Bulletin of the Belgian Mathematical Society - Simon Stevin

The Bundle Structure of Noncommutative Tori over $UHF$-Algebras

Chun-Gil Park

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The noncommutative torus $C^*(\mathbb Z^n,\,\omega)$ of rank $n$ is realized as the $C^*$-algebra of sections of a locally trivial continuous $C^*$-algebra bundle over $\widehat{S_{\omega}}$ with fibres $C^*(\mathbb Z^n/S_{\omega},\, \omega_1)$ for some totally skew multiplier $\omega_1$ on $\mathbb Z^n/S_{\omega}$. It is shown that $C^*(\mathbb Z^n/S_{\omega},\,\omega_1)$ is isomorphic to $A_{\varphi}\otimes M_k(\mathbb C)$ for some completely irrational noncommutative torus $A_{\varphi}$ and some positive integer $k$, and that $A_{\omega} \otimes M_{l^{\infty}}$ has the trivial bundle structure if and only if the set of prime factors of $k$ is a subset of the set of prime factors of $l$. This is applied to understand the bundle structure of the tensor products of Cuntz algebras with noncommutative tori.

Article information

Bull. Belg. Math. Soc. Simon Stevin, Volume 10, Number 3 (2003), 321-328.

First available in Project Euclid: 12 September 2003

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L87: Noncommutative differential geometry [See also 58B32, 58B34, 58J22] 46L05: General theory of $C^*$-algebras

$C^*$-algebra bundle twisted group $C^*$-algebra $K$-theory Cuntz algebra


Park, Chun-Gil. The Bundle Structure of Noncommutative Tori over $UHF$-Algebras. Bull. Belg. Math. Soc. Simon Stevin 10 (2003), no. 3, 321--328. doi:10.36045/bbms/1063372339. https://projecteuclid.org/euclid.bbms/1063372339

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