CCR and CAR algebras are connected via a path of Cuntz-Toeplitz algebras

Abstract

For $q \in \mathbb{R}$, $|q| < 1$ we consider the universal enveloping $C^*$-algebra of a $*$-algebra of $q$-canonical commutation relations ($q$-CCR), which is generated by $a_1, \ldots, a_n$ subject to the relations \[ a_i^* a_j = \delta_{ij} 1 + q a_j a_i^* . \] It has a distinguished representation $\pi_F$ called the Fock representation, which is believed to be faithful. In this article we denote the image of the universal enveloping $C^*$-algebra of $q$-CCR in the Fock representation by $\beth_q$. The question whether $C^*$-isomorphism $\beth_q \simeq \beth_0$ holds has been considered in the literature and proved for $|q| < 0.44$. In this article we show that $\beth_q \simeq \beth_0$ for $|q| < 1$.