Elliott, George A.; Li, Hanfeng Morita equivalence of smooth noncommutative tori. (English) Zbl 1137.46030 Acta Math. 199, No. 1, 1-27 (2007). Let \(T_n\), \(n\geq 2\), denote the set of \(n{\times}n\) real skew-symmetric matrices. Each \(\theta\in T_n\) determines a non-commutative torus, i.e., the universal \(C^*\)-algebra \(A_\theta\) generated by unitaries \(U_i\), \(i=1,\dots,n\), satisfying the relations \(U_kU_j=\exp(2\pi i\theta_{kj})U_jU_k\) and its ‘smooth subalgebra’ \(A_\theta^\infty\), which consists of formal series \(\sum c_{j_1,\dots,j_n}U_1^{j_1}\ldots U_n^{j_n}\), where the coefficients are of rapid decay on \(\mathbb Z^n\). The authors have recently classified the non-commutative tori up to Rieffel-Morita equivalence and the present paper deals with classification of smooth non-commutative tori up to Morita equivalence as unital algebras and solves this classification problem in the generic case.The authors define a subset \(T'_n\subset T_n\) such that (i) the complement \(T_n\setminus T'_n\) has Lebesgue measure zero and (ii) \(T'_n\) is closed under Morita equivalence of the associated smooth non-commutative tori. The main result is that two matrices in \(T'_n\) give rise to Morita equivalent smooth non-commutative tori iff they are in the same orbit of the natural action of \({\mathrm S}{\mathrm O}(n,\mathbb Z)\). Reviewer: Vladimir M. Manuilov (Moskva) Cited in 13 Documents MSC: 46L05 General theory of \(C^*\)-algebras Keywords:non-commutative tori; Morita equivalence PDFBibTeX XMLCite \textit{G. A. Elliott} and \textit{H. 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