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We give a characterization of 
${{C}^{*}}$-normed algebras, among certain involutive normed ones. This is done through the existence of enough specific positive functionals. The same question is also examined in some non normed (topological) algebras.
$^{\ast }$-ALGEBRAS
It is an open question whether every derivation of a Fréchet GB
$^{\ast }$-algebra 
$A[{\it\tau}]$ is continuous. We give an affirmative answer for the case where 
$A[{\it\tau}]$ is a smooth Fréchet nuclear GB
$^{\ast }$-algebra. Motivated by this result, we give examples of smooth Fréchet nuclear GB
$^{\ast }$-algebras which are not pro-C
$^{\ast }$-algebras.
