Hermitian and algebraic *-algebras,representable extensions of positive functionals

We introduce the concept of E*-algebras in the context of representable extensions of positive functionals, and we show that the class of E*-algebras coincides with the class of hermitian algebraic *-algebras. Results on Banach E*-algebras and preC*-algebras which are E*-algebras are also included.As applications we give a characterization of discrete locally finite groups via E*algebras and answer Kurosh's problem in the affirmative in the case of the convolution *-algebras of compactly supported continuous functions on an arbitrary locally compact group.