On *-representations of λ-deformations of canonical commutation relations

Abstract

We study irreducible integrable *-representations of the algebra $\mathfrak{U}_{\lambda, 2}$ generated by the following relations: $$\mathfrak{U}_{\lambda, 2} = \mathbb{C} \langle a_j, a_j^{*} \,| \,a_j^{*} a_j = 1 + a_ja_j^{*},\; a_1^{*}a_2 = \lambda a_2a_1^{*},\; a_2a_1 = \lambda a_1 a_2,\; j = 1, 2 \rangle .$$ For this *-algebra, we prove an analog of the von Neumann theorem on the uniqueness of an irreducible integrable representation.