  Bosons and fermions are often written by elements of other algebras. M. Abe
gave a recursive realization of the boson by formal infinite sums of the
canonical generators of the Cuntz algebra ${\cal O}_{\infty}$. We show that
such formal infinite sum always makes sense on a certain dense subspace of any
permutative representation of ${\cal O}_{\infty}$. In this meaning, we can
regard as if the algebra ${\cal B}$ of bosons was a unital $*$-subalgebra of
${\cal O}_{\infty}$ on a given permutative representation by keeping their
unboundedness. By this relation, we compute branching laws arising from
restrictions of representations of ${\cal O}_{\infty}$ on ${\cal B}$. For
example, it is shown that the Fock representation of ${\cal B}$ is given as the
restriction of the standard representation of ${\cal O}_{\infty}$ on ${\cal
B}$.
