  The concept of logarithmic representation of infinitesimal generators is
introduced, and it is applied to clarify the algebraic structure of bounded and
unbounded infinitesimal generators. In particular, by means of the logarithmic
representation, the bounded components can be extracted from
generally-unbounded infinitesimal generators. In conclusion the concept of
module over a Banach algebra is proposed as the generalization of Banach
algebra. As an application to mathematical physics, the rigorous formulation of
rotation group, which consists of unbounded operators being written by
differential operators, is provided using the module over a Banach algebra.
