Classification of irrational $\Theta$-deformed CAR $C^*$-algebras

TL;DR

This paper classifies irrational $ heta$-deformed CAR $C^*$-algebras, describing their structure, irreducible representations, and when two such algebras are isomorphic, especially fully for the two-dimensional case.

Contribution

It provides a detailed classification of irrational $ heta$-deformed CAR algebras, including their structure, representation theory, and isomorphism conditions, with a complete case for $n=2$.

Findings

$ ext{CAR}_ heta$ has a $C(K_n)$-structure with described fibers.

Classification of irreducible representations via noncommutative tori.

Finiteness of isomorphism classes for fixed irrational $ heta$.

Abstract

Given a skew-symmetric real $n\times n$ matrix $\Theta$ we consider the universal enveloping $C^*$-algebra $\mathsf{CAR}_\Theta$ of the $*$-algebra generated by $a_1, \ldots, a_n$ subject to the relations \[ a_i^* a_i + a_i a_i^* = 1, \ \] \[ a_i^* a_j = e^{2 \pi i \Theta_{i,j}} a_j a_i^*, \] \[ a_i a_j = e^{-2 \pi i \Theta_{i,j}} a_j a_i. \] 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 $(\Theta_1)_{ij} = \pm (\Theta_2)_{\sigma(i,j)} \mod \mathbb{Z}$ for a bijection $\sigma$ of the set $\{(i,j) : i < j, \ i, j = 1, \ldots, n\}$. For $n = 2$ we give a full classification: $\mathsf{CAR}_{\theta_1} \simeq \mathsf{CAR}_{\theta_2}$ iff $\theta_1 = \pm \theta_2 \mod \mathbb{Z}$.