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

Abstract

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

18 pp

References in corpus (1)

• Recursive Fermion System in Cuntz Algebra. II -- Endomorphism, Automorphism and Branching of Representation --

Cited by in corpus (2)

• Unitary isomorphism of Fock spaces of bosons and fermions arising from a representation of the Cuntz algebra $\co{2}$
• Universal fermionization of bosons on permutative representations of the Cuntz algebra ${\mathcal{O}}_2$