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

Abstract

Given a skew-symmetric matrix Θ we consider the universal enveloping C * -algebra CARΘ of the * -algebra generated by a1, . . . , an subject to relations a * i ai + aia * i = 1, a * i aj = e 2πiΘ i,j aja * i , aiaj = e −2πiΘ i,j ajai. 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 irreducible representations of higherdimensional noncommutative tori. We prove that for a given irrational Θ1 there are only finitely many Θ2 such that CARΘ 1 ≃ CARΘ 2 . Namely, CARΘ 1 ≃ CARΘ 2 implies (Θ1)ij = ±(Θ2) σ(i,j) mod Z for a bijection σ of the set {(i, j) : i < j, i, j = 1, . . . , n}. For n = 2 it means that CAR θ 1 ≃ CAR θ 2 iff θ1 = ±θ2 mod Z.