Recursive boson system in the Cuntz algebra O∞

Abstract

Bosons and fermions are often written by elements of other algebras. Abe (private communication) gave a realization of bosons by formal infinite sums of the canonical generators of the Cuntz algebra O∞. We show that such formal infinite sum always makes sense on a certain dense subspace of any permutative representation of O∞. In this meaning, we can regard as if the algebra B of bosons was a unital ∗-subalgebra of O∞ on a given permutative representation. According to this relation, we compute branching laws arising from restrictions of representations of O∞ on B. For example, it is shown that the Fock representation of B is given as the restriction of the standard representation of O∞ on B.