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

DOI10.17879/06089640368zbMATH Open1493.46074arXiv2010.15660MaRDI QIDQ5034336

Lyudmyla Turowska, Aleksey Kuzmin

Publication date: 24 February 2022

Abstract: Given a skew-symmetric real $nimesn$ matrix $Theta$ we consider the universal enveloping $C^{*}$-algebra $mathsf{CAR}_{T}heta$ 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 $mathsf{CAR}_{T}heta$ has a $C(K_n)$-structure, where $K_n=left[0,frac12ight{]}^n$ is the hypercube and describe the fibers. We classify irreducible representations of $mathsf{CAR}_{T}heta$ in terms of irreducible representations of a higher-dimensional noncommutative torus. We prove that for a given irrational skew-symmetric $Thet{a}_1$ there are only finitely many $Thet{a}_2$ such that $mathsf{CAR}_{Thet{a}_1}simeqmathsf{CAR}_{Thet{a}_2}$. Namely, $mathsf{CAR}_{Thet{a}_1}simeqmathsf{CAR}_{Thet{a}_2}$ implies $(Thet{a}_1{)}_{ij}=pm(Thet{a}_2{)}_{sigma(i,j)}modmathbbZ$ for a bijection $sigma$ of the set $(i,j):i<j,i,j=1,ldots,n$. For $n=2$ we give a full classification: $mathsf{CAR}_{het{a}_1}simeqmathsf{CAR}_{het{a}_2}$ iff $het{a}_1=pmhet{a}_{2}modmathbbZ$.

Full work available at URL: https://arxiv.org/abs/2010.15660

zbMATH Keywords

deformationCAR algebra

Mathematics Subject Classification ID

General theory of (C^*)-algebras (46L05) Quantizations, deformations for selfadjoint operator algebras (46L65)

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