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 $C^{\ast }$-algebra of a $\ast$-algebra of $q$-canonical commutation relations ($q$-CCR), which is generated by $a_1,\dots ,a_n$ subject to the relations \[ a_i^* a_j = δ_{ij} 1 + q a_j a_i^* . \] 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 $C^{\ast }$-algebra of $q$-CCR in the Fock representation by ${\beth }_q$. The question whether $C^{\ast }$-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 $|q|<1$.

References in corpus (1)

• On C*-algebras generated by pairs of q-commuting isometries