Classification of irrational $\Theta$-deformed CAR $C^*$-algebras
Alexey Kuzmin, Lyudmila Turowska

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 matrix we consider the universal enveloping -algebra of the -algebra generated by 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 has a -structure, where is the hypercube and describe the fibers. We classify irreducible representations of in terms of irreducible representations of a higher-dimensional noncommutative torus. We prove that for a given irrational skew-symmetric there are only finitely many such that . Namely, implies…
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Operator Algebra Research · Advanced Topics in Algebra
