Classification of irrational $Θ$-deformed CAR $C^*$-algebras

Abstract

Given a skew-symmetric real $n\times n$ matrix $Θ$ we consider the universal enveloping $C^*$-algebra $\mathsf{CAR}_Θ$ of the $*$-algebra generated by $a_1, \ldots, a_n$ subject to the relations \[ a_i^* a_i + a_i a_i^* = 1, \ \] \[ a_i^* a_j = e^{2 πi Θ_{i,j}} a_j a_i^*, \] \[ a_i a_j = e^{-2 πi Θ_{i,j}} a_j a_i. \] We prove that $\mathsf{CAR}_Θ$ has a $C(K_n)$-structure, where $K_n = \left[ 0,\frac{1}{2} \right]^n$ is the hypercube and describe the fibers. We classify irreducible representations of $\mathsf{CAR}_Θ$ in terms of irreducible representations of a higher-dimensional noncommutative torus. We prove that for a given irrational skew-symmetric $Θ_1$ there are only finitely many $Θ_2$ such that $\mathsf{CAR}_{Θ_1} \simeq \mathsf{CAR}_{Θ_2}$. Namely, $\mathsf{CAR}_{Θ_1} \simeq \mathsf{CAR}_{Θ_2}$ implies $(Θ_1)_{ij} = \pm (Θ_2)_{σ(i,j)} \mod \mathbb{Z}$ for a bijection $σ$ of the set $\{(i,j) : i < j, \ i, j = 1, \ldots, n\}$. For $n = 2$ we give a full classification: $\mathsf{CAR}_{θ_1} \simeq \mathsf{CAR}_{θ_2}$ iff $θ_1 = \pm θ_2 \mod \mathbb{Z}$.