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.
