It is well known that for any irrational rotation number α, the noncommutative torus Aα must have representations π such that the generated von Neumann algebra π(Aα)″ is of type III. Therefore, it could be of interest to exhibit and investigate such kind of representations, together with the associated spectral triples whose twist of the Dirac operator and the corresponding derivation arises from the Tomita modular operator. In the present paper, we show that this program can be carried out, at least when α is a Liouville number satisfying a faster approximation property by rationals. In this case, we exhibit several type II∞ and IIIλ, λ∈[0,1], factor representations and modular spectral triples. The method developed in the present paper can be generalised to CCR algebras based on a locally compact abelian group equipped with a symplectic form.

Fidaleo, F., Suriano, L. (2018). Type III representation and modular spectral triples for the noncommutative torus. JOURNAL OF FUNCTIONAL ANALYSIS, 275(6), 1484-1531 [10.1016/j.jfa.2018.06.009].

Type III representation and modular spectral triples for the noncommutative torus

Fidaleo, Francesco;Suriano, Luca
2018-01-01

Abstract

It is well known that for any irrational rotation number α, the noncommutative torus Aα must have representations π such that the generated von Neumann algebra π(Aα)″ is of type III. Therefore, it could be of interest to exhibit and investigate such kind of representations, together with the associated spectral triples whose twist of the Dirac operator and the corresponding derivation arises from the Tomita modular operator. In the present paper, we show that this program can be carried out, at least when α is a Liouville number satisfying a faster approximation property by rationals. In this case, we exhibit several type II∞ and IIIλ, λ∈[0,1], factor representations and modular spectral triples. The method developed in the present paper can be generalised to CCR algebras based on a locally compact abelian group equipped with a symplectic form.
2018
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/05 - ANALISI MATEMATICA
English
Fidaleo, F., Suriano, L. (2018). Type III representation and modular spectral triples for the noncommutative torus. JOURNAL OF FUNCTIONAL ANALYSIS, 275(6), 1484-1531 [10.1016/j.jfa.2018.06.009].
Fidaleo, F; Suriano, L
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/205974
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