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Algebra di Banach: differenze tra le versioni

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Questo assicura che l'operazione di moltiplicazione è una [[funzione continua]].
 
<!--IfSe in the above we relax [[Banhttp://it.wikipedia.org/skins-1.5/common/images/button_media.png
Collegamento a file multimedialeach space]] to [[normed space]] the analogous structure isè calleddetta auna '''normed algebra'''
AUn'algebra di Banach algebraè is calleddetta "unital" ifse itha has anun [[identity element]] for the multiplication whosela cui normnorma isè 1, ande "commutative" ifse its multiplication isè [[commutative]].
 
BanachLe algebrasalgebre candi alsoBanach bepossono definedessere definite anche over fields of [[p-adic number]]s. This is part of [[p-adic analysis]].-->
 
== Esempi ==
* L'insieme di numeri reali (o complessi) è un'algebra di Banach con la norma del [[valore assoluto]].
<!--* L'insieme di tuttitutte le [[matrice|matrici]] reali o complesse ''n'' per ''n'' è un'algebra di Banach se gli si associa una norma
* TheL'insieme setdi oftutte allle real[[matrice or(matematica)|matrici]] reali o complexcomplesse ''n''-by- x ''n'' [[matrixdiventa (mathematics)|matrices]]un'algebra becomesdi Banach a [[unital]] Banach algebra ifse we equip it with a sub-multiplicative [[matrix norm]].-->
<!--* Take the Banach space '''R'''<sup>''n''</sup> (oro '''C'''<sup>''n''</sup>) withcon normnorma ||''x''|| = max |''x''<sub>''i''</sub>| ande 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>).
* 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- oro complex-valued functions defined on some set (with pointwise multiplication ande the [[supremum]] norm) is a unital Banach algebra.
* The algebra of all bounded [[continuous function (topology)|continuous]] real- oro complex-valued functions on some [[locally compact space]] (again with pointwise operations ande supremum norm) is a Banach algebra.
* AnyOgni [[C*-algebra]] isè aun'algebra Banachdi algebraBanach.
* The algebra of all [[continuous function (topology)|continuous]] [[linear transformation|linear]] operators on a Banach space E (with functional composition as multiplication ande the [[operator norm]] as norm) is a unital Banach algebra. The set of all compact operators on E is a closed ideal in this algebra.-->
* Gli operatori lineari continui su uno [[spazio di Hilbert]] formano una C-star-algebra e quindi un'algebra di Banach.
<!--* IfSe ''G'' isè a [[locally compact]] [[Hausdorff space|Hausdorff]] [[topological group]] ande μ itsla sua [[Haar measure]], thenallora thelo spazio di Banach space L<sup>1</sup>(''G'') ofdi alltutte le funzioni μ-integrableintegrabili functions onsu ''G'' becomesdiventa aun'algebra Banachdi algebraBanach undersotto the [[convolution]] ''xy''(''g'') = ∫ ''x''(''h'') ''y''(''h''<sup>-1</sup>''g'') dμ(''h'') for ''x'', ''y'' in L<sup>1</sup>(''G'').-->
<!--
== Properties ==
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. The formula for the [[geometric series]] and the [[binomial theorem]] also remain valid in general unital Banach algebras.
 
== Proprietà ==
The set of [[invertible element]]s in any unital Banach algebra is an [[open set]], and the inversion operation on this set is continuous, so that it forms a [[topological group]] under multiplication.▼
Molte [[elenco di funzioni|funzioni elementari]] che sono definite attraverso [[serie di potenze]] possono essere definite in ogni algebra di Banach unital; gli esempi includono la [[funzione esponenziale]] e le [[funzioni trigonometriche]]. La formula per le [[serie geometriche]] e il [[teorema binomiale]] also remain valid in general unital Banach algebras.
 
▲TheL'insieme set ofdi [[invertible element]]s in anyogni unital Banach algebra isè anun [[openinsieme setaperto]], ande the inversion operation on this set is continuous, so that it forms a [[topological group]] under multiplication.
Unital Banach algebras provide a natural setting to study general spectral theory. The ''spectrum'' of an element ''x'' consists of all those [[scalar (mathematics)|scalar]]s λ such that ''x'' -λ1 is not invertible. (In the Banach algebra of all ''n''-by-''n'' matrices mentioned above, the spectrum of a matrix coincides with the set of all its [[eigenvalue]]s.) The spectrum of any element is [[Compact space|compact]]. If the base field is the field of [[complex number]]s, then the spectrum of any element is [[non-empty]].▼
 
▲Unital Banach algebras provide a natural setting to study general spectral theory. The ''spectrum'' of an element ''x'' consists of all those [[scalar (mathematicsmatematica)|scalar]]s λ such that ''x'' -λ1 is not invertible. (InNell'algebra thedi Banach algebradi tutte ofle allmatrici ''n''-by-x''n'' matricessu mentioned abovemenzionate, thelo spectrumspettro ofdi auna matrixmatrice coincidescoincide withcon thel'insieme setdi oftutti alli itssuoi [[eigenvalue]]s.) TheLo spectrumspettro ofdi anyogni elementelemento isè [[Compact space|compact]]. IfSe the base field isè theil fieldcampo ofdei [[complexnumeri numbercomplessi]]s, thenallora the spectrum of any element is [[non-empty]].
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. ForAd exampleesempio:
* 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.▼
 
▲* Every real Banach algebra which is a [[division algebra]] is isomorphic to the reals, the complexes, oro the quaternions. Hence, the only complex Banach algebra which is a division algebra is the complexes.
* 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.
* Every commutative real unital [[noetherian ring|noetherian]] Banach algebra with no zero divisors is isomorphic to the real or complex numbers.
Estratto da "https://it.wikipedia.org/wiki/Algebra_di_Banach"