Exactness of the Fock space representation of the q-commutation relations

TL;DR

This paper demonstrates that for all q in (-1,1), the Fock representation of q-commutation relations can be embedded into the extended Cuntz algebra's Fock space, implying exactness of the associated C*-algebra.

Contribution

It establishes the unitary embedding of q-commutation Fock representations into the extended Cuntz algebra, proving the exactness of the generated C*-algebra and weak exactness of q-Gaussian von Neumann algebras.

Findings

C*-algebra generated by q-commutation Fock representation is exact

q-Gaussian von Neumann algebra is weakly exact for all q in (-1,1)

Fock representations can be unitarily embedded into the extended Cuntz algebra

Abstract

We show that for all q in the interval (-1,1), the Fock representation of the q-commutation relations can be unitarily embedded into the Fock representation of the extended Cuntz algebra. In particular, this implies that the C*-algebra generated by the Fock representation of the q-commutation relations is exact. An immediate consequence is that the q-Gaussian von Neumann algebra is weakly exact for all q in the interval (-1,1).