Banach algebra | Wikipedia audio article
Channel: wikipedia tts
Date: 26/03/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Banach_algebra


00:02:28 1 Examples
00:03:56 2 Counterexamples
00:06:24 3 Properties
00:08:39 4 Spectral theory
00:11:32 5 Ideals and characters
00:14:53 6 See also
00:15:03 7 Remarks
00:16:14 8 Notes
00:19:07 9 References
00:19:29 Remarks



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SUMMARY
=======
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach space, i.e. a normed space and complete in the metric induced by the norm. The norm is required to satisfy

  
    
      
        ∀
        x
        ,
        y
        ∈
        A
        :
        ‖
        x
        
        y
        ‖
         
        ≤
        ‖
        x
        ‖
        
        ‖
        y
        ‖
        .
      
    
    {\displaystyle \forall x,y\in A:\|x\,y\|\ \leq \|x\|\,\|y\|.}
  This ensures that the multiplication operation is continuous.
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.
Any Banach algebra 
  
    
      
        A
      
    
    {\displaystyle A}
   (whether it has an identity element or not) can be embedded isometrically into a unital Banach algebra 
  
    
      
        
          A
          
            e
          
        
      
    
    {\displaystyle A_{e}}
   so as to form a closed ideal of 
  
    
      
        
          A
          
            e
          
        
      
    
    {\displaystyle A_{e}}
  . Often one assumes a priori that the algebra under consideration is unital: for 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 trigonometric functions 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 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 of p-adic numbers. This is part of p-adic analysis.