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 ai∗aj=δij1+qajai∗.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} a_i^* a_j = \\delta _{ij} 1 + q a_j a_i^* . \\end{aligned}$$\\end{document}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 _{n,q}. The question whether C^*-isomorphism beth _{n,q} simeq beth _{n,0} holds has been considered in the literature and proved for |q| < 0.44. In this article we show that beth _{n,q} simeq beth _{n,0} for |q| < 1.