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  Volume 7, Issue 2, Article 68
 
A Note on Commutative Banach Algebras
    Authors: Takashi Sano,  
    Keywords: Commutative Banach algebra; Norm; Similarity transformation; Spectral radius.  
    Date Received: 09/03/06  
    Date Accepted: 06/04/06  
    Subject Codes:

46J99, 47A30.

  
    Editors: Chi-Kwong Li,  
 
    Abstract:

Let $mathcal{A}$ be a unital Banach algebra over $mathbb{C}$ with norm $Vert cdot Vert$. In this note, several characterizations of commutativity of $mathcal{A}$ are given. For instance, it is shown that $mathcal{A}$ is commutative if

begin{displaymath} Vert A B Vert = Vert B A Vert end{displaymath}

for all $A, B in mathcal{A}$, or if the spectral radius on $mathcal{A}$ is a norm.

         
       
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