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Strong Morita equivalence of higher-dimensional noncommutative tori. (English) Zbl 1063.46057

Let \(T_n\) be the set of \(n{\times}n\) real skew-symmetric matrices. For each \(\theta\in T_n\), the correspoding non-commutative torus \(A_\theta\) is defined as the universal \(C^*\)-algebra generated by unitaries \(U_1,\ldots, U_n\) satisfying the relations \(U_kU_j=\exp(2\pi i\theta_{kj})U_jU_k\). The smooth subalgebra \(A_\theta^\infty\) consists of all formal series \(\sum c_{j_1,\ldots,j_n}U^{j_1}\cdots U^{j_n}\) such that the coefficient function \(\mathbb Z\ni(j_1,\ldots,j_n)\mapsto c_{j_1,\ldots,j_n}\) vanishes at infinity more rapidly than any polynomial growth. Let \(O(n,n;\mathbb R)\) be the group of linear transformations of \(\mathbb R^{2n}\) preserving the form \(x_1x_{n+1}+x_2x_{n+2}+\ldots+x_nx_{2n}\) and let \(O(n,n;\mathbb Z)\) be its subgroup consisting of matrices with integer entries and determinant 1. For each \(g\in O(n,n;\mathbb Z)\), the action on a dense open subset of \(T_n\) is defined by \(g(\theta)=(A\theta+B)(C\theta+D)^{-1}\), where \(g=\left(\begin{smallmatrix} A & B\cr C & D\end{smallmatrix}\right)\).
It is proved that, for any \(\theta\in T_n\) and \(g\in O(n,n;\mathbb Z)\), if \(g(\theta)\) is defined, then \(A_\theta\) and \(A_{g(\theta)}\) are strongly Morita equivalent and \(A_\theta^\infty\) and \(A_{g(\theta)}^\infty\) are completely Morita equivalent. This was conjectured by M. Rieffel and A. Schwarz [Int. J. Math. 10, 289–299 (1999; Zbl 0968.46060)].

MSC:

46L80 \(K\)-theory and operator algebras (including cyclic theory)
46L05 General theory of \(C^*\)-algebras

Citations:

Zbl 0968.46060
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References:

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