Recursive boson system in the Cuntz algebra ${\cal O}_{\infty}$

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 ${\cal B}$.