Generalized non-commutative tori

Chun-Gil Park

Generalized non-commutative tori

Chun-Gil Park

Studia Mathematica (2002)

Volume: 149, Issue: 2, page 101-108
ISSN: 0039-3223

Access Full Article

top

Access to full text

Full (PDF)

Abstract

top

The generalized non-commutative torus

T_{ϱ}^{k}

of rank n is defined by the crossed product

A_{m / k} \times_{α ₃} ℤ \times_{α ₄} . . . \times_{α ₙ} ℤ

, where the actions

α_{i}

of ℤ on the fibre

M_{k} (ℂ)

of a rational rotation algebra

A_{m / k}

are trivial, and

C * (k ℤ \times k ℤ) \times_{α ₃} ℤ \times_{α ₄} . . . \times_{α ₙ} ℤ

is a non-commutative torus

A_{ϱ}

. It is shown that

T_{ϱ}^{k}

is strongly Morita equivalent to

A_{ϱ}

, and that

T_{ϱ}^{k} \otimes M_{p^{\infty}}

is isomorphic to

A_{ϱ} \otimes M_{k} (ℂ) \otimes M_{p^{\infty}}

if and only if the set of prime factors of k is a subset of the set of prime factors of p.

How to cite

top

MLA
BibTeX
RIS

Chun-Gil Park. "Generalized non-commutative tori." Studia Mathematica 149.2 (2002): 101-108. <http://eudml.org/doc/284857>.

@article{Chun2002,
abstract = {The generalized non-commutative torus $T_\{ϱ\}^\{k\}$ of rank n is defined by the crossed product $A_\{m/k\} ×_\{α₃\} ℤ ×_\{α₄\}... ×_\{αₙ\} ℤ$, where the actions $α_\{i\}$ of ℤ on the fibre $M_\{k\}(ℂ)$ of a rational rotation algebra $A_\{m/k\}$ are trivial, and $C*(kℤ × kℤ) ×_\{α₃\} ℤ ×_\{α₄\} ... ×_\{αₙ\} ℤ$ is a non-commutative torus $A_\{ϱ\}$. It is shown that $T^\{k\}_\{ϱ\}$ is strongly Morita equivalent to $A_\{ϱ\}$, and that $T_\{ϱ\}^\{k\} ⊗ M_\{p^\{∞\}\}$ is isomorphic to $A_\{ϱ\} ⊗ M_\{k\}(ℂ) ⊗ M_\{p^\{∞\}\}$ if and only if the set of prime factors of k is a subset of the set of prime factors of p.},
author = {Chun-Gil Park},
journal = {Studia Mathematica},
keywords = {non-commutative torus; equivalence bimodule; circle algebra; real rank 0; crossed product; UHF-algebra; Cuntz algebra; fibre; strongly Morita equivalent; prime factors},
language = {eng},
number = {2},
pages = {101-108},
title = {Generalized non-commutative tori},
url = {http://eudml.org/doc/284857},
volume = {149},
year = {2002},
}

TY - JOUR
AU - Chun-Gil Park
TI - Generalized non-commutative tori
JO - Studia Mathematica
PY - 2002
VL - 149
IS - 2
SP - 101
EP - 108
AB - The generalized non-commutative torus $T_{ϱ}^{k}$ of rank n is defined by the crossed product $A_{m/k} ×_{α₃} ℤ ×_{α₄}... ×_{αₙ} ℤ$, where the actions $α_{i}$ of ℤ on the fibre $M_{k}(ℂ)$ of a rational rotation algebra $A_{m/k}$ are trivial, and $C*(kℤ × kℤ) ×_{α₃} ℤ ×_{α₄} ... ×_{αₙ} ℤ$ is a non-commutative torus $A_{ϱ}$. It is shown that $T^{k}_{ϱ}$ is strongly Morita equivalent to $A_{ϱ}$, and that $T_{ϱ}^{k} ⊗ M_{p^{∞}}$ is isomorphic to $A_{ϱ} ⊗ M_{k}(ℂ) ⊗ M_{p^{∞}}$ if and only if the set of prime factors of k is a subset of the set of prime factors of p.
LA - eng
KW - non-commutative torus; equivalence bimodule; circle algebra; real rank 0; crossed product; UHF-algebra; Cuntz algebra; fibre; strongly Morita equivalent; prime factors
UR - http://eudml.org/doc/284857
ER -

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Language to use for this widget.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Number of notes per page

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.