  It is known that the generator of a strictly continuous one parameter unitary
group in the multiplier algebra of a $C^{*}$-algebra is affiliated to that
$C^{*}$-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 $C^{*}$-algebra.
