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Morita equivalence of multidimensional noncommutative tori. (English) Zbl 0968.46060

An \(n\)-dimensional noncommutative torus is an associative algebra with involution having unitary generators \(U_1,\dots ,U_n\) obeying the relations \(U_kU_j=e(\theta_{kj})U_jU_k,\) where \(e(t)=e^{2\pi it}\) and \(\theta\) is antisymmetric matrix. The same name is used for \(C^*\)-algebra \(A_{\theta}\) generated by \(U_1,\dots ,U_n.\) Let \(O(n,n|{\mathbb R})\) be a group of linear transformations of the space \({\mathbb R}^{2n}\) preserving the quadratic form \(x_1x_{n+1}+\dots +x_nx_{2n}\) and \(SO(n,n|{\mathbb R})\) be a subgroup of \(O(n,n|{\mathbb R})\) consisting of matrices with integer entries and determinant \(+1.\) The action of \(O(n,n|{\mathbb R})\) on the space \({\mathcal T}_n\) of antisymmetric matrices is defined by the formula \(\theta '=(g_{11}\theta +g_{12})(g_{21}\theta +g_{22})^{-1},\) where \(g=(g_{ij})_{ij=1,2}\in O(n,n|{\mathbb R})\) and \(n\times n\)-matrices \(g_{ij}\) satisfy \(g_{11}^tg_{21}+g_{21}^tg_{11}=g_{12}^tg_{22}+g_{22}^tg_{12}=0,\) \(g_{11}^tg_{22}+g_{21}^tg_{12}=1.\) (This action is defined only on the dense subset of \({\mathcal T}_n\) where \((g_{21}\theta +g_{22})\) is invertible.) Let \[ {\mathcal T}_n^0=\{\theta \in {\mathcal T}_n: g\theta \text{ is defined for all }g\in SO(n,n|{\mathbb R})\}. \] \({\mathcal T}_n^0\) is dense in \({\mathcal T}_n\) and a simple calculation shows that \({\mathcal T}_n^0\) is carried into itself by the action of \(SO(n,n|{\mathbb R}).\) The main result of the paper is the following.
Theorem. For \(\theta \in {\mathcal T}_n^0\) and \(g\in SO(n,n|{\mathbb R})\) the noncommutative torus corresponding to \(g\theta\) is Morita equivalent to the noncommutative torus corresponding to \(\theta.\) This remains true for the smooth versions of the noncommutative tori.

MSC:

46L85 Noncommutative topology
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References:

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