Classification of irrational $\Theta$-deformed CAR $C^{\ast }$-algebras

Abstract

Given a skew-symmetric real $n\times n$ matrix $\Theta$ we consider the universal enveloping $C^{\ast }$-algebra ${\mathsf{CAR}}_{\Theta }$ of the $\ast$-algebra generated by $a_1,\dots ,a_n$ 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 ${\mathsf{CAR}}_{\Theta }$ 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 ${\mathsf{CAR}}_{\Theta }$ in terms of irreducible representations of a higher-dimensional noncommutative torus. We prove that for a given irrational skew-symmetric ${\Theta }_1$ there are only finitely many ${\Theta }_2$ such that ${\mathsf{CAR}}_{{\Theta }_1}\simeq {\mathsf{CAR}}_{{\Theta }_2}$. Namely, ${\mathsf{CAR}}_{{\Theta }_1}\simeq {\mathsf{CAR}}_{{\Theta }_2}$ implies $({\Theta }_1{)}_{ij}=\pm ({\Theta }_2{)}_{\sigma (i,j)}\mathrm{mod}\mathbb{Z}$ for a bijection $\sigma$ of the set $\{(i,j):i<j,i,j=1,\dots ,n\}$. For $n=2$ we give a full classification: ${\mathsf{CAR}}_{{\theta }_1}\simeq {\mathsf{CAR}}_{{\theta }_2}$ iff ${\theta }_1=\pm {\theta }_2\mathrm{mod}\mathbb{Z}$.