In this paper, we show that if <mml:math alttext="$A$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math> is a unital semisimple complex Banach algebra with
only the trivial idempotents and if <mml:math alttext="$\sigma _A \left( x \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#963;</mml:mi><mml:mi>A</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> is countable for each <mml:math alttext="$x \in Fr\left( {G\left( A \right)} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>x</mml:mi><mml:mo>&#8712;</mml:mo><mml:mi>F</mml:mi><mml:mi>r</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>G</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>A</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>, then <mml:math alttext="$A \cong C$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi><mml:mo>&#8773;</mml:mo><mml:mi>C</mml:mi></mml:math>, this
generalizes the Gelfand-Mazur theorem.

International Journal of Mathematics and Mathematical Sciences

A generalization of Gelfand-Mazur theorem

Abstract

Copyright