Classification of irrational Θ-deformed CAR C*-algebras

Journal article, 2021

Given a skew-symmetric matrix Θ we consider the universal enveloping C*-algebra CARΘ of the ∗-algebra generated by a_1,…,a_n subject to relations
a_i*a_i+a_ia_i*=1,
a_i*a_j=e^{2πiΘij}a_ja_i*,
a_ia_j=e^{−2πiΘij}a_ja_i.
We prove that CARΘ has a C(Kn)-structure, where Kn=[0,1/2]^n is the hypercube and describe the fibers. We classify irreducible representations of CARΘ in terms of representations of higher-dimensional noncommutative tori. We prove that for a given irrational Θ1 there are only finitely many Θ2 such that CARΘ1≃CARΘ2. Namely, CARΘ1≃CARΘ2 implies (Θ1)ij=±(Θ2)σ(i,j)modZ for a bijection σ of the set {(i,j):i<j, i,j=1,…,n}. For n=2 it means that CARθ1≃CARθ2 iff θ1=±θ2modZ.