On generalized Stone's Theorem

It is known that the generator of a strictly continuous one parameter unitary group in the multiplier algebra of a $C^{\ast }$-algebra is affiliated to that $C^{\ast }$-algebra . We show that under natural non degeneracy conditions, this self adjoint unbounded operator lies indeed in the (unbounded) multiplier algebra of the Pedersen’s ideal of the $C^{\ast }$-algebra.