Equivalence bimodule between non-commutative tori

Sei-Qwon Oh; Chun-Gil Park

Czechoslovak Mathematical Journal (2003)

  • Volume: 53, Issue: 2, page 289-294
  • ISSN: 0011-4642

Abstract

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The non-commutative torus C * ( ℤ n , ω ) is realized as the C * -algebra of sections of a locally trivial C * -algebra bundle over S ω ^ with fibres isomorphic to C * ( ℤ n / S ω , ω 1 ) for a totally skew multiplier ω 1 on ℤ n / S ω . D. Poguntke [9] proved that A ω is stably isomorphic to C ( S ω ^ ) ⊗ C * ( ℤ n / S ω , ω 1 ) ≅ C ( S ω ^ ) ⊗ A ϕ ⊗ M k l ( ℂ ) for a simple non-commutative torus A ϕ and an integer k l . It is well-known that a stable isomorphism of two separable C * -algebras is equivalent to the existence of equivalence bimodule between them. We construct an A ω - C ( S ω ^ ) ⊗ A ϕ -equivalence bimodule.

How to cite

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Oh, Sei-Qwon, and Park, Chun-Gil. "Equivalence bimodule between non-commutative tori." Czechoslovak Mathematical Journal 53.2 (2003): 289-294. <http://eudml.org/doc/30777>.

@article{Oh2003,
abstract = {The non-commutative torus $C^*(\mathbb \{Z\}^n,\omega )$ is realized as the $C^*$-algebra of sections of a locally trivial $C^*$-algebra bundle over $\widehat\{S_\{\omega \}\}$ with fibres isomorphic to $C^*(\mathbb \{Z\}^n/S_\{\omega \}, \omega _1)$ for a totally skew multiplier $\omega _1$ on $\mathbb \{Z\}^n/S_\{\omega \}$. D. Poguntke [9] proved that $A_\{\omega \}$ is stably isomorphic to $C(\widehat\{S_\{\omega \}\}) \otimes C^*(\mathbb \{Z\}^n/S_\{\omega \}, \omega _1) \cong C(\widehat\{S_\{\omega \}\}) \otimes A_\{\varphi \} \otimes M_\{kl\}(\mathbb \{C\})$ for a simple non-commutative torus $A_\{\varphi \}$ and an integer $kl$. It is well-known that a stable isomorphism of two separable $C^*$-algebras is equivalent to the existence of equivalence bimodule between them. We construct an $A_\{\omega \}$-$C(\widehat\{S_\{\omega \}\}) \otimes A_\{\varphi \}$-equivalence bimodule.},
author = {Oh, Sei-Qwon, Park, Chun-Gil},
journal = {Czechoslovak Mathematical Journal},
keywords = {Morita equivalent; twisted group $C^*$-algebra; crossed product; Morita equivalent; twisted group -algebra; crossed product},
language = {eng},
number = {2},
pages = {289-294},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Equivalence bimodule between non-commutative tori},
url = {http://eudml.org/doc/30777},
volume = {53},
year = {2003},
}

TY - JOUR
AU - Oh, Sei-Qwon
AU - Park, Chun-Gil
TI - Equivalence bimodule between non-commutative tori
JO - Czechoslovak Mathematical Journal
PY - 2003
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 53
IS - 2
SP - 289
EP - 294
AB - The non-commutative torus $C^*(\mathbb {Z}^n,\omega )$ is realized as the $C^*$-algebra of sections of a locally trivial $C^*$-algebra bundle over $\widehat{S_{\omega }}$ with fibres isomorphic to $C^*(\mathbb {Z}^n/S_{\omega }, \omega _1)$ for a totally skew multiplier $\omega _1$ on $\mathbb {Z}^n/S_{\omega }$. D. Poguntke [9] proved that $A_{\omega }$ is stably isomorphic to $C(\widehat{S_{\omega }}) \otimes C^*(\mathbb {Z}^n/S_{\omega }, \omega _1) \cong C(\widehat{S_{\omega }}) \otimes A_{\varphi } \otimes M_{kl}(\mathbb {C})$ for a simple non-commutative torus $A_{\varphi }$ and an integer $kl$. It is well-known that a stable isomorphism of two separable $C^*$-algebras is equivalent to the existence of equivalence bimodule between them. We construct an $A_{\omega }$-$C(\widehat{S_{\omega }}) \otimes A_{\varphi }$-equivalence bimodule.
LA - eng
KW - Morita equivalent; twisted group $C^*$-algebra; crossed product; Morita equivalent; twisted group -algebra; crossed product
UR - http://eudml.org/doc/30777
ER -

References

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  9. The structure of twisted convolution C * -algebras on abelian groups, J.  Operator Theory 38 (1997), 3–18. (1997) Zbl0924.46046MR1462012
  10. Morita equivalence for operator algebras, Operator Algebras and Applications. Proc. Symp. Pure Math. Vol. 38, R. V.  Kadison (ed.), Amer. Math. Soc., Providence, R. I., 1982, pp. 285–298. (1982) Zbl0541.46044MR0679708

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