbundleBochner spaceConvex seriesDifferentiation in Fréchet spacesDerivativesGateauxfunctionalIntegralsBochnerDunfordGelfand–PettisregulatedPaley–WienerMeasuresLebesgueProjection-valuedVectorWeaklyStronglyAbsolutely convexAbsorbingAffineBalanced/CircledConvexConvex cone (subset)Linear cone (subset)RadialRadially convex/Star-shapedSymmetricZonotopeAffine hullAlgebraic interior (core)Bounding pointsConvex hullExtreme pointInteriorLinear spanMinkowski additionAbsolute continuity AC b a ( Σ ) {\displaystyle ba(\Sigma )} c spaceBanach coordinate BKBesov B p , q s ( R ) {\displaystyle B_{p,q}^{s}(\mathbb {R} )} Birnbaum–OrliczBounded variation BVBs spaceContinuous C(K) with K compact HausdorffHardy HpMorrey–Campanato L λ , p ( Ω ) {\displaystyle L^{\lambda ,p}(\Omega )} Schwartz S ( R n ) {\displaystyle S\left(\mathbb {R} ^{n}\right)} Segal–Bargmann FSequence spaceSobolev Wk,pSobolev inequalityTriebel–LizorkinWiener amalgam W ( X , L p ) {\displaystyle W(X,L^{p})} Differential operatorFinite element methodMathematical formulation of quantum mechanicsOrdinary Differential Equations (ODEs)Validated numerics