Banach algebra

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, especially <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>, a Banach algebra, named after <a href="/facts/Stefan_Banach/b1qHSNg2">Stefan Banach</a>, is an <a href="/facts/Associative_algebra/DRfoF6KV">associative algebra</a> 
 
 
 
 A
 
 
 {\displaystyle A}
 
 over the <a href="/facts/Real_number/R02gw5Pb">real</a> or <a href="/facts/Complex_number/2w7mqI7e">complex</a> numbers (or over a <a href="/facts/Nonarchimedean_field/61pC7UP6">non-Archimedean</a> complete <a href="/facts/Norm_(mathematics)/xIbR4uE1">normed field</a>) that at the same time is also a <a href="/facts/Banach_space/sDEIk7EC">Banach space</a>, that is, a <a href="/facts/Normed_space/rmDljfyE">normed space</a> that is <a href="/facts/Complete_metric_space/lsckmU8G">complete</a> in the <a href="/facts/Metric_(mathematics)/RkUptSo8">metric</a> induced by the norm. The norm is required to satisfy

‖
        x
        
        y
        ‖
         
        ≤
        ‖
        x
        ‖
        
        ‖
        y
        ‖
        
        
           for all 
        
        x
        ,
        y
        ∈
        A
        .
      
    
    {\displaystyle \|x\,y\|\ \leq \|x\|\,\|y\|\quad {\text{ for all }}x,y\in A.}

This ensures that the multiplication operation is <a href="/facts/Continuous_function_(topology)/sKbl02pB">continuous</a> with respect to the metric topology.
A Banach algebra is called unital if it has an <a href="/facts/Identity_element/9nss3xHY">identity element</a> for the multiplication whose norm is 
 
 
 
 1
 ,
 
 
 {\displaystyle 1,}
 
 and commutative if its multiplication is <a href="/facts/Commutative/WaYxJLxd">commutative</a>.
Any Banach algebra 
 
 
 
 A
 
 
 {\displaystyle A}
 
 (whether it is unital or not) can be embedded <a href="/facts/Isometry/K5wX8ah6">isometrically</a> into a unital Banach algebra 
 
 
 
 
 A
 
 e
 
 
 
 
 {\displaystyle A_{e}}
 
 so as to form a <a href="/facts/Closed_set/upYnRTUj">closed</a> <a href="/facts/Ideal_(algebra)/vCvytduX">ideal</a> of 
 
 
 
 
 A
 
 e
 
 
 
 
 {\displaystyle A_{e}}
 
. Often one assumes a priori that the algebra under consideration is unital because one can develop much of the theory by considering 
 
 
 
 
 A
 
 e
 
 
 
 
 {\displaystyle A_{e}}
 
 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 <a href="/facts/Trigonometric_function/czej7ur0">trigonometric functions</a> 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 <a href="/facts/Spectrum_(functional_analysis)/cA1evTo6">spectrum</a> 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 <a href="/facts/P-adic_number/SDgDbkLF">
 
 
 
 p
 
 
 {\displaystyle p}
 
-adic numbers</a>. This is part of <a href="/facts/P-adic_analysis/VrdVVN43">
 
 
 
 p
 
 
 {\displaystyle p}
 
-adic analysis</a>.

Banach algebra open-in-new

Banach algebra