|
|
| |
| THE FUNDAMENTAL GROUP OF THE AUTOMORPHISM GROUP OF A NONCOMMUTATIVE TORUS |
| |
Citation： |
D. H. BOO,C. G. PARK.THE FUNDAMENTAL GROUP OF THE AUTOMORPHISM GROUP OF A NONCOMMUTATIVE TORUS[J].Chinese Annals of Mathematics B,2000,21(4):441~452 |
| Page view： 1144
Net amount： 883 |
Authors： |
D. H. BOO; C. G. PARK |
Foundation： |
Project supported by the grant No. 1999-2-102-001-3 from the Interdisciplinary Research Program Year of the KOSEF. |
|
|
| Abstract： |
Assume that each completely irrational noncommutative torus is realized as an inductive limit of circle algebras, and that for a completely irrational noncommutative torus $A_{\omega}$ of rank $m$ there are a completely irrational noncommutative torus $A_{\rho}$ of rank $m$ and a positive integer $d$ such that $\operatorname{tr}(A_{\omega}) = \frac{1}{d}\cdot \operatorname{tr}(A_{\rho})$. It is proved that the set of all $C^*$-algebras of sections of locally trivial $C^*$-algebra bundles over $S^{2}$ with fibres $A_{\omega}$ has a group structure, denoted by $\pi^s_{1}(\operatorname{Aut}(A_{\omega}))$, which is isomorphic to $\Bbb Z$ if $\exists d>1$ and $\{0\}$ if $\nexists d>1$.
Let $B_{cd}$ be a $cd$-homogeneous $C^*$-algebra over $S^{2} \times \Bbb T^2$ of which no non-trivial matrix algebra can be factored out. The spherical noncommutative torus $\Bbb S_{\rho}^{cd}$ is defined by twisting $C^*(\widehat{\Bbb T^2} \times \Bbb Z^{m-2})$ in $B_{cd} \otimes C^*(\Bbb Z^{m-2})$ by a totally skew multiplier $\rho$ on $\widehat{\Bbb T^2} \times \Bbb Z^{m-2}$. It is shown that $\Bbb S_{\rho}^{cd} \otimes M_{p^{\infty}}$ is isomorphic to $C(S^{2}) \otimes C^*(\widehat{\Bbb T^2} \times \Bbb Z^{m-2}, \rho) \otimes M_{cd}(\Bbb C) \otimes M_{p^{\infty}}$ if and only if the set of prime factors of $cd$ is a subset of the set of prime factors of $p$. |
Keywords： |
C^?-algebra bundle, Homogeneous C^?-algebra, Crossed product, UHF-algebra, Cuntz algebra |
Classification： |
46L05, 46L87, 55R15 |
|
|
Download PDF Full-Text
|
|
|
|
