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

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 ${\mathcal{O}}_{\mathrm{\infty }}$. We show that such formal infinite sum always makes sense on a certain dense subspace of any permutative representation of ${\mathcal{O}}_{\mathrm{\infty }}$. In this meaning, we can regard as if the algebra $\mathcal{B}$ of bosons was a unital $\ast$-subalgebra of ${\mathcal{O}}_{\mathrm{\infty }}$ on a given permutative representation by keeping their unboundedness. By this relation, we compute branching laws arising from restrictions of representations of ${\mathcal{O}}_{\mathrm{\infty }}$ on $\mathcal{B}$. For example, it is shown that the Fock representation of $\mathcal{B}$ is given as the restriction of the standard representation of ${\mathcal{O}}_{\mathrm{\infty }}$ on ${\cal B}$.