Algebra di Banach: differenze tra le versioni
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Questo assicura che l'operazione di moltiplicazione è una [[funzione continua]].
<!--If in the above we relax [[Banach space]] to [[normed space]] the analogous structure is called a '''normed algebra'''
A Banach algebra is called "unital" if it has an [[identity element]] for the multiplication whose norm is 1,
Banach algebras can also be defined 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 tutti le [[matrice|matrici]] reali o complesse ''n'' per ''n'' è un'algebra di Banach se gli si associa una norma
* The set of all real or complex ''n''-by-''n'' [[matrix (mathematics)|matrices]] becomes a [[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>|
* 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
* The algebra of all bounded [[continuous function (topology)|continuous]] real- or complex-valued functions on some [[locally compact space]] (again with pointwise operations
* Any [[C*-algebra]] 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
* Gli operatori lineari continui su uno [[spazio di Hilbert]] formano una C-star-algebra e quindi un'algebra di Banach.
<!--* If ''G'' is a [[locally compact]] [[Hausdorff space|Hausdorff]] [[topological group]]
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== 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]]
The set of [[invertible element]]s in any unital Banach algebra is an [[open set]],
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]].
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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.
* Every unital real Banach algebra with no [[zero divisor]]s,
* 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.
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== See also ==
* [[Uniform algebra]] A Banach algebra that is a subalgebra of C(X) with the supremum norm
* [[Uniform algebra|Natural Banach function algebra]] A uniform algebra whose all characters are evaluations at points of X.
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[[Categoria:Teoria delle algebre]]
[[Categoria:Spazi di Banach]]
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